C₃²×#1291 #11768340

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₃²×#1291

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	#95
commutator	C₂×C₄	C₃²×C₄²
Frattini, cosocle	C₂×C₄²	C₂²×C₃²
prime radical	C₂³×C₃²	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂×C₃²×C₄² ◁ G

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

normal subgroups only

This group has 414 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) = 30
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 2 generators on 30 points:

(1,2,3)(7,8)(9,11)(10,13)(12,14)(15,16,18,23,20,22,19,24)(17,25,26,30,27,21,28,29);(4,5,6)(8,11,13,12)(15,17,19,26,20,27,18,28)(16,21,24,29,22,25,23,30)

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, 2351);