C₃×#11758087 #72243713

order	2916
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₃×#11758087

Splittings

Fitting, Sylow	C₃×H(5,𝔽₃) ⋊ C₂²
derived	H(5,𝔽₃) ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, socle, hypercenter	C₃²	C₃⋊D₃²
commutator, cosocle, hypocenter	H(5,𝔽₃)	C₂²×C₃
Frattini	C₃	C₃×C₃⋊D₃²
Fitting	C₃×H(5,𝔽₃)	C₂²
LCS	G ▷ H(5,𝔽₃)
UCS	C₁ ◁ C₃²

2300 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 2300 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 30
transitive degree	μ_t(G) = 81

A minimal faithful permutation representation — 3 generators on 30 points:

(1,2,3)(5,16)(6,24)(8,10)(9,18)(11,17)(12,29)(13,14)(19,22)(20,28)(21,27)(23,30)(25,26);(5,8)(6,23,26,12,14,22)(9,20,21)(10,17)(11,16)(13,30,25,19,24,29)(18,27,28);(4,12,19)(5,6,18)(7,23,29)(8,9,13)(10,26,27)(11,14,28)(15,22,30)(16,20,25)(17,21,24)

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243728, #72243729, C₃×#72214502, …
maximal normal subgroup of	#72243728, #72243729, C₃×#72214502, …

GAP

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

gap> G := Group((1,2,3)(5,16)(6,24)(8,10)(9,18)(11,17)(12,29)(13,14)(19,22)(20,28)(21,27)(23,30)(25,26), (5,8)(6,23,26,12,14,22)(9,20,21)(10,17)(11,16)(13,30,25,19,24,29)(18,27,28), (4,12,19)(5,6,18)(7,23,29)(8,9,13)(10,26,27)(11,14,28)(15,22,30)(16,20,25)(17,21,24));