C₂₃×#1502 #15669121

order	2944
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂₃×#1502

Subgroups

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂₃×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

284 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 39
transitive degree	μ_t(G) = 736

A minimal faithful permutation representation — 3 generators on 39 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)(26,28)(27,30)(29,31)(33,37,38,39);(25,28)(26,31)(29,30)(32,33)(34,37)(35,39)(36,38);(32,34,36,35)(33,37,38,39)

Conjugacy classes & character table

874 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₃×#8251, C₂₃×#8253, C₂₃×#8255, …
maximal normal subgroup of	C₂₃×#8251, C₂₃×#8253, C₂₃×#8255, …
maximal quotient of	C₂₃×#7560, C₂₃×#7569, C₂₃×#7570, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2944, 234);