C₂₃×#5997 #17985207

order	4508
isoclinism	isoclinic class of #5997
automorphism	?
homology	Abelianization (H₁) = C₄×C₂₃ · Schur multiplier (H₂) = C₇

Aliases

direct product

C₂₃×#5997

Splittings

Fitting, socle, Hall	C₇²×C₂₃ ⋊ C₄
derived, Sylow	C₇² ⋊ C₄×C₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂₃	#5997
commutator, hypocenter	C₇²	C₄×C₂₃
Frattini	C₁	G
Fitting, socle	C₇²×C₂₃	C₄
prime radical	C₂₃×C₇⋊D₇	C₂
cosocle	C₇⋊D₇	C₂×C₂₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂₃

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 322

A minimal faithful permutation representation — 2 generators on 37 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)(26,28,29,33)(27,31,32,36)(30,34,35,37);(24,26,30,36,31,35,29)(25,32,28,37,34,33,27)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂₃×#10596885, C₂₃×#5997⋊C₃, C₃×C₂₃×#5997, …
maximal normal subgroup of	C₂₃×#5997⋊C₃, C₃×C₂₃×#5997, C₅×C₂₃×#5997
maximal quotient of	C₃×C₂₃×#5997, C₂₃×C₃⋊#5997, C₅×C₂₃×#5997, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4508, 22);

C23×#5997 #17985207