C₂₃×#2115 #15669734

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

Aliases

direct product

C₂₃×#2115

Subgroups

	subgroup	G/H
center	C₈×C₂₃	C₂²⋊C₄
commutator	C₂²	C₂²×C₈×C₂₃
Frattini, cosocle	C₂×C₈	C₂³×C₂₃
socle	C₂×C₂₃	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₂₃	C₄
LCS	G ▷ C₂² ▷ 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)(24,25,27,32,28,33,39,47,29,31,40,44,41,45,52,54)(26,34,36,46,37,48,50,55,35,30,38,42,49,43,51,53);(24,26)(25,30)(27,38)(28,37)(29,35)(31,34)(32,46)(33,43)(36,40)(39,51)(41,49)(42,44)(45,48)(47,55)(50,52)(53,54);(26,35)(30,34)(36,38)(37,49)(42,46)(43,48)(50,51)(53,55)

Conjugacy classes & character table

1012 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₃×#7207, C₂₃×#7208, C₂₃×#11483, …
maximal normal subgroup of	C₂₃×#7207, C₂₃×#7208, C₂₃×#11483, …
maximal quotient of	C₂₃×#11419, C₂₃×#11441, C₂₃×#11461, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2944, 847);