C₃²×#1364 #11768413

order	1152
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₃²×#1364

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₃²	#102
commutator	C₂×C₈	C₂×C₃²×C₄
Frattini, cosocle	C₄×C₈	C₂²×C₃²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×C₄×C₈ ◁ G

258 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 144 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 — 2 generators on 30 points:

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

Conjugacy classes & character table

261 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2424);