D₁₁₅₄₃ #37528142

order	23086
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₅₄₃
unambiguous semidirect product	C₇⋊D₁₆₄₉
unambiguous semidirect product	C₁₇⋊D₆₇₉
unambiguous semidirect product	C₉₇⋊D₁₁₉

Splittings

Fitting, socle, derived, Hall	C₇×C₁₇×C₉₇ ⋊ C₂
Hall	C₁₇×C₉₇ ⋊ D₇
Hall	C₇×C₉₇ ⋊ D₁₇
Hall	C₇×C₁₇ ⋊ D₉₇
Sylow	C₉₇ ⋊ D₁₁₉
Sylow	C₁₇ ⋊ D₆₇₉
Sylow	C₇ ⋊ D₁₆₄₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₇×C₉₇	C₂
LCS	G ▷ C₇×C₁₇×C₉₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 11543

A minimal faithful permutation representation — 2 generators on 121 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);(2,7)(3,6)(4,5)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(26,121)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,111)(37,110)(38,109)(39,108)(40,107)(41,106)(42,105)(43,104)(44,103)(45,102)(46,101)(47,100)(48,99)(49,98)(50,97)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)

Polycyclic presentation:

2,7,17,97|b^a=b^6,c^a=c^16,d^a=d^96

Conjugacy classes & character table

5773 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23086, 8);

D11543 #37528142