D₁₁₇₅₃ #37555074

order	23506
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₇₅₃
unambiguous semidirect product	C₇⋊D₁₆₇₉
unambiguous semidirect product	C₂₃⋊D₅₁₁
unambiguous semidirect product	C₇₃⋊D₁₆₁

Splittings

Fitting, socle, derived, Hall	C₇×C₂₃×C₇₃ ⋊ C₂
Hall	C₂₃×C₇₃ ⋊ D₇
Hall	C₇×C₇₃ ⋊ D₂₃
Hall	C₇×C₂₃ ⋊ D₇₃
Sylow	C₇₃ ⋊ D₁₆₁
Sylow	C₂₃ ⋊ D₅₁₁
Sylow	C₇ ⋊ D₁₆₇₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₂₃×C₇₃	C₂
LCS	G ▷ C₇×C₂₃×C₇₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 103
transitive degree	μ_t(G) = 11753

A minimal faithful permutation representation — 2 generators on 103 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);(2,7)(3,6)(4,5)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(32,103)(33,102)(34,101)(35,100)(36,99)(37,98)(38,97)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)

Polycyclic presentation:

2,7,23,73|b^a=b^6,c^a=c^22,d^a=d^72

Conjugacy classes & character table

5878 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23506, 8);

D11753 #37555074