C₃²×#1720 #11768769

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₃×C₈

Aliases

direct product

C₃²×#1720

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₃²
prime radical	C₃²×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ G

558 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 576

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

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

Conjugacy classes & character table

234 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2780);