C₃×#13091643 #72277539

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃⁴

Aliases

direct product

C₃×#13091643

Splittings

Fitting, Sylow

C₃×#10624930 ⋊ C₂

Subgroups

	subgroup	G/H
center, socle	C₃²	C₃⋊D₃×H(3,𝔽₃)
commutator, cosocle	C₃×H(3,𝔽₃)	C₂×C₃³
Frattini	C₃²	C₃³×C₃⋊D₃
Fitting	C₃×#10624930	C₂
hypercenter	C₃×AΣL(1,𝔽₂₇)	C₃⋊D₃
hypocenter	H(3,𝔽₃)	C₂×C₃×H(3,𝔽₃)
LCS	G ▷ C₃×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×AΣL(1,𝔽₂₇)

4624 conjugacy classes of subgroups; only the 228 normal subgroups are shown.

normal subgroups only

This group has 4624 conjugacy classes of subgroups — too many to draw. Only its 228 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)(5,9,10)(6,12)(8,16,17)(11,27,20,21,19,28)(13,23)(14,22)(18,29,25,26,24,30);(5,9,10)(6,13,14)(11,20,19)(12,22,23)(18,25,24)(26,29,30);(4,20,30,7,19,26,15,11,29)(5,24,12,10,25,22,9,18,23)(6,28,17,14,27,8,13,21,16)

Conjugacy classes & character table

234 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72277547, C₃×#72243788
maximal normal subgroup of	C₃×#72243788

GAP

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

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