D₁₆₄₈₅ #45682136

order	32970
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₄₈₅
unambiguous semidirect product	C₃⋊D₅₄₉₅
unambiguous semidirect product	C₅⋊D₃₂₉₇
unambiguous semidirect product	C₇⋊D₂₃₅₅
unambiguous semidirect product	C₁₅₇⋊D₁₀₅

Splittings

Fitting, socle, derived, Hall	C₃×C₅×C₇×C₁₅₇ ⋊ C₂
Hall	C₅×C₇×C₁₅₇ ⋊ D₃
Hall	C₃×C₇×C₁₅₇ ⋊ D₅
Hall	C₃×C₅×C₁₅₇ ⋊ D₇
Hall	C₇×C₁₅₇ ⋊ D₁₅
Hall	C₅×C₁₅₇ ⋊ D₂₁
Hall	C₃×C₁₅₇ ⋊ D₃₅
Sylow	C₁₅₇ ⋊ D₁₀₅
Hall	C₃×C₅×C₇ ⋊ D₁₅₇
Hall	C₅×C₇ ⋊ D₄₇₁
Hall	C₃×C₇ ⋊ D₇₈₅
Hall	C₃×C₅ ⋊ D₁₀₉₉
Sylow	C₇ ⋊ D₂₃₅₅
Sylow	C₅ ⋊ D₃₂₉₇
Sylow	C₃ ⋊ D₅₄₉₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₇×C₁₅₇	C₂
LCS	G ▷ C₃×C₅×C₇×C₁₅₇
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 172
transitive degree	μ_t(G) = 16485

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);(2,3)(5,8)(6,7)(10,15)(11,14)(12,13)(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:

2,3,5,7,157|b^a=b^2,c^a=c^4,d^a=d^6,e^a=e^156

Conjugacy classes & character table

8244 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32970, 20);

D16485 #45682136