C₃²×#2161 #11769210

order	1152
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#2161

Subgroups

	subgroup	G/H
center	C₃²×C₄	C₂×D₈
commutator	C₈	C₂²×C₃²×C₄
Frattini, cosocle	C₂×C₈	C₂³×C₃²
socle	C₂×C₃²	C₂×#107
prime radical	C₂³×C₃²	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃²×C₄ ◁ C₂²×C₃²×C₄ ◁ C₂²×C₃²×C₈ ◁ G

420 conjugacy classes of subgroups; only the 300 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 300 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)(7,8,11,16,13,17,12,27)(9,18,22,34,20,32,23,37)(10,24,25,31,26,38,19,15)(14,28,30,36,29,35,33,21);(4,5,6)(7,9,19,35,12,23,26,36,13,20,25,28,11,22,10,21)(8,14,24,34,16,30,31,32,17,29,38,37,27,33,15,18);(9,20)(14,29)(18,32)(21,36)(22,23)(28,35)(30,33)(34,37)

Conjugacy classes & character table

288 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 3221);