D₁₁₅₅₇ #37528810

order	23114
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₅₅₇
unambiguous semidirect product	C₇⋊D₁₆₅₁
unambiguous semidirect product	C₁₃⋊D₈₈₉
unambiguous semidirect product	C₁₂₇⋊D₉₁

Splittings

Fitting, socle, derived, Hall	C₇×C₁₃×C₁₂₇ ⋊ C₂
Hall	C₁₃×C₁₂₇ ⋊ D₇
Hall	C₇×C₁₂₇ ⋊ D₁₃
Sylow	C₁₂₇ ⋊ D₉₁
Hall	C₇×C₁₃ ⋊ D₁₂₇
Sylow	C₁₃ ⋊ D₈₈₉
Sylow	C₇ ⋊ D₁₆₅₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₃×C₁₂₇	C₂
LCS	G ▷ C₇×C₁₃×C₁₂₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 147
transitive degree	μ_t(G) = 11557

A minimal faithful permutation representation — 2 generators on 147 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);(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(22,147)(23,146)(24,145)(25,144)(26,143)(27,142)(28,141)(29,140)(30,139)(31,138)(32,137)(33,136)(34,135)(35,134)(36,133)(37,132)(38,131)(39,130)(40,129)(41,128)(42,127)(43,126)(44,125)(45,124)(46,123)(47,122)(48,121)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,112)(58,111)(59,110)(60,109)(61,108)(62,107)(63,106)(64,105)(65,104)(66,103)(67,102)(68,101)(69,100)(70,99)(71,98)(72,97)(73,96)(74,95)(75,94)(76,93)(77,92)(78,91)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)

Polycyclic presentation:

2,7,13,127|b^a=b^6,c^a=c^12,d^a=d^126

Conjugacy classes & character table

5780 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23114, 9);

D11557 #37528810