D₃×#10624749 #72244332

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃⁵

Aliases

direct product

D₃×#10624749

Splittings

Fitting, Sylow

C₃×#10624749 ⋊ C₂

Subgroups

	subgroup	G/H
center	C₃	D₃×C₃²⋊H(3,𝔽₃)
commutator, cosocle	C₃⁴	C₂×C₃³
Frattini	C₃³	C₃³×D₃
Fitting	C₃×#10624749	C₂
socle	C₃²	C₂×C₃²⋊H(3,𝔽₃)
hypercenter	#10624749	D₃
hypocenter	C₃	C₂×#10624749
LCS	G ▷ C₃⁴ ▷ C₃² ▷ C₃
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ #10624749

1969 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1969 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 54

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

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

Conjugacy classes & character table

171 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃⋊D₃×#10624749, S₄×#10624749
maximal quotient of	C₃⋊D₃×#10624749, S₄×#10624749

GAP

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

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