C₃²×#2141 #11769190

order	1152
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

C₃²×#2141

Subgroups

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

660 conjugacy classes of subgroups; only the 312 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 312 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)(13,22)(14,19)(15,28)(16,32)(17,21)(18,24)(20,30)(23,36)(25,42)(26,39)(27,37)(29,35)(31,38)(33,40)(34,41);(4,5,6)(11,13,17,26)(12,18,21,34)(14,25,16,23)(15,29,31,40)(19,35,32,33)(20,36,37,42)(22,38,39,28)(24,27,41,30);(11,14,17,16)(12,19,21,32)(13,23,26,25)(15,27,31,30)(18,33,34,35)(20,28,37,38)(22,36,39,42)(24,40,41,29)

Conjugacy classes & character table

288 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 3201);