C₂₃×#2117 #15669736

order	2944
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₂₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₃×#2117

Subgroups

	subgroup	G/H
center	C₈×C₂₃	C₂×D₄
commutator	C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂×C₈	C₂³×C₂₃
socle	C₂×C₂₃	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₂₃	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ G

220 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 55
transitive degree	μ_t(G) = 736

A minimal faithful permutation representation — 3 generators on 55 points:

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

Conjugacy classes & character table

1012 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₃×#12609, C₂₃×#12618, C₂₃×#12685, …
maximal normal subgroup of	C₂₃×#12609, C₂₃×#12618, C₂₃×#12685, …
maximal quotient of	C₂₃×#11421, C₂₃×#11422, C₂₃×#11443, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2944, 849);