C₃²×#1353 #11768402

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

Aliases

direct product

C₃²×#1353

Subgroups

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

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

normal subgroups only

This group has 348 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)(8,11,12,13)(15,16,18,22,19,21,25,28)(17,20,24,26,23,27,30,29);(4,5,6)(7,8)(9,13)(10,12)(11,14)(15,17)(16,20,21,27)(18,24)(19,23)(22,29,28,26)(25,30)

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, 2413);