C₂₃×#2240 #15669859

order	2944
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₂₃×#2240

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₂₃
prime radical	C₂₃×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄²×C₂₃ ◁ C₄×C₈×C₂₃ ◁ G

160 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 72 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)(26,29)(27,28)(30,31)(33,39)(34,44)(35,46)(36,47)(37,40)(41,42)(43,45);(25,28)(29,31)(32,33,36,41,37,42,35,39,38,43,46,34,40,44,47,45);(24,26,27,30)(25,29,28,31)(32,34,35,33,40,39,36,44,38,41,47,43,37,45,46,42)

Conjugacy classes & character table

874 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₃×#12340, C₂₃×#12341, C₂₃×#12342, …
maximal normal subgroup of	C₂₃×#12340, C₂₃×#12341, C₂₃×#12342, …
maximal quotient of	C₂₃×#12888, C₂₃×#13147, C₂₃×#13202, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2944, 972);