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