C₂₃×#1564 #15669183

order	2944
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₂₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₂₃×#1564

Subgroups

	subgroup	G/H
center	C₄²×C₂₃	D₄
commutator	C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×C₄⋊C₄
prime radical	C₂³×C₂₃	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

180 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 47
transitive degree	μ_t(G) = 1472

A minimal faithful permutation representation — 3 generators on 47 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23)(24,25,26,28,27,29,30,31)(32,33)(34,41)(35,39)(36,45)(37,40)(38,43)(42,46)(44,47);(32,34,36,43,37,44,35,42)(33,38,39,41,40,46,45,47);(32,35,37,36)(33,39,40,45)(34,42,44,43)(38,41,46,47)

Conjugacy classes & character table

1288 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₃×#7155, C₂₃×#7157, C₂₃×#7158, …
maximal normal subgroup of	C₂₃×#7155, C₂₃×#7157, C₂₃×#7158, …
maximal quotient of	C₂₃×#7872, C₄×C₂₃×#334, C₄×C₂₃×#335, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2944, 296);