C₃²×#2233 #11769282

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#2233

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃²
prime radical	C₃²×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₃²×C₄ ◁ C₂²×C₃²×C₈ ◁ G

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

normal subgroups only

This group has 450 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)(15,16)(17,20)(18,21)(19,27)(22,26)(23,25)(24,28)(29,30);(4,5,6)(8,11)(12,14)(16,20)(21,28)(22,29)(23,27);(7,9,10,13)(8,12,11,14)(16,21)(18,26)(19,25)(20,28)(22,23)(24,30)(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, 3293);