C₃²×#2191 #11769240

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2190
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

C₃²×#2191

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	#447
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃²
prime radical	C₃²×#575	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ C₂×C₃²×M₁₆ ◁ G

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

normal subgroups only

This group has 648 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) = 38
transitive degree	μ_t(G) = 288

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

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

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 3251);