C₃²×#2221 #11769270

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×#2221

Subgroups

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

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

normal subgroups only

This group has 444 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)(10,15)(11,12)(16,31)(17,28)(18,27)(19,29)(21,35)(22,25)(23,24)(26,32)(33,38)(34,37);(4,5,6)(7,8,15,28,12,18,24,29,13,19,23,27,11,17,10,14)(9,16,30,31,20,32,35,37,22,33,36,38,25,34,21,26);(7,9)(8,16)(10,21)(11,25)(12,20)(13,22)(14,26)(15,30)(17,34)(18,32)(19,33)(23,36)(24,35)(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, 3281);