(C₃⋊D₃)⋊C₃×C₃⋊AGL(1,ℤ/9) #72243980

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

Aliases

direct product

(C₃⋊D₃)⋊C₃×C₃⋊AGL(1,ℤ/9)

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	C₃²×D₃²
LCS	G ▷ C₃³×C₉
UCS	C₁

4156 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:

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

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243996, D₃×(C₃⋊D₃)⋊C₃×AGL(1,ℤ/9)
maximal normal subgroup of	D₃×(C₃⋊D₃)⋊C₃×AGL(1,ℤ/9)
maximal quotient of	#72243996

GAP

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

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

(C3⋊D3)⋊C3×C3⋊AGL(1,ℤ/9) #72243980