C₂₃×#1991 #15669610

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

Aliases

direct product

C₂₃×#1991

Subgroups

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

252 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 51
transitive degree	μ_t(G) = 2944

A minimal faithful permutation representation — 3 generators on 51 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)(36,37)(38,44)(39,48)(40,42)(41,46)(43,45)(47,50)(49,51);(25,27)(29,32)(33,34)(36,38,45,51,40,47,39,46)(37,41,48,50,42,49,43,44);(28,30,31,35)(29,33,32,34)(37,42)(41,49)(43,48)(44,50)

Conjugacy classes & character table

1012 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₃×#7135, C₂₃×#7359, C₂₃×#8529, …
maximal normal subgroup of	C₂₃×#7135, C₂₃×#7359, C₂₃×#8529, …
maximal quotient of	C₂₃×#9871, C₂₃×#9875, C₂₃×#9886, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2944, 723);