D₆₆₈₀ #72212342

order	13360
automorphism	Aut(G) order 17742080
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₆₆₈₀

Splittings

Fitting	C₅×C₈×C₁₆₇ ⋊ C₂
Hall	C₅×C₁₆₇ ⋊ D₈
Sylow	C₁₆₇ ⋊ D₄₀
Sylow	C₅ ⋊ D₁₃₃₆

Subgroups

	subgroup	G/H
center	C₂	D₃₃₄₀
commutator, cosocle	C₄×C₅×C₁₆₇	C₂²
Frattini	C₄	C₂×D₈₃₅
Fitting	C₅×C₈×C₁₆₇	C₂
socle	C₂×C₅×C₁₆₇	D₄
hypercenter	C₈	D₈₃₅
hypocenter	C₅×C₁₆₇	D₈
LCS	G ▷ C₄×C₅×C₁₆₇ ▷ C₂×C₅×C₁₆₇ ▷ C₅×C₁₆₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 180
transitive degree	μ_t(G) = 6680

A minimal faithful permutation representation — 2 generators on 180 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);(2,8)(3,7)(4,6)(10,13)(11,12)(15,180)(16,179)(17,178)(18,177)(19,176)(20,175)(21,174)(22,173)(23,172)(24,171)(25,170)(26,169)(27,168)(28,167)(29,166)(30,165)(31,164)(32,163)(33,162)(34,161)(35,160)(36,159)(37,158)(38,157)(39,156)(40,155)(41,154)(42,153)(43,152)(44,151)(45,150)(46,149)(47,148)(48,147)(49,146)(50,145)(51,144)(52,143)(53,142)(54,141)(55,140)(56,139)(57,138)(58,137)(59,136)(60,135)(61,134)(62,133)(63,132)(64,131)(65,130)(66,129)(67,128)(68,127)(69,126)(70,125)(71,124)(72,123)(73,122)(74,121)(75,120)(76,119)(77,118)(78,117)(79,116)(80,115)(81,114)(82,113)(83,112)(84,111)(85,110)(86,109)(87,108)(88,107)(89,106)(90,105)(91,104)(92,103)(93,102)(94,101)(95,100)(96,99)(97,98)

Polycyclic presentation:

2^4,5,167|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^166

Conjugacy classes & character table

3343 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₃₃₆₀, D₂₀₀₄₀
maximal normal subgroup of	D₁₃₃₆₀
maximal quotient of	D₁₃₃₆₀, Dic₆₆₈₀, D₂₀₀₄₀

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, 5, 167]);;
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.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^166);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D6680 #72212342