C₃²×#2241 #11769290

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₃×C₄

Aliases

direct product

C₃²×#2241

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₃²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×C₄×C₈ ◁ G

420 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 216 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)(7,8)(9,12)(10,11)(13,14)(15,16,22,17)(18,19,28,30)(20,27,23,24)(21,29,26,25);(4,5,6)(8,11)(12,14)(15,17,20,28,21,29,19,27,22,16,23,18,26,25,30,24);(7,9,10,13)(8,12,11,14)(15,18,19,17,26,27,20,25,22,28,30,16,21,24,23,29)

Conjugacy classes & character table

342 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 3301);