C₃²×#1606 #11768655

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

C₃²×#1606

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃²
prime radical	C₃²×#538	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₃²×C₄ ◁ G

780 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 288

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

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

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, 2666);