C₃×#72253805 #72269386

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

Aliases

direct product

C₃×#72253805

Splittings

Fitting, Sylow	C₃²×#6891 ⋊ C₂²
derived	C₃×C₃²⋊C₉ ⋊ C₂²×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#72253805
commutator, cosocle, hypocenter	C₃×C₃²⋊C₉	C₂²×C₃²
Frattini	C₃²	C₃²×D₃×C₃⋊D₃
Fitting	C₃²×#6891	C₂²
socle	C₃⁴	C₃×D₃²
LCS	G ▷ C₃×C₃²⋊C₉
UCS	C₁ ◁ C₃

9008 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 9008 conjugacy classes of subgroups — too many to draw. Only its 176 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 — 3 generators on 24 points:

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

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