C₃²×#2255 #11769304

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₃²×#2255

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

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

normal subgroups only

This group has 360 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)(7,8,10,12)(9,13,14,11)(16,20)(17,22)(18,24)(21,26)(23,30)(25,29)(27,28);(4,5,6)(7,9,10,14)(8,11,12,13)(16,21)(20,26)(23,27)(28,30);(15,16,22,28,18,23,25,26,19,21,29,30,24,27,17,20)

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, 3315);