D₂₃₁₂ #72210343

order	4624
automorphism	Aut(G) order 2515456
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₃₁₂

Splittings

Fitting	C₈×C₂₈₉ ⋊ C₂
Sylow	C₂₈₉ ⋊ D₈

Subgroups

	subgroup	G/H
center	C₂	D₁₁₅₆
commutator, cosocle	C₄×C₂₈₉	C₂²
Frattini	C₄×C₁₇	C₂×D₁₇
Fitting	C₈×C₂₈₉	C₂
socle	C₂×C₁₇	D₆₈
hypercenter	C₈	D₂₈₉
hypocenter	C₂₈₉	D₈
LCS	G ▷ C₄×C₂₈₉ ▷ C₂×C₂₈₉ ▷ C₂₈₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

33 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 297
transitive degree	μ_t(G) = 2312

A minimal faithful permutation representation — 2 generators on 297 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);(2,8)(3,7)(4,6)(10,297)(11,296)(12,295)(13,294)(14,293)(15,292)(16,291)(17,290)(18,289)(19,288)(20,287)(21,286)(22,285)(23,284)(24,283)(25,282)(26,281)(27,280)(28,279)(29,278)(30,277)(31,276)(32,275)(33,274)(34,273)(35,272)(36,271)(37,270)(38,269)(39,268)(40,267)(41,266)(42,265)(43,264)(44,263)(45,262)(46,261)(47,260)(48,259)(49,258)(50,257)(51,256)(52,255)(53,254)(54,253)(55,252)(56,251)(57,250)(58,249)(59,248)(60,247)(61,246)(62,245)(63,244)(64,243)(65,242)(66,241)(67,240)(68,239)(69,238)(70,237)(71,236)(72,235)(73,234)(74,233)(75,232)(76,231)(77,230)(78,229)(79,228)(80,227)(81,226)(82,225)(83,224)(84,223)(85,222)(86,221)(87,220)(88,219)(89,218)(90,217)(91,216)(92,215)(93,214)(94,213)(95,212)(96,211)(97,210)(98,209)(99,208)(100,207)(101,206)(102,205)(103,204)(104,203)(105,202)(106,201)(107,200)(108,199)(109,198)(110,197)(111,196)(112,195)(113,194)(114,193)(115,192)(116,191)(117,190)(118,189)(119,188)(120,187)(121,186)(122,185)(123,184)(124,183)(125,182)(126,181)(127,180)(128,179)(129,178)(130,177)(131,176)(132,175)(133,174)(134,173)(135,172)(136,171)(137,170)(138,169)(139,168)(140,167)(141,166)(142,165)(143,164)(144,163)(145,162)(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)

Polycyclic presentation:

2^4,17^2|b^2=cd,c^2=d,e^17=f,b^a=bc,c^a=cd,e^a=e^16f^16,f^a=f^16

Conjugacy classes & character table

1159 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₄₆₂₄, D₆₉₃₆, D₁₁₅₆₀, …
maximal normal subgroup of	D₄₆₂₄
maximal quotient of	D₄₆₂₄, D₆₉₃₆, Dic₂₃₁₂, …

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 17, 17]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^16*F.6^16);;
gap> SetConjugate(coll, 6, 1, F.6^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D2312 #72210343