C₃²×#2222 #11769271

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#2222

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	#466
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃²
prime radical	C₃²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×#131 ◁ G

420 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 180 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) = 70
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 70 points:

(1,2,3)(7,8,14,20)(9,21,27,49)(10,28,31,45)(11,32,34,43)(12,19,13,18)(15,36,40,24)(16,35,42,30)(17,33,44,22)(23,52,55,69)(25,48,26,47)(29,60,61,41)(37,64,65,70)(38,56,39,50)(46,57,68,54)(51,59,62,67)(53,58,63,66);(4,5,6)(7,9,22,50,13,26,34,36,14,27,33,56,12,25,11,24)(8,15,32,48,19,39,44,49,20,40,43,47,18,38,17,21)(10,23,51,70,30,54,61,63,31,55,62,64,35,57,29,53)(16,37,59,52,28,58,60,68,42,65,67,69,45,66,41,46);(7,10)(8,16)(9,23)(11,29)(12,35)(13,30)(14,31)(15,37)(17,41)(18,45)(19,28)(20,42)(21,46)(22,51)(24,53)(25,57)(26,54)(27,55)(32,59)(33,62)(34,61)(36,63)(38,66)(39,58)(40,65)(43,67)(44,60)(47,69)(48,52)(49,68)(50,70)(56,64)

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, 3282);