C₃²×#1660 #11768709

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×#1660

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

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

normal subgroups only

This group has 588 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) = 30
transitive degree	μ_t(G) = 576

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

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

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, 2720);