D₃×#13091513 #72254070

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃³

Aliases

direct product

D₃×#13091513

Splittings

Fitting, Sylow	C₃×H(3,𝔽₃)×3₋¹⁺² ⋊ C₂²
derived	C₃³×C₉ ⋊ C₂²×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃³×C₉	C₂²×C₃²
Frattini	C₃²	C₃²×D₃×C₃⋊D₃
Fitting	C₃×H(3,𝔽₃)×3₋¹⁺²	C₂²
socle	C₃³	C₂×C₃²×C₃⋊D₃
LCS	G ▷ C₃³×C₉
UCS	C₁

4519 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

222 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72254071, D₃×(C₃⋊D₃)⋊C₃×AGL(1,ℤ/9)
maximal normal subgroup of	D₃×(C₃⋊D₃)⋊C₃×AGL(1,ℤ/9)
maximal quotient of	#72254071

GAP

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

gap> G := Group((1,2)(5,7)(6,9)(8,10)(11,12)(14,20,21,16,17,18)(15,19), (2,3)(5,8,12)(7,10,11)(13,17,16,15,21,18,19,14,20), (4,5,7)(6,8,11)(9,12,10));

D3×#13091513 #72254070