C₂₃×#1517 #15669136

order	2944
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂₃×#1517

Subgroups

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

264 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 736

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)(26,30)(27,29)(28,31)(32,33)(34,40,42,37)(35,38)(36,39)(41,46,43,45)(44,47);(25,29)(26,27)(30,31)(32,34)(33,37)(35,43)(36,42)(38,46)(39,40)(41,44)(45,47);(24,26,28,27)(25,29,31,30)(32,35,36,44)(33,38,39,47)(34,41,42,43)(37,45,40,46)

Conjugacy classes & character table

736 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₃×#8428, C₂₃×#8429, C₂₃×#8430, …
maximal normal subgroup of	C₂₃×#8428, C₂₃×#8429, C₂₃×#8430, …
maximal quotient of	C₂₃×#7574, C₂₃×#7587, C₂₃×#7588, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2944, 249);