D₈₇₅₆ #72213601

order	17512
automorphism	Aut(G) order 34673760
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₈₇₅₆

Splittings

Fitting	C₄×C₁₁×C₁₉₉ ⋊ C₂
Hall	C₁₁×C₁₉₉ ⋊ D₄
Sylow	C₁₉₉ ⋊ D₄₄
Sylow	C₁₁ ⋊ D₇₉₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₁₈₉
commutator, socle, cosocle	C₂×C₁₁×C₁₉₉	C₂²
Fitting	C₄×C₁₁×C₁₉₉	C₂
hypercenter	C₄	D₂₁₈₉
hypocenter	C₁₁×C₁₉₉	D₄
LCS	G ▷ C₂×C₁₁×C₁₉₉ ▷ C₁₁×C₁₉₉
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 214
transitive degree	μ_t(G) = 8756

A minimal faithful permutation representation — 2 generators on 214 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);(2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,214)(18,213)(19,212)(20,211)(21,210)(22,209)(23,208)(24,207)(25,206)(26,205)(27,204)(28,203)(29,202)(30,201)(31,200)(32,199)(33,198)(34,197)(35,196)(36,195)(37,194)(38,193)(39,192)(40,191)(41,190)(42,189)(43,188)(44,187)(45,186)(46,185)(47,184)(48,183)(49,182)(50,181)(51,180)(52,179)(53,178)(54,177)(55,176)(56,175)(57,174)(58,173)(59,172)(60,171)(61,170)(62,169)(63,168)(64,167)(65,166)(66,165)(67,164)(68,163)(69,162)(70,161)(71,160)(72,159)(73,158)(74,157)(75,156)(76,155)(77,154)(78,153)(79,152)(80,151)(81,150)(82,149)(83,148)(84,147)(85,146)(86,145)(87,144)(88,143)(89,142)(90,141)(91,140)(92,139)(93,138)(94,137)(95,136)(96,135)(97,134)(98,133)(99,132)(100,131)(101,130)(102,129)(103,128)(104,127)(105,126)(106,125)(107,124)(108,123)(109,122)(110,121)(111,120)(112,119)(113,118)(114,117)(115,116)

Polycyclic presentation:

2^3,11,199|b^2=c,b^a=bc,d^a=d^10,e^a=e^198

Conjugacy classes & character table

4381 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₇₅₁₂, D₂₆₂₆₈
maximal normal subgroup of	D₁₇₅₁₂
maximal quotient of	Dic₈₇₅₆, D₁₇₅₁₂, D₂₆₂₆₈

GAP

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

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 11, 199]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^10);;
gap> SetConjugate(coll, 5, 1, F.5^198);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D8756 #72213601