C₂₃×#2312 #15669931

order	2944
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₂₃ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂₃×#2312

Subgroups

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂⁵×C₂₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

672 conjugacy classes of subgroups; only the 280 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 280 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) = 43
transitive degree	μ_t(G) = 736

A minimal faithful permutation representation — 4 generators on 43 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)(30,34)(31,32)(33,35)(37,40)(38,42);(29,32)(33,34)(36,37,39,40)(38,41,42,43);(28,29)(30,33)(31,32)(34,35)(36,37)(38,41)(39,40)(42,43);(28,30)(29,33)(31,35)(32,34)(36,38)(37,41)(39,42)(40,43)

Conjugacy classes & character table

1012 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₃×#8736, C₂₃×#8750, C₂₃×#8759, …
maximal normal subgroup of	C₂₃×#8736, C₂₃×#8750, C₂₃×#8759, …
maximal quotient of	C₂₃×#13775, C₂₃×#13778, C₂₃×#13788, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2944, 1044);