C₂₃×#4223 #16836826

order	3772
isoclinism	isoclinic class of #4223
automorphism	?
homology	Abelianization (H₁) = C₄×C₂₃ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₂₃×#4223

Splittings

Fitting, socle, Hall	C₂₃×C₄₁ ⋊ C₄
derived, Sylow	C₄₁ ⋊ C₄×C₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂₃	#4223
commutator, hypocenter	C₄₁	C₄×C₂₃
Frattini	C₁	G
Fitting, socle	C₂₃×C₄₁	C₄
prime radical	C₂₃×D₄₁	C₂
cosocle	D₄₁	C₂×C₂₃
LCS	G ▷ C₄₁
UCS	C₁ ◁ C₂₃

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 943

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23)(25,26,27,29)(28,31,32,38)(30,35,36,34)(33,40,41,51)(37,44,45,47)(39,48,49,43)(42,53,54,58)(46,56,57,63)(50,60,61,59)(52,64,62,55);(24,25,28,33,42,53,57,45,36,29,34,43,55,56,40,52,61,49,38,47,59,60,44,31,39,50,62,51,63,64,48,35,26,30,37,46,58,54,41,32,27)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×C₂₃×#4223, C₂₃×C₃⋊#4223, C₂₃×#4223⋊C₅, …
maximal normal subgroup of	C₃×C₂₃×#4223, C₂₃×#4223⋊C₅, C₅×C₂₃×#4223, …
maximal quotient of	C₃×C₂₃×#4223, C₂₃×C₃⋊#4223, C₅×C₂₃×#4223, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3772, 6);

C23×#4223 #16836826