C₃²×#1593 #11768642

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

Aliases

direct product

C₃²×#1593

Subgroups

	subgroup	G/H
center	C₂×C₃²×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₃²×C₄
Frattini, cosocle	C₄²	C₂³×C₃²
socle, prime radical	C₂²×C₃²	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₃²×C₄×C₈ ◁ G

444 conjugacy classes of subgroups; only the 252 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 252 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) = 1152

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

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

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, 2653);