C₃²×#2194 #11769243

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×#2194

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	#447
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃²
prime radical	C₃²×#576	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×#131 ◁ 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)(7,8,13,19)(9,20,23,36)(10,24,28,35)(11,18,12,17)(14,30,33,21)(15,29,34,22)(16,26,31,27)(25,37,38,32);(4,5,6)(7,9)(8,14)(10,25)(11,22)(12,29)(13,23)(15,31)(16,34)(17,32)(18,37)(19,33)(20,24)(21,27)(26,30)(28,38)(35,36);(7,9)(8,15)(10,21)(11,29)(12,22)(13,23)(14,31)(16,33)(17,36)(18,20)(19,34)(24,37)(25,27)(26,38)(28,30)(32,35)

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, 3254);