D₁₁₁₇₅ #36375310

order	22350
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₁₇₅
unambiguous semidirect product	C₃⋊D₃₇₂₅
unambiguous semidirect product	C₂₅⋊D₄₄₇
unambiguous semidirect product	C₁₄₉⋊D₇₅

Splittings

Fitting, derived, Hall	C₃×C₂₅×C₁₄₉ ⋊ C₂
Hall	C₂₅×C₁₄₉ ⋊ D₃
Hall	C₃×C₁₄₉ ⋊ D₂₅
Sylow	C₁₄₉ ⋊ D₇₅
Hall	C₃×C₂₅ ⋊ D₁₄₉
Sylow	C₂₅ ⋊ D₄₄₇
Sylow	C₃ ⋊ D₃₇₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₂₅×C₁₄₉	C₂
Frattini	C₅	D₂₂₃₅
socle	C₃×C₅×C₁₄₉	D₅
LCS	G ▷ C₃×C₂₅×C₁₄₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 177
transitive degree	μ_t(G) = 11175

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

Polycyclic presentation:

2,3,5,149,5|c^5=e^4,b^a=b^2,c^a=c^4e,d^a=d^148,e^a=e^4

Conjugacy classes & character table

5589 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22350, 8);

D11175 #36375310