C₃×#60399⋊C₃ #72275104

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂⁴×C₃

Aliases

direct product

C₃×#60399⋊C₃

Splittings

Fitting	C₃×#60399 ⋊ C₃
Sylow	#60399 ⋊ C₃²

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₂²×#120⋊C₃
commutator, cosocle, hypocenter	H(5,𝔽₂)	C₂³×C₃²
Frattini	C₂³	C₂³×C₃×A₄
Fitting	C₃×#60399	C₃
hypercenter	C₃×D₄	#120⋊C₃
LCS	G ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂×C₃ ◁ C₃×D₄

2152 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 2152 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 19
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#72217027, D₃×#60399⋊C₃, #72275114
maximal normal subgroup of	C₃×#72217027, D₃×#60399⋊C₃, #72275114

GAP

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

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