Finite Group DatabasebetaFinite Group Databasebeta| order | 15888 |
| automorphism | Aut(G) order 20972160 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D7944 |
| Fitting | C3×C8×C331 ⋊ C2 |
| Hall | C3×C331 ⋊ D8 |
| Sylow | C331 ⋊ D24 |
| Sylow | C3 ⋊ D2648 |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 342 |
| transitive degree | μt(G) = 7944 |
A minimal faithful permutation representation — 2 generators on 342 points:
(1,2,3,4,5,6,7,8)(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);(2,8)(3,7)(4,6)(10,11)(13,342)(14,341)(15,340)(16,339)(17,338)(18,337)(19,336)(20,335)(21,334)(22,333)(23,332)(24,331)(25,330)(26,329)(27,328)(28,327)(29,326)(30,325)(31,324)(32,323)(33,322)(34,321)(35,320)(36,319)(37,318)(38,317)(39,316)(40,315)(41,314)(42,313)(43,312)(44,311)(45,310)(46,309)(47,308)(48,307)(49,306)(50,305)(51,304)(52,303)(53,302)(54,301)(55,300)(56,299)(57,298)(58,297)(59,296)(60,295)(61,294)(62,293)(63,292)(64,291)(65,290)(66,289)(67,288)(68,287)(69,286)(70,285)(71,284)(72,283)(73,282)(74,281)(75,280)(76,279)(77,278)(78,277)(79,276)(80,275)(81,274)(82,273)(83,272)(84,271)(85,270)(86,269)(87,268)(88,267)(89,266)(90,265)(91,264)(92,263)(93,262)(94,261)(95,260)(96,259)(97,258)(98,257)(99,256)(100,255)(101,254)(102,253)(103,252)(104,251)(105,250)(106,249)(107,248)(108,247)(109,246)(110,245)(111,244)(112,243)(113,242)(114,241)(115,240)(116,239)(117,238)(118,237)(119,236)(120,235)(121,234)(122,233)(123,232)(124,231)(125,230)(126,229)(127,228)(128,227)(129,226)(130,225)(131,224)(132,223)(133,222)(134,221)(135,220)(136,219)(137,218)(138,217)(139,216)(140,215)(141,214)(142,213)(143,212)(144,211)(145,210)(146,209)(147,208)(148,207)(149,206)(150,205)(151,204)(152,203)(153,202)(154,201)(155,200)(156,199)(157,198)(158,197)(159,196)(160,195)(161,194)(162,193)(163,192)(164,191)(165,190)(166,189)(167,188)(168,187)(169,186)(170,185)(171,184)(172,183)(173,182)(174,181)(175,180)(176,179)(177,178)Polycyclic presentation:
2^4,3,331|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^3303975 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D15888, D23832 |
| maximal normal subgroup of | D15888 |
| maximal quotient of | Dic7944, D15888, D23832 |
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, 2, 3, 331]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^330);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);