D₁₉₅₆₀ #72274059

order	39120
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₉₅₆₀

Splittings

Fitting	C₃×C₅×C₈×C₁₆₃ ⋊ C₂
Hall	C₃×C₅×C₁₆₃ ⋊ D₈
Hall	C₅×C₁₆₃ ⋊ D₂₄
Hall	C₃×C₁₆₃ ⋊ D₄₀
Sylow	C₁₆₃ ⋊ D₁₂₀
Hall	C₃×C₅ ⋊ D₁₃₀₄
Sylow	C₅ ⋊ D₃₉₁₂
Sylow	C₃ ⋊ D₆₅₂₀

Subgroups

	subgroup	G/H
center	C₂	D₉₇₈₀
commutator, cosocle	C₃×C₄×C₅×C₁₆₃	C₂²
Frattini	C₄	C₂×D₂₄₄₅
Fitting	C₃×C₅×C₈×C₁₆₃	C₂
socle	C₂×C₃×C₅×C₁₆₃	D₄
hypercenter	C₈	D₂₄₄₅
hypocenter	C₃×C₅×C₁₆₃	D₈
LCS	G ▷ C₃×C₄×C₅×C₁₆₃ ▷ C₂×C₃×C₅×C₁₆₃ ▷ C₃×C₅×C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 179
transitive degree	μ_t(G) = 19560

A minimal faithful permutation representation — 2 generators on 179 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);(2,8)(3,7)(4,6)(10,11)(13,16)(14,15)(18,179)(19,178)(20,177)(21,176)(22,175)(23,174)(24,173)(25,172)(26,171)(27,170)(28,169)(29,168)(30,167)(31,166)(32,165)(33,164)(34,163)(35,162)(36,161)(37,160)(38,159)(39,158)(40,157)(41,156)(42,155)(43,154)(44,153)(45,152)(46,151)(47,150)(48,149)(49,148)(50,147)(51,146)(52,145)(53,144)(54,143)(55,142)(56,141)(57,140)(58,139)(59,138)(60,137)(61,136)(62,135)(63,134)(64,133)(65,132)(66,131)(67,130)(68,129)(69,128)(70,127)(71,126)(72,125)(73,124)(74,123)(75,122)(76,121)(77,120)(78,119)(79,118)(80,117)(81,116)(82,115)(83,114)(84,113)(85,112)(86,111)(87,110)(88,109)(89,108)(90,107)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,99)

Polycyclic presentation:

2^4,3,5,163|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^4,g^a=g^162

Conjugacy classes & character table

9783 conjugacy classes. The character table is too large to display.

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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 5, 163]);;
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.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^162);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D19560 #72274059