AGL(1,ℤ/9)×C₃⋊((C₃⋊D₃)⋊C₃) #72254068

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

Aliases

direct product

AGL(1,ℤ/9)×C₃⋊((C₃⋊D₃)⋊C₃)

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

4433 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)(5,7)(6,9)(8,10)(11,12)(13,18,15,20,16,14)(19,21);(1,2,3)(5,8,12)(7,10,11)(13,15)(14,19,20,21,18,17);(4,5,7)(6,8,11)(9,12,10)

Conjugacy classes & character table

210 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((2,3)(5,7)(6,9)(8,10)(11,12)(13,18,15,20,16,14)(19,21), (1,2,3)(5,8,12)(7,10,11)(13,15)(14,19,20,21,18,17), (4,5,7)(6,8,11)(9,12,10));

AGL(1,ℤ/9)×C3⋊((C3⋊D3)⋊C3) #72254068