D₁₁₇₇₅ #37555485

order	23550
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₇₇₅
unambiguous semidirect product	C₃⋊D₃₉₂₅
unambiguous semidirect product	C₂₅⋊D₄₇₁
unambiguous semidirect product	C₁₅₇⋊D₇₅

Splittings

Fitting, derived, Hall	C₃×C₂₅×C₁₅₇ ⋊ C₂
Hall	C₂₅×C₁₅₇ ⋊ D₃
Hall	C₃×C₁₅₇ ⋊ D₂₅
Sylow	C₁₅₇ ⋊ D₇₅
Hall	C₃×C₂₅ ⋊ D₁₅₇
Sylow	C₂₅ ⋊ D₄₇₁
Sylow	C₃ ⋊ D₃₉₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₂₅×C₁₅₇	C₂
Frattini	C₅	D₂₃₅₅
socle	C₃×C₅×C₁₅₇	D₅
LCS	G ▷ C₃×C₂₅×C₁₅₇
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 185
transitive degree	μ_t(G) = 11775

A minimal faithful permutation representation — 2 generators on 185 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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185);(2,3)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(30,185)(31,184)(32,183)(33,182)(34,181)(35,180)(36,179)(37,178)(38,177)(39,176)(40,175)(41,174)(42,173)(43,172)(44,171)(45,170)(46,169)(47,168)(48,167)(49,166)(50,165)(51,164)(52,163)(53,162)(54,161)(55,160)(56,159)(57,158)(58,157)(59,156)(60,155)(61,154)(62,153)(63,152)(64,151)(65,150)(66,149)(67,148)(68,147)(69,146)(70,145)(71,144)(72,143)(73,142)(74,141)(75,140)(76,139)(77,138)(78,137)(79,136)(80,135)(81,134)(82,133)(83,132)(84,131)(85,130)(86,129)(87,128)(88,127)(89,126)(90,125)(91,124)(92,123)(93,122)(94,121)(95,120)(96,119)(97,118)(98,117)(99,116)(100,115)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)

Polycyclic presentation:

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

Conjugacy classes & character table

5889 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23550, 9);

D11775 #37555485