Finite Group DatabasebetaFinite Group Databasebeta| order | 56616 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic14154 |
| Hall | C3×C7×C337 ⋊ Q8 |
| Hall | C7×C337 ⋊ Dic6 |
| Hall | C3×C337 ⋊ Dic14 |
| Sylow | C337 ⋊ Dic42 |
| Hall | C3×C7 ⋊ Dic674 |
| Sylow | C7 ⋊ Dic2022 |
| Sylow | C3 ⋊ Dic4718 |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 355 |
| transitive degree | μt(G) = 56616 |
A minimal faithful permutation representation — 2 generators on 355 points:
(1,2,3,4)(5,8,7,6)(9,10,11)(12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355);(1,5,3,7)(2,6,4,8)(10,11)(13,18)(14,17)(15,16)(20,355)(21,354)(22,353)(23,352)(24,351)(25,350)(26,349)(27,348)(28,347)(29,346)(30,345)(31,344)(32,343)(33,342)(34,341)(35,340)(36,339)(37,338)(38,337)(39,336)(40,335)(41,334)(42,333)(43,332)(44,331)(45,330)(46,329)(47,328)(48,327)(49,326)(50,325)(51,324)(52,323)(53,322)(54,321)(55,320)(56,319)(57,318)(58,317)(59,316)(60,315)(61,314)(62,313)(63,312)(64,311)(65,310)(66,309)(67,308)(68,307)(69,306)(70,305)(71,304)(72,303)(73,302)(74,301)(75,300)(76,299)(77,298)(78,297)(79,296)(80,295)(81,294)(82,293)(83,292)(84,291)(85,290)(86,289)(87,288)(88,287)(89,286)(90,285)(91,284)(92,283)(93,282)(94,281)(95,280)(96,279)(97,278)(98,277)(99,276)(100,275)(101,274)(102,273)(103,272)(104,271)(105,270)(106,269)(107,268)(108,267)(109,266)(110,265)(111,264)(112,263)(113,262)(114,261)(115,260)(116,259)(117,258)(118,257)(119,256)(120,255)(121,254)(122,253)(123,252)(124,251)(125,250)(126,249)(127,248)(128,247)(129,246)(130,245)(131,244)(132,243)(133,242)(134,241)(135,240)(136,239)(137,238)(138,237)(139,236)(140,235)(141,234)(142,233)(143,232)(144,231)(145,230)(146,229)(147,228)(148,227)(149,226)(150,225)(151,224)(152,223)(153,222)(154,221)(155,220)(156,219)(157,218)(158,217)(159,216)(160,215)(161,214)(162,213)(163,212)(164,211)(165,210)(166,209)(167,208)(168,207)(169,206)(170,205)(171,204)(172,203)(173,202)(174,201)(175,200)(176,199)(177,198)(178,197)(179,196)(180,195)(181,194)(182,193)(183,192)(184,191)(185,190)(186,189)(187,188)Polycyclic presentation:
2^3,3,7,337|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^6,f^a=f^33614157 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 7, 337]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^336);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);