D₆₉₀₈ #72212435

order	13816
automorphism	Aut(G) order 21552960
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₆₉₀₈

Splittings

Fitting	C₄×C₁₁×C₁₅₇ ⋊ C₂
Hall	C₁₁×C₁₅₇ ⋊ D₄
Sylow	C₁₅₇ ⋊ D₄₄
Sylow	C₁₁ ⋊ D₆₂₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₇₂₇
commutator, socle, cosocle	C₂×C₁₁×C₁₅₇	C₂²
Fitting	C₄×C₁₁×C₁₅₇	C₂
hypercenter	C₄	D₁₇₂₇
hypocenter	C₁₁×C₁₅₇	D₄
LCS	G ▷ C₂×C₁₁×C₁₅₇ ▷ C₁₁×C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 172
transitive degree	μ_t(G) = 6908

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

Polycyclic presentation:

2^3,11,157|b^2=c,b^a=bc,d^a=d^10,e^a=e^156

Conjugacy classes & character table

3457 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₃₈₁₆, D₂₀₇₂₄
maximal normal subgroup of	D₁₃₈₁₆
maximal quotient of	D₁₃₈₁₆, Dic₆₉₀₈, D₂₀₇₂₄

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, 157]);;
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.5^156);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D6908 #72212435