D₃₆₂₄ #72211162

order	7248
automorphism	Aut(G) order 4348800
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₆₂₄

Splittings

Fitting	C₃×C₈×C₁₅₁ ⋊ C₂
Hall	C₃×C₁₅₁ ⋊ D₈
Sylow	C₁₅₁ ⋊ D₂₄
Sylow	C₃ ⋊ D₁₂₀₈

Subgroups

	subgroup	G/H
center	C₂	D₁₈₁₂
commutator, cosocle	C₃×C₄×C₁₅₁	C₂²
Frattini	C₄	C₂×D₄₅₃
Fitting	C₃×C₈×C₁₅₁	C₂
socle	C₂×C₃×C₁₅₁	D₄
hypercenter	C₈	D₄₅₃
hypocenter	C₃×C₁₅₁	D₈
LCS	G ▷ C₃×C₄×C₁₅₁ ▷ C₂×C₃×C₁₅₁ ▷ C₃×C₁₅₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 162
transitive degree	μ_t(G) = 3624

A minimal faithful permutation representation — 2 generators on 162 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);(2,8)(3,7)(4,6)(10,11)(13,162)(14,161)(15,160)(16,159)(17,158)(18,157)(19,156)(20,155)(21,154)(22,153)(23,152)(24,151)(25,150)(26,149)(27,148)(28,147)(29,146)(30,145)(31,144)(32,143)(33,142)(34,141)(35,140)(36,139)(37,138)(38,137)(39,136)(40,135)(41,134)(42,133)(43,132)(44,131)(45,130)(46,129)(47,128)(48,127)(49,126)(50,125)(51,124)(52,123)(53,122)(54,121)(55,120)(56,119)(57,118)(58,117)(59,116)(60,115)(61,114)(62,113)(63,112)(64,111)(65,110)(66,109)(67,108)(68,107)(69,106)(70,105)(71,104)(72,103)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)

Polycyclic presentation:

2^4,3,151|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^150

Conjugacy classes & character table

1815 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₇₂₄₈, D₁₀₈₇₂, D₁₈₁₂₀, …
maximal normal subgroup of	D₇₂₄₈
maximal quotient of	D₇₂₄₈, Dic₃₆₂₄, D₁₀₈₇₂, …

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, 3, 151]);;
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.6^150);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D3624 #72211162