C₃²×#1838 #11768887

order	1152
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂×C₃×C₄

Aliases

direct product

C₃²×#1838

Subgroups

	subgroup	G/H
center	C₂×C₃²×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₃²×C₄
Frattini, cosocle	C₂²×C₄	C₂³×C₃²
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃²	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₂×C₃²×C₄² ◁ G

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

normal subgroups only

This group has 540 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) = 576

A minimal faithful permutation representation — 3 generators on 30 points:

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

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, 2898);