Polytope of Type {14,4,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {14,4,4}*1792
Also Known As : {{14,4|2},{4,4|4}}. if this polytope has another name.
Group : SmallGroup(1792,323566)
Rank : 4
Schlafli Type : {14,4,4}
Number of vertices, edges, etc : 14, 112, 32, 16
Order of s0s1s2s3 : 56
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {14,4,4}*896
4-fold quotients : {14,4,4}*448
7-fold quotients : {2,4,4}*256
8-fold quotients : {14,2,4}*224, {14,4,2}*224
14-fold quotients : {2,4,4}*128
16-fold quotients : {7,2,4}*112, {14,2,2}*112
28-fold quotients : {2,4,4}*64
32-fold quotients : {7,2,2}*56
56-fold quotients : {2,2,4}*32, {2,4,2}*32
112-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 57)( 2, 63)( 3, 62)( 4, 61)( 5, 60)( 6, 59)( 7, 58)( 8, 64)
( 9, 70)( 10, 69)( 11, 68)( 12, 67)( 13, 66)( 14, 65)( 15, 71)( 16, 77)
( 17, 76)( 18, 75)( 19, 74)( 20, 73)( 21, 72)( 22, 78)( 23, 84)( 24, 83)
( 25, 82)( 26, 81)( 27, 80)( 28, 79)( 29, 85)( 30, 91)( 31, 90)( 32, 89)
( 33, 88)( 34, 87)( 35, 86)( 36, 92)( 37, 98)( 38, 97)( 39, 96)( 40, 95)
( 41, 94)( 42, 93)( 43, 99)( 44,105)( 45,104)( 46,103)( 47,102)( 48,101)
( 49,100)( 50,106)( 51,112)( 52,111)( 53,110)( 54,109)( 55,108)( 56,107)
(113,169)(114,175)(115,174)(116,173)(117,172)(118,171)(119,170)(120,176)
(121,182)(122,181)(123,180)(124,179)(125,178)(126,177)(127,183)(128,189)
(129,188)(130,187)(131,186)(132,185)(133,184)(134,190)(135,196)(136,195)
(137,194)(138,193)(139,192)(140,191)(141,197)(142,203)(143,202)(144,201)
(145,200)(146,199)(147,198)(148,204)(149,210)(150,209)(151,208)(152,207)
(153,206)(154,205)(155,211)(156,217)(157,216)(158,215)(159,214)(160,213)
(161,212)(162,218)(163,224)(164,223)(165,222)(166,221)(167,220)(168,219);;
s1 := ( 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)( 29,100)( 30, 99)( 31,105)( 32,104)
( 33,103)( 34,102)( 35,101)( 36,107)( 37,106)( 38,112)( 39,111)( 40,110)
( 41,109)( 42,108)( 43, 86)( 44, 85)( 45, 91)( 46, 90)( 47, 89)( 48, 88)
( 49, 87)( 50, 93)( 51, 92)( 52, 98)( 53, 97)( 54, 96)( 55, 95)( 56, 94)
(113,184)(114,183)(115,189)(116,188)(117,187)(118,186)(119,185)(120,191)
(121,190)(122,196)(123,195)(124,194)(125,193)(126,192)(127,170)(128,169)
(129,175)(130,174)(131,173)(132,172)(133,171)(134,177)(135,176)(136,182)
(137,181)(138,180)(139,179)(140,178)(141,198)(142,197)(143,203)(144,202)
(145,201)(146,200)(147,199)(148,205)(149,204)(150,210)(151,209)(152,208)
(153,207)(154,206)(155,212)(156,211)(157,217)(158,216)(159,215)(160,214)
(161,213)(162,219)(163,218)(164,224)(165,223)(166,222)(167,221)(168,220);;
s2 := ( 15, 22)( 16, 23)( 17, 24)( 18, 25)( 19, 26)( 20, 27)( 21, 28)( 43, 50)
( 44, 51)( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 71, 78)( 72, 79)
( 73, 80)( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 99,106)(100,107)(101,108)
(102,109)(103,110)(104,111)(105,112)(113,141)(114,142)(115,143)(116,144)
(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)
(125,153)(126,154)(127,162)(128,163)(129,164)(130,165)(131,166)(132,167)
(133,168)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)
(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)
(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,218)(184,219)
(185,220)(186,221)(187,222)(188,223)(189,224)(190,211)(191,212)(192,213)
(193,214)(194,215)(195,216)(196,217);;
s3 := ( 1,141)( 2,142)( 3,143)( 4,144)( 5,145)( 6,146)( 7,147)( 8,148)
( 9,149)( 10,150)( 11,151)( 12,152)( 13,153)( 14,154)( 15,155)( 16,156)
( 17,157)( 18,158)( 19,159)( 20,160)( 21,161)( 22,162)( 23,163)( 24,164)
( 25,165)( 26,166)( 27,167)( 28,168)( 29,113)( 30,114)( 31,115)( 32,116)
( 33,117)( 34,118)( 35,119)( 36,120)( 37,121)( 38,122)( 39,123)( 40,124)
( 41,125)( 42,126)( 43,127)( 44,128)( 45,129)( 46,130)( 47,131)( 48,132)
( 49,133)( 50,134)( 51,135)( 52,136)( 53,137)( 54,138)( 55,139)( 56,140)
( 57,197)( 58,198)( 59,199)( 60,200)( 61,201)( 62,202)( 63,203)( 64,204)
( 65,205)( 66,206)( 67,207)( 68,208)( 69,209)( 70,210)( 71,211)( 72,212)
( 73,213)( 74,214)( 75,215)( 76,216)( 77,217)( 78,218)( 79,219)( 80,220)
( 81,221)( 82,222)( 83,223)( 84,224)( 85,169)( 86,170)( 87,171)( 88,172)
