D₁₂₂₁₃ #37567665

order	24426
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₂₂₁₃
unambiguous semidirect product	C₉⋊D₁₃₅₇
unambiguous semidirect product	C₂₃⋊D₅₃₁
unambiguous semidirect product	C₅₉⋊D₂₀₇

Splittings

Fitting, derived, Hall	C₉×C₂₃×C₅₉ ⋊ C₂
Hall	C₂₃×C₅₉ ⋊ D₉
Hall	C₉×C₅₉ ⋊ D₂₃
Hall	C₉×C₂₃ ⋊ D₅₉
Sylow	C₅₉ ⋊ D₂₀₇
Sylow	C₂₃ ⋊ D₅₃₁
Sylow	C₉ ⋊ D₁₃₅₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₂₃×C₅₉	C₂
Frattini	C₃	D₄₀₇₁
socle	C₃×C₂₃×C₅₉	D₃
LCS	G ▷ C₉×C₂₃×C₅₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 12213

A minimal faithful permutation representation — 2 generators on 91 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);(2,9)(3,8)(4,7)(5,6)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(34,91)(35,90)(36,89)(37,88)(38,87)(39,86)(40,85)(41,84)(42,83)(43,82)(44,81)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)

Polycyclic presentation:

2,3,23,59,3|b^3=e^2,b^a=b^2e,c^a=c^22,d^a=d^58,e^a=e^2

Conjugacy classes & character table

6108 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24426, 8);

D12213 #37567665