Polytope of Type {4,4,28} with group SmallGroup(1792,201151)

Overview

Group: SmallGroup(1792,201151)
Rank: 4
Schläfli Type: {4,4,28}
Vertices, edges, …: 4, 16, 112, 56
Order of s₀s₁s₂s₃: 28
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

2-fold

4-fold

7-fold

{4,4,4}*256a

8-fold

14-fold

16-fold

28-fold

32-fold

{2,2,7}*56

56-fold

112-fold

{2,2,2}*16

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s1*s2)^2> of order 2

42 facets

28 of {4,2}*16
14 of {4,4}*32

4 vertex figures

4 of 2-fold non-regular quotient of {4,28}*448

P/N, where N=<(s2*s3)^14> of order 2

28 facets

28 of {4,4}*32

4 vertex figures

4 of 2-fold non-regular quotient of {4,28}*448

Representations

Permutation Representation (GAP)

s0 := (  1, 57)(  2, 58)(  3, 59)(  4, 60)(  5, 61)(  6, 62)(  7, 63)(  8, 64)(  9, 65)( 10, 66)( 11, 67)( 12, 68)( 13, 69)( 14, 70)( 15, 71)( 16, 72)( 17, 73)( 18, 74)( 19, 75)( 20, 76)( 21, 77)( 22, 78)( 23, 79)( 24, 80)( 25, 81)( 26, 82)( 27, 83)( 28, 84)( 29, 85)( 30, 86)( 31, 87)( 32, 88)( 33, 89)( 34, 90)( 35, 91)( 36, 92)( 37, 93)( 38, 94)( 39, 95)( 40, 96)( 41, 97)( 42, 98)( 43, 99)( 44,100)( 45,101)( 46,102)( 47,103)( 48,104)( 49,105)( 50,106)( 51,107)( 52,108)( 53,109)( 54,110)( 55,111)( 56,112)(113,183)(114,184)(115,185)(116,186)(117,187)(118,188)(119,189)(120,190)(121,191)(122,192)(123,193)(124,194)(125,195)(126,196)(127,169)(128,170)(129,171)(130,172)(131,173)(132,174)(133,175)(134,176)(135,177)(136,178)(137,179)(138,180)(139,181)(140,182)(141,211)(142,212)(143,213)(144,214)(145,215)(146,216)(147,217)(148,218)(149,219)(150,220)(151,221)(152,222)(153,223)(154,224)(155,197)(156,198)(157,199)(158,200)(159,201)(160,202)(161,203)(162,204)(163,205)(164,206)(165,207)(166,208)(167,209)(168,210);;
s1 := (  1,113)(  2,114)(  3,115)(  4,116)(  5,117)(  6,118)(  7,119)(  8,120)(  9,121)( 10,122)( 11,123)( 12,124)( 13,125)( 14,126)( 15,127)( 16,128)( 17,129)( 18,130)( 19,131)( 20,132)( 21,133)( 22,134)( 23,135)( 24,136)( 25,137)( 26,138)( 27,139)( 28,140)( 29,141)( 30,142)( 31,143)( 32,144)( 33,145)( 34,146)( 35,147)( 36,148)( 37,149)( 38,150)( 39,151)( 40,152)( 41,153)( 42,154)( 43,155)( 44,156)( 45,157)( 46,158)( 47,159)( 48,160)( 49,161)( 50,162)( 51,163)( 52,164)( 53,165)( 54,166)( 55,167)( 56,168)( 57,169)( 58,170)( 59,171)( 60,172)( 61,173)( 62,174)( 63,175)( 64,176)( 65,177)( 66,178)( 67,179)( 68,180)( 69,181)( 70,182)( 71,183)( 72,184)( 73,185)( 74,186)( 75,187)( 76,188)( 77,189)( 78,190)( 79,191)( 80,192)( 81,193)( 82,194)( 83,195)( 84,196)( 85,197)( 86,198)( 87,199)( 88,200)( 89,201)( 90,202)( 91,203)( 92,204)( 93,205)( 94,206)( 95,207)( 96,208)( 97,209)( 98,210)( 99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224);;
