C₂₃×#1388 #15669007

order	2944
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₂₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₃×#1388

Subgroups

	subgroup	G/H
center	C₄×C₂₃	#102
commutator	C₈	C₄²×C₂₃
Frattini, cosocle	C₂²×C₈	C₂²×C₂₃
socle	C₂×C₂₃	#340
prime radical	C₂³×C₂₃	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ G

124 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 76 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 — 2 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)(24,25,28,34,31,36,30,47)(26,37,39,49,42,51,41,55)(27,43,45,52,46,35,29,33)(32,40,48,44,50,54,53,38);(24,26,29,40,30,41,46,38,31,42,45,54,28,39,27,44)(25,32,35,49,34,48,33,51,36,50,43,55,47,53,52,37)

Conjugacy classes & character table

736 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₂₃×#1388, C₂₃×#12646, C₂₃×#12651, …
maximal normal subgroup of	C₂×C₂₃×#1388, C₂₃×#12646, C₂₃×#12651, …
maximal quotient of	C₂₃×#7158, C₂₃×#7176, C₂₃×#7177, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2944, 120);