C₃²×#2219 #11769268

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#2219

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃²
prime radical	C₂³×C₃²	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₃²×C₄ ◁ C₂²×C₃²×C₈ ◁ G

450 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 42
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 42 points:

(1,2,3)(7,8)(9,10)(11,12,16,20)(13,21,23,35)(14,26,28,41)(15,29,27,19)(17,30,31,24)(18,33,34,22)(25,40,39,37)(32,38,36,42);(4,5,6)(11,13,22,38,15,25,28,30,16,23,33,42,27,39,14,24)(12,17,26,37,19,32,34,35,20,31,41,40,29,36,18,21);(8,10)(13,23)(17,31)(21,35)(24,30)(25,39)(32,36)(37,40)(38,42)

Conjugacy classes & character table

234 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 3279);