C₃²×#1344 #11768393

order	1152
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₄ · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#1344

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	#99
commutator	C₂²×C₄	C₂×C₃²×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂²×C₃²
prime radical	C₂⁴×C₃²	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂³×C₃² ◁ C₂×C₃²×C₂²⋊C₄ ◁ G

438 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 120 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 — 2 generators on 30 points:

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

Conjugacy classes & character table

207 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2404);