D₁₁₇₁₃ #37552836

order	23426
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₇₁₃
unambiguous semidirect product	C₁₃⋊D₉₀₁
unambiguous semidirect product	C₁₇⋊D₆₈₉
unambiguous semidirect product	C₅₃⋊D₂₂₁

Splittings

Fitting, socle, derived, Hall	C₁₃×C₁₇×C₅₃ ⋊ C₂
Hall	C₁₇×C₅₃ ⋊ D₁₃
Hall	C₁₃×C₅₃ ⋊ D₁₇
Hall	C₁₃×C₁₇ ⋊ D₅₃
Sylow	C₅₃ ⋊ D₂₂₁
Sylow	C₁₇ ⋊ D₆₈₉
Sylow	C₁₃ ⋊ D₉₀₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₃×C₁₇×C₅₃	C₂
LCS	G ▷ C₁₃×C₁₇×C₅₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 11713

A minimal faithful permutation representation — 2 generators on 83 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);(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(32,83)(33,82)(34,81)(35,80)(36,79)(37,78)(38,77)(39,76)(40,75)(41,74)(42,73)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)

Polycyclic presentation:

2,13,17,53|b^a=b^12,c^a=c^16,d^a=d^52

Conjugacy classes & character table

5858 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23426, 9);

D11713 #37552836