C₂×C₂₃×#364 #15669771

order	2944
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₂₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×C₂₃×#364

Subgroups

	subgroup	G/H
center	C₂×C₈×C₂₃	D₄
commutator	C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₄⋊C₈
prime radical	C₂³×C₂₃	M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ G

168 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 41
transitive degree	μ_t(G) = 736

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41);(3,4,7,12,8,11,16,18,9,13,15,10,6,17,5,14);(3,5,6,15,9,16,8,7)(4,10,11,14,13,12,17,18)

Conjugacy classes & character table

1288 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂₃×#7160, C₂₃×#7161, C₂₃×#7162, …
maximal normal subgroup of	C₂₃×#7160, C₂₃×#7161, C₂₃×#7162, …
maximal quotient of	C₂×C₂₃×#1367, C₂₃×#11996, C₂×C₂₃×#1370, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2944, 884);