C₃²×#1680 #11768729

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₃²×#1680

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

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

normal subgroups only

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

Conjugacy classes & character table

261 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2740);