C₃²×#2214 #11769263

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2212
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#2214

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	#459
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

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

normal subgroups only

This group has 450 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)(7,8,14,20)(9,18,23,29)(10,24,26,37)(11,19,13,17)(12,15,22,30)(16,31,33,21)(25,38,28,34)(27,35,36,32);(4,5,6)(7,9,13,22,14,23,11,12)(8,15,19,29,20,30,17,18)(10,21,28,36,26,31,25,27)(16,24,35,38,33,37,32,34);(7,10)(8,16)(9,21)(11,25)(12,27)(13,28)(14,26)(15,24)(17,32)(18,34)(19,35)(20,33)(22,36)(23,31)(29,38)(30,37)

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, 3274);