D₂₁₀₉₆ #72274840

order	42192
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₁₀₉₆

Splittings

Fitting	C₈×C₉×C₂₉₃ ⋊ C₂
Hall	C₉×C₂₉₃ ⋊ D₈
Sylow	C₂₉₃ ⋊ D₇₂
Sylow	C₉ ⋊ D₂₃₄₄

Subgroups

	subgroup	G/H
center	C₂	D₁₀₅₄₈
commutator, cosocle	C₄×C₉×C₂₉₃	C₂²
Frattini	C₃×C₄	C₂×D₈₇₉
Fitting	C₈×C₉×C₂₉₃	C₂
socle	C₂×C₃×C₂₉₃	D₁₂
hypercenter	C₈	D₂₆₃₇
hypocenter	C₉×C₂₉₃	D₈
LCS	G ▷ C₄×C₉×C₂₉₃ ▷ C₂×C₉×C₂₉₃ ▷ C₉×C₂₉₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 310
transitive degree	μ_t(G) = 21096

A minimal faithful permutation representation — 2 generators on 310 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);(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,310)(20,309)(21,308)(22,307)(23,306)(24,305)(25,304)(26,303)(27,302)(28,301)(29,300)(30,299)(31,298)(32,297)(33,296)(34,295)(35,294)(36,293)(37,292)(38,291)(39,290)(40,289)(41,288)(42,287)(43,286)(44,285)(45,284)(46,283)(47,282)(48,281)(49,280)(50,279)(51,278)(52,277)(53,276)(54,275)(55,274)(56,273)(57,272)(58,271)(59,270)(60,269)(61,268)(62,267)(63,266)(64,265)(65,264)(66,263)(67,262)(68,261)(69,260)(70,259)(71,258)(72,257)(73,256)(74,255)(75,254)(76,253)(77,252)(78,251)(79,250)(80,249)(81,248)(82,247)(83,246)(84,245)(85,244)(86,243)(87,242)(88,241)(89,240)(90,239)(91,238)(92,237)(93,236)(94,235)(95,234)(96,233)(97,232)(98,231)(99,230)(100,229)(101,228)(102,227)(103,226)(104,225)(105,224)(106,223)(107,222)(108,221)(109,220)(110,219)(111,218)(112,217)(113,216)(114,215)(115,214)(116,213)(117,212)(118,211)(119,210)(120,209)(121,208)(122,207)(123,206)(124,205)(125,204)(126,203)(127,202)(128,201)(129,200)(130,199)(131,198)(132,197)(133,196)(134,195)(135,194)(136,193)(137,192)(138,191)(139,190)(140,189)(141,188)(142,187)(143,186)(144,185)(145,184)(146,183)(147,182)(148,181)(149,180)(150,179)(151,178)(152,177)(153,176)(154,175)(155,174)(156,173)(157,172)(158,171)(159,170)(160,169)(161,168)(162,167)(163,166)(164,165)

Polycyclic presentation:

2^4,3,293,3|b^2=cd,c^2=d,e^3=g,b^a=bc,c^a=cd,e^a=e^2g^2,f^a=f^292,g^a=g^2

Conjugacy classes & character table

10551 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 293, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
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*F.7^2);;
gap> SetConjugate(coll, 6, 1, F.6^292);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D21096 #72274840