s2 := (  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 16, 21)( 17, 20)( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)( 37, 42)( 38, 41)( 39, 40)( 44, 49)( 45, 48)( 46, 47)( 51, 56)( 52, 55)( 53, 54)( 58, 63)( 59, 62)( 60, 61)( 65, 70)( 66, 69)( 67, 68)( 72, 77)( 73, 76)( 74, 75)( 79, 84)( 80, 83)( 81, 82)( 86, 91)( 87, 90)( 88, 89)( 93, 98)( 94, 97)( 95, 96)(100,105)(101,104)(102,103)(107,112)(108,111)(109,110)(113,141)(114,147)(115,146)(116,145)(117,144)(118,143)(119,142)(120,148)(121,154)(122,153)(123,152)(124,151)(125,150)(126,149)(127,155)(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,162)(135,168)(136,167)(137,166)(138,165)(139,164)(140,163)(169,197)(170,203)(171,202)(172,201)(173,200)(174,199)(175,198)(176,204)(177,210)(178,209)(179,208)(180,207)(181,206)(182,205)(183,211)(184,217)(185,216)(186,215)(187,214)(188,213)(189,212)(190,218)(191,224)(192,223)(193,222)(194,221)(195,220)(196,219);;
s3 := (  1, 58)(  2, 57)(  3, 63)(  4, 62)(  5, 61)(  6, 60)(  7, 59)(  8, 65)(  9, 64)( 10, 70)( 11, 69)( 12, 68)( 13, 67)( 14, 66)( 15, 72)( 16, 71)( 17, 77)( 18, 76)( 19, 75)( 20, 74)( 21, 73)( 22, 79)( 23, 78)( 24, 84)( 25, 83)( 26, 82)( 27, 81)( 28, 80)( 29, 93)( 30, 92)( 31, 98)( 32, 97)( 33, 96)( 34, 95)( 35, 94)( 36, 86)( 37, 85)( 38, 91)( 39, 90)( 40, 89)( 41, 88)( 42, 87)( 43,107)( 44,106)( 45,112)( 46,111)( 47,110)( 48,109)( 49,108)( 50,100)( 51, 99)( 52,105)( 53,104)( 54,103)( 55,102)( 56,101)(113,170)(114,169)(115,175)(116,174)(117,173)(118,172)(119,171)(120,177)(121,176)(122,182)(123,181)(124,180)(125,179)(126,178)(127,184)(128,183)(129,189)(130,188)(131,187)(132,186)(133,185)(134,191)(135,190)(136,196)(137,195)(138,194)(139,193)(140,192)(141,205)(142,204)(143,210)(144,209)(145,208)(146,207)(147,206)(148,198)(149,197)(150,203)(151,202)(152,201)(153,200)(154,199)(155,219)(156,218)(157,224)(158,223)(159,222)(160,221)(161,220)(162,212)(163,211)(164,217)(165,216)(166,215)(167,214)(168,213);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP)

