D₁₁₉₉₇ #37562004

order	23994
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₉₉₇
unambiguous semidirect product	C₉⋊D₁₃₃₃
unambiguous semidirect product	C₃₁⋊D₃₈₇
unambiguous semidirect product	C₄₃⋊D₂₇₉

Splittings

Fitting, derived, Hall	C₉×C₃₁×C₄₃ ⋊ C₂
Hall	C₃₁×C₄₃ ⋊ D₉
Hall	C₉×C₄₃ ⋊ D₃₁
Hall	C₉×C₃₁ ⋊ D₄₃
Sylow	C₄₃ ⋊ D₂₇₉
Sylow	C₃₁ ⋊ D₃₈₇
Sylow	C₉ ⋊ D₁₃₃₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₃₁×C₄₃	C₂
Frattini	C₃	D₃₉₉₉
socle	C₃×C₃₁×C₄₃	D₃
LCS	G ▷ C₉×C₃₁×C₄₃
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 11997

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);(2,9)(3,8)(4,7)(5,6)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(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:

2,3,31,43,3|b^3=e^2,b^a=b^2e,c^a=c^30,d^a=d^42,e^a=e^2

Conjugacy classes & character table

6000 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23994, 12);

D11997 #37562004