C₃²×#1849 #11768898

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×#1849

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

636 conjugacy classes of subgroups; only the 408 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 408 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) = 26
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 26 points:

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

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, 2909);