C₃²×#1722 #11768771

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂×C₃×C₄

Aliases

direct product

C₃²×#1722

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

594 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 240 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)(8,11)(9,12)(13,14)(15,16,19,22)(17,23,24,30)(18,26,27,21)(20,28,29,25);(4,5,6)(15,17,18,25,19,24,27,28)(16,20,21,23,22,29,26,30);(7,8,12,14,10,13,9,11)(17,24)(20,29)(23,30)(25,28)

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, 2782);