C₃²×#2223 #11769272

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×#2223

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	#466
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃²
prime radical	C₃²×#472	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×#131 ◁ G

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

normal subgroups only

This group has 432 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) = 38
transitive degree	μ_t(G) = 288

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

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

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, 3283);