C₃²×#2192 #11769241

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

C₃²×#2192

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	#447
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×#131 ◁ G

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

normal subgroups only

This group has 660 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)(9,20)(10,24)(11,18)(12,17)(13,19)(14,21)(15,22)(16,27)(23,35)(25,36)(26,30)(28,34)(29,33)(31,38)(32,37);(4,5,6)(7,9)(8,14)(10,25)(11,22)(12,29)(13,23)(15,30)(16,33)(17,31)(18,36)(19,32)(20,24)(21,27)(26,37)(28,38)(34,35);(7,9)(8,15)(10,21)(11,29)(12,22)(13,23)(14,30)(16,32)(17,35)(18,20)(19,33)(24,36)(25,27)(26,38)(28,37)(31,34)

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, 3252);