C₃²×#1692 #11768741

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₃²×#1692

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₃²	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ G

438 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 204 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) = 30
transitive degree	μ_t(G) = 1152

A minimal faithful permutation representation — 3 generators on 30 points:

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

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, 2752);