D₁₁₁₈₇ #36375412

order	22374
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₁₈₇
unambiguous semidirect product	C₉⋊D₁₂₄₃
unambiguous semidirect product	C₁₁⋊D₁₀₁₇
unambiguous semidirect product	C₁₁₃⋊D₉₉

Splittings

Fitting, derived, Hall	C₉×C₁₁×C₁₁₃ ⋊ C₂
Hall	C₁₁×C₁₁₃ ⋊ D₉
Hall	C₉×C₁₁₃ ⋊ D₁₁
Sylow	C₁₁₃ ⋊ D₉₉
Hall	C₉×C₁₁ ⋊ D₁₁₃
Sylow	C₁₁ ⋊ D₁₀₁₇
Sylow	C₉ ⋊ D₁₂₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₁×C₁₁₃	C₂
Frattini	C₃	D₃₇₂₉
socle	C₃×C₁₁×C₁₁₃	D₃
LCS	G ▷ C₉×C₁₁×C₁₁₃
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 133
transitive degree	μ_t(G) = 11187

A minimal faithful permutation representation — 2 generators on 133 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);(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(22,133)(23,132)(24,131)(25,130)(26,129)(27,128)(28,127)(29,126)(30,125)(31,124)(32,123)(33,122)(34,121)(35,120)(36,119)(37,118)(38,117)(39,116)(40,115)(41,114)(42,113)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)

Polycyclic presentation:

2,3,11,113,3|b^3=e^2,b^a=b^2e,c^a=c^10,d^a=d^112,e^a=e^2

Conjugacy classes & character table

5595 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22374, 8);

D11187 #36375412