( 89,173)( 90,174)( 91,175)( 92,176)( 93,177)( 94,178)( 95,179)( 96,180)
( 97,181)( 98,182)( 99,183)(100,184)(101,185)(102,186)(103,187)(104,188)
(105,189)(106,190)(107,191)(108,192)(109,193)(110,194)(111,195)(112,196);;
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*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(224)!( 1, 57)( 2, 63)( 3, 62)( 4, 61)( 5, 60)( 6, 59)( 7, 58)
( 8, 64)( 9, 70)( 10, 69)( 11, 68)( 12, 67)( 13, 66)( 14, 65)( 15, 71)
( 16, 77)( 17, 76)( 18, 75)( 19, 74)( 20, 73)( 21, 72)( 22, 78)( 23, 84)
( 24, 83)( 25, 82)( 26, 81)( 27, 80)( 28, 79)( 29, 85)( 30, 91)( 31, 90)
( 32, 89)( 33, 88)( 34, 87)( 35, 86)( 36, 92)( 37, 98)( 38, 97)( 39, 96)
( 40, 95)( 41, 94)( 42, 93)( 43, 99)( 44,105)( 45,104)( 46,103)( 47,102)
( 48,101)( 49,100)( 50,106)( 51,112)( 52,111)( 53,110)( 54,109)( 55,108)
( 56,107)(113,169)(114,175)(115,174)(116,173)(117,172)(118,171)(119,170)
(120,176)(121,182)(122,181)(123,180)(124,179)(125,178)(126,177)(127,183)
(128,189)(129,188)(130,187)(131,186)(132,185)(133,184)(134,190)(135,196)
(136,195)(137,194)(138,193)(139,192)(140,191)(141,197)(142,203)(143,202)
(144,201)(145,200)(146,199)(147,198)(148,204)(149,210)(150,209)(151,208)
(152,207)(153,206)(154,205)(155,211)(156,217)(157,216)(158,215)(159,214)
(160,213)(161,212)(162,218)(163,224)(164,223)(165,222)(166,221)(167,220)
(168,219);
s1 := 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)( 29,100)( 30, 99)( 31,105)
( 32,104)( 33,103)( 34,102)( 35,101)( 36,107)( 37,106)( 38,112)( 39,111)
( 40,110)( 41,109)( 42,108)( 43, 86)( 44, 85)( 45, 91)( 46, 90)( 47, 89)
( 48, 88)( 49, 87)( 50, 93)( 51, 92)( 52, 98)( 53, 97)( 54, 96)( 55, 95)
( 56, 94)(113,184)(114,183)(115,189)(116,188)(117,187)(118,186)(119,185)
(120,191)(121,190)(122,196)(123,195)(124,194)(125,193)(126,192)(127,170)
(128,169)(129,175)(130,174)(131,173)(132,172)(133,171)(134,177)(135,176)
(136,182)(137,181)(138,180)(139,179)(140,178)(141,198)(142,197)(143,203)
(144,202)(145,201)(146,200)(147,199)(148,205)(149,204)(150,210)(151,209)
(152,208)(153,207)(154,206)(155,212)(156,211)(157,217)(158,216)(159,215)
(160,214)(161,213)(162,219)(163,218)(164,224)(165,223)(166,222)(167,221)
(168,220);
s2 := Sym(224)!( 15, 22)( 16, 23)( 17, 24)( 18, 25)( 19, 26)( 20, 27)( 21, 28)
( 43, 50)( 44, 51)( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 71, 78)
( 72, 79)( 73, 80)( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 99,106)(100,107)
(101,108)(102,109)(103,110)(104,111)(105,112)(113,141)(114,142)(115,143)
(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)
(124,152)(125,153)(126,154)(127,162)(128,163)(129,164)(130,165)(131,166)
(132,167)(133,168)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)
(140,161)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)
(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,218)
(184,219)(185,220)(186,221)(187,222)(188,223)(189,224)(190,211)(191,212)
(192,213)(193,214)(194,215)(195,216)(196,217);
s3 := Sym(224)!( 1,141)( 2,142)( 3,143)( 4,144)( 5,145)( 6,146)( 7,147)
( 8,148)( 9,149)( 10,150)( 11,151)( 12,152)( 13,153)( 14,154)( 15,155)
( 16,156)( 17,157)( 18,158)( 19,159)( 20,160)( 21,161)( 22,162)( 23,163)
( 24,164)( 25,165)( 26,166)( 27,167)( 28,168)( 29,113)( 30,114)( 31,115)
( 32,116)( 33,117)( 34,118)( 35,119)( 36,120)( 37,121)( 38,122)( 39,123)
( 40,124)( 41,125)( 42,126)( 43,127)( 44,128)( 45,129)( 46,130)( 47,131)
( 48,132)( 49,133)( 50,134)( 51,135)( 52,136)( 53,137)( 54,138)( 55,139)
( 56,140)( 57,197)( 58,198)( 59,199)( 60,200)( 61,201)( 62,202)( 63,203)
( 64,204)( 65,205)( 66,206)( 67,207)( 68,208)( 69,209)( 70,210)( 71,211)
( 72,212)( 73,213)( 74,214)( 75,215)( 76,216)( 77,217)( 78,218)( 79,219)
( 80,220)( 81,221)( 82,222)( 83,223)( 84,224)( 85,169)( 86,170)( 87,171)
( 88,172)( 89,173)( 90,174)( 91,175)( 92,176)( 93,177)( 94,178)( 95,179)
( 96,180)( 97,181)( 98,182)( 99,183)(100,184)(101,185)(102,186)(103,187)
(104,188)(105,189)(106,190)(107,191)(108,192)(109,193)(110,194)(111,195)
(112,196);
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*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope