C₃²×#1820 #11768869

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂⁴×C₃

Aliases

direct product

C₃²×#1820

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₃²×C₄
Frattini, cosocle	C₂⁴	C₂³×C₃²
prime radical	C₂⁴×C₃²	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂³×C₃²×C₄ ◁ G

900 conjugacy classes of subgroups; only the 312 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 312 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) = 288

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

Conjugacy classes & character table

288 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2880);