C₃²×#2166 #11769215

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₃²×#2166

Subgroups

	subgroup	G/H
center	C₃²×C₄	#117
commutator	C₂×C₄	C₂²×C₃²×C₄
Frattini, cosocle	C₂×C₈	C₂³×C₃²
socle	C₂×C₃²	#432
prime radical	C₂³×C₃²	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃²×C₄ ◁ C₃²×#97 ◁ G

366 conjugacy classes of subgroups; only the 228 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 228 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) = 38
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 38 points:

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

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, 3226);