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s1*s2*s3*s2*s1*s2*s1*s3*s2*s3*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(224)!(  1, 57)(  2, 58)(  3, 59)(  4, 60)(  5, 61)(  6, 62)(  7, 63)(  8, 64)(  9, 65)( 10, 66)( 11, 67)( 12, 68)( 13, 69)( 14, 70)( 15, 71)( 16, 72)( 17, 73)( 18, 74)( 19, 75)( 20, 76)( 21, 77)( 22, 78)( 23, 79)( 24, 80)( 25, 81)( 26, 82)( 27, 83)( 28, 84)( 29, 85)( 30, 86)( 31, 87)( 32, 88)( 33, 89)( 34, 90)( 35, 91)( 36, 92)( 37, 93)( 38, 94)( 39, 95)( 40, 96)( 41, 97)( 42, 98)( 43, 99)( 44,100)( 45,101)( 46,102)( 47,103)( 48,104)( 49,105)( 50,106)( 51,107)( 52,108)( 53,109)( 54,110)( 55,111)( 56,112)(113,183)(114,184)(115,185)(116,186)(117,187)(118,188)(119,189)(120,190)(121,191)(122,192)(123,193)(124,194)(125,195)(126,196)(127,169)(128,170)(129,171)(130,172)(131,173)(132,174)(133,175)(134,176)(135,177)(136,178)(137,179)(138,180)(139,181)(140,182)(141,211)(142,212)(143,213)(144,214)(145,215)(146,216)(147,217)(148,218)(149,219)(150,220)(151,221)(152,222)(153,223)(154,224)(155,197)(156,198)(157,199)(158,200)(159,201)(160,202)(161,203)(162,204)(163,205)(164,206)(165,207)(166,208)(167,209)(168,210);
s1 := Sym(224)!(  1,113)(  2,114)(  3,115)(  4,116)(  5,117)(  6,118)(  7,119)(  8,120)(  9,121)( 10,122)( 11,123)( 12,124)( 13,125)( 14,126)( 15,127)( 16,128)( 17,129)( 18,130)( 19,131)( 20,132)( 21,133)( 22,134)( 23,135)( 24,136)( 25,137)( 26,138)( 27,139)( 28,140)( 29,141)( 30,142)( 31,143)( 32,144)( 33,145)( 34,146)( 35,147)( 36,148)( 37,149)( 38,150)( 39,151)( 40,152)( 41,153)( 42,154)( 43,155)( 44,156)( 45,157)( 46,158)( 47,159)( 48,160)( 49,161)( 50,162)( 51,163)( 52,164)( 53,165)( 54,166)( 55,167)( 56,168)( 57,169)( 58,170)( 59,171)( 60,172)( 61,173)( 62,174)( 63,175)( 64,176)( 65,177)( 66,178)( 67,179)( 68,180)( 69,181)( 70,182)( 71,183)( 72,184)( 73,185)( 74,186)( 75,187)( 76,188)( 77,189)( 78,190)( 79,191)( 80,192)( 81,193)( 82,194)( 83,195)( 84,196)( 85,197)( 86,198)( 87,199)( 88,200)( 89,201)( 90,202)( 91,203)( 92,204)( 93,205)( 94,206)( 95,207)( 96,208)( 97,209)( 98,210)( 99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224);
s2 := Sym(224)!(  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 16, 21)( 17, 20)( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)( 37, 42)( 38, 41)( 39, 40)( 44, 49)( 45, 48)( 46, 47)( 51, 56)( 52, 55)( 53, 54)( 58, 63)( 59, 62)( 60, 61)( 65, 70)( 66, 69)( 67, 68)( 72, 77)( 73, 76)( 74, 75)( 79, 84)( 80, 83)( 81, 82)( 86, 91)( 87, 90)( 88, 89)( 93, 98)( 94, 97)( 95, 96)(100,105)(101,104)(102,103)(107,112)(108,111)(109,110)(113,141)(114,147)(115,146)(116,145)(117,144)(118,143)(119,142)(120,148)(121,154)(122,153)(123,152)(124,151)(125,150)(126,149)(127,155)(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,162)(135,168)(136,167)(137,166)(138,165)(139,164)(140,163)(169,197)(170,203)(171,202)(172,201)(173,200)(174,199)(175,198)(176,204)(177,210)(178,209)(179,208)(180,207)(181,206)(182,205)(183,211)(184,217)(185,216)(186,215)(187,214)(188,213)(189,212)(190,218)(191,224)(192,223)(193,222)(194,221)(195,220)(196,219);
s3 := Sym(224)!(  1, 58)(  2, 57)(  3, 63)(  4, 62)(  5, 61)(  6, 60)(  7, 59)(  8, 65)(  9, 64)( 10, 70)( 11, 69)( 12, 68)( 13, 67)( 14, 66)( 15, 72)( 16, 71)( 17, 77)( 18, 76)( 19, 75)( 20, 74)( 21, 73)( 22, 79)( 23, 78)( 24, 84)( 25, 83)( 26, 82)( 27, 81)( 28, 80)( 29, 93)( 30, 92)( 31, 98)( 32, 97)( 33, 96)( 34, 95)( 35, 94)( 36, 86)( 37, 85)( 38, 91)( 39, 90)( 40, 89)( 41, 88)( 42, 87)( 43,107)( 44,106)( 45,112)( 46,111)( 47,110)( 48,109)( 49,108)( 50,100)( 51, 99)( 52,105)( 53,104)( 54,103)( 55,102)( 56,101)(113,170)(114,169)(115,175)(116,174)(117,173)(118,172)(119,171)(120,177)(121,176)(122,182)(123,181)(124,180)(125,179)(126,178)(127,184)(128,183)(129,189)(130,188)(131,187)(132,186)(133,185)(134,191)(135,190)(136,196)(137,195)(138,194)(139,193)(140,192)(141,205)(142,204)(143,210)(144,209)(145,208)(146,207)(147,206)(148,198)(149,197)(150,203)(151,202)(152,201)(153,200)(154,199)(155,219)(156,218)(157,224)(158,223)(159,222)(160,221)(161,220)(162,212)(163,211)(164,217)(165,216)(166,215)(167,214)(168,213);
poly := sub<Sym(224)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s1*s2*s3*s2*s1*s2*s1*s3*s2*s3*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References

None.

to this polytope.