C₃²×#2231 #11769280

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2230
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₃²×#2231

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	#476
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃²
prime radical	C₃²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×C₄×C₈ ◁ G

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

normal subgroups only

This group has 360 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)(7,8,13,18)(9,19,22,17)(10,23)(11,27,20,16)(12,14,29,31)(15,32)(21,35)(24,37)(25,36)(26,30)(28,33)(34,38);(4,5,6)(7,9,20,29,13,22,11,12)(8,14,27,17,18,31,16,19)(10,21,25,38,26,32,24,28)(15,30,34,36,35,23,33,37);(7,10)(8,15)(9,21)(11,24)(12,28)(13,26)(14,30)(16,33)(17,36)(18,35)(19,37)(20,25)(22,32)(23,31)(27,34)(29,38)

Conjugacy classes & character table

207 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 3291);