C₃×#72224516 #72269399

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃²

Aliases

direct product

C₃×#72224516

Splittings

Fitting, Sylow	C₃²×C₃²⋊H(3,𝔽₃) ⋊ C₂²
derived	C₃×C₃²⋊H(3,𝔽₃) ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃²	((C₃⋊D₃)⋊C₃)⋊(C₃⋊D₃)
commutator, cosocle, hypocenter	C₃×C₃²⋊H(3,𝔽₃)	C₂²×C₃
Frattini	C₃²	C₃×D₃×C₃⋊(C₃⋊D₃)
Fitting	C₃²×C₃²⋊H(3,𝔽₃)	C₂²
socle	C₃⁴	D₃×C₃⋊D₃
LCS	G ▷ C₃×C₃²⋊H(3,𝔽₃)
UCS	C₁ ◁ C₃²

13391 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 13391 conjugacy classes of subgroups — too many to draw. Only its 220 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) = 108

A minimal faithful permutation representation — 4 generators on 24 points:

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

Conjugacy classes & character table

243 conjugacy classes — view the full character table.

GAP

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

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