C₃²×#92302 #72244474

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃³

Aliases

direct product

C₃²×#92302

Splittings

Fitting, Sylow	C₃²×#6891 ⋊ C₂
derived	C₃²⋊C₉ ⋊ C₂×C₃³

Subgroups

	subgroup	G/H
center, hypercenter	C₃³	C₃⋊AGL(1,ℤ/9)
commutator, cosocle, hypocenter	C₃²⋊C₉	C₂×C₃³
Frattini	C₃²	C₃³×C₃⋊D₃
Fitting	C₃²×#6891	C₂
socle	C₃⁴	C₃×C₃⋊D₃
LCS	G ▷ C₃²⋊C₉
UCS	C₁ ◁ C₃³

15058 conjugacy classes of subgroups; only the 426 normal subgroups are shown.

normal subgroups only

This group has 15058 conjugacy classes of subgroups — too many to draw. Only its 426 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) = 24
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 3 generators on 24 points:

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

Conjugacy classes & character table

351 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of

C₃²×#13091120

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((7,8)(9,13,11,15,14,12)(17,19)(20,24)(21,23), (4,5,6)(17,20,21)(19,23,24), (1,2,3)(7,14,12,10,11,13,8,9,15)(16,17,19)(18,21,23)(20,24,22));