C₃²×#3292 #11770341

order	1152
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁵×C₃

Aliases

direct product

C₃²×#3292

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ G

1428 conjugacy classes of subgroups; only the 552 normal subgroups are shown.

normal subgroups only

This group has 1428 conjugacy classes of subgroups — too many to draw. Only its 552 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) = 22
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 4 generators on 22 points:

(1,2,3)(7,8)(9,20)(10,16)(11,18)(12,17)(13,19)(14,21)(15,22);(4,5,6)(7,9,13,15)(8,14,19,10)(11,16,12,21)(17,22,18,20);(7,10)(8,15)(9,19)(11,22)(12,20)(13,14)(16,17)(18,21);(7,10)(8,16)(9,18)(11,20)(12,22)(13,14)(15,17)(19,21)

Conjugacy classes & character table

261 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 4352);