C₃²×#2180 #11769229

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2180
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#2180

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	#442
commutator	C₈	C₂²×C₃²×C₄
Frattini, cosocle	C₂×C₈	C₂³×C₃²
prime radical	C₃²×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×#97 ◁ G

426 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 216 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)(9,19,21,30)(10,23)(11,15)(12,27)(13,18)(14,20,31,37)(16,24)(17,29)(22,34,28,36)(25,35,38,32)(26,33);(4,5,6)(7,9)(8,14)(10,20)(11,25)(12,28)(13,21)(15,30)(16,34)(17,35)(18,31)(19,33)(22,29)(23,36)(24,37)(26,38)(27,32);(7,10,12,11,13,24,29,26)(8,15,17,23,18,33,27,16)(9,20,28,25,21,37,22,38)(14,30,35,36,31,19,32,34)

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, 3240);