C₃²×#1633 #11768682

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#1633

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₃²	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₃²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ G

618 conjugacy classes of subgroups; only the 228 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 228 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)(8,11)(9,13)(12,14)(15,16,19,22)(17,23,25,30)(18,26,28,21)(20,27,29,24);(4,5,6)(9,13)(11,14)(15,17,19,25)(16,20,22,29)(18,27,28,24)(21,30,26,23);(7,8,10,12)(9,14,13,11)

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, 2693);