C₂₃×#2138 #15669757

order	2944
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₃×#2138

Subgroups

	subgroup	G/H
center	C₂×C₄×C₂₃	D₈
commutator	C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×D₈
prime radical	C₂×C₂₃×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ G

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

normal subgroups only

This group has 228 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) = 59
transitive degree	μ_t(G) = 1472

A minimal faithful permutation representation — 3 generators on 59 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)(30,39)(31,36)(32,44)(33,48)(34,38)(35,41)(37,46)(40,55)(42,56)(43,54)(45,50)(47,58)(49,53)(51,59)(52,57);(28,30)(29,35)(31,40)(32,45)(33,49)(34,42)(36,50)(37,53)(38,52)(39,44)(41,46)(43,57)(47,59)(48,51)(54,55)(56,58);(28,31,34,33)(29,36,38,48)(30,40,42,49)(32,43,47,46)(35,50,52,51)(37,44,54,58)(39,55,56,53)(41,45,57,59)

Conjugacy classes & character table

1012 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₃×#12641, C₂₃×#12642, C₂₃×#12745, …
maximal normal subgroup of	C₂₃×#12641, C₂₃×#12642, C₂₃×#12745, …
maximal quotient of	C₂₃×#11591, C₂₃×#11647, C₂₃×#11659, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2944, 870);