C₃²×#2244 #11769293

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₃×C₄

Aliases

direct product

C₃²×#2244

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃²
prime radical	C₃²×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×C₄×C₈ ◁ G

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

normal subgroups only

This group has 456 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) = 576

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

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

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, 3304);