C₃²×#1664 #11768713

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#1664

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₃²
prime radical	C₂×C₃²×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ G

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

normal subgroups only

This group has 552 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) = 576

A minimal faithful permutation representation — 3 generators on 38 points:

(1,2,3)(7,8,12,16)(9,17,19,22)(10,15,11,14)(13,20,21,18)(23,24,28,26)(25,31,34,38)(27,35,37,30)(29,36,32,33);(4,5,6)(7,9)(8,13)(10,20)(11,18)(12,19)(14,22)(15,17)(16,21)(23,25,28,34)(24,29,26,32)(27,36,37,33)(30,38,35,31);(7,9)(8,13)(10,18)(11,20)(12,19)(14,17)(15,22)(16,21)(23,26,28,24)(25,32,34,29)(27,30,37,35)(31,36,38,33)

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2724);