C₃²×#1293 #11768342

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄² · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#1293

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	#95
commutator	C₂×C₄	C₃²×C₄²
Frattini, cosocle	C₂³×C₄	C₂²×C₃²
prime radical	C₂⁴×C₃²	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂³×C₃²×C₄ ◁ G

552 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 204 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)(9,13,14,11)(10,12)(15,16,18,23,20,24,19,22)(17,25,26,21,28,29,27,30);(4,5,6)(7,9)(8,11)(10,14)(12,13)(15,17,19,27,20,28,18,26)(16,21,23,29,24,30,22,25)

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, 2353);