C₂₃×#1496 #15669115

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

Aliases

direct product

C₂₃×#1496

Subgroups

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂₃×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

246 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 92 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)(26,30)(27,29)(28,31)(32,33,35,39,36,38,42,45)(34,37,41,43,40,44,46,47);(25,29)(26,27)(30,31)(32,34)(33,37)(35,41)(36,40)(38,44)(39,43)(42,46)(45,47);(24,26,28,27)(25,30,31,29)(33,38)(34,40)(39,45)(41,46)

Conjugacy classes & character table

736 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₃×#8489, C₂₃×#8500, C₂₃×#8503, …
maximal normal subgroup of	C₂₃×#8489, C₂₃×#8500, C₂₃×#8503, …
maximal quotient of	C₂₃×#7698, C₂₃×#7699, C₂₃×#7700, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2944, 228);