AGL(1,ℤ/9)×#546 #72268818

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

AGL(1,ℤ/9)×#546

Splittings

Fitting	3₋¹⁺²×#546 ⋊ C₂
Sylow	3₋¹⁺² ⋊ C₂×#546

Subgroups

	subgroup	G/H
center	C₂²	C₂⁴×AGL(1,ℤ/9)
commutator, cosocle	C₂²×C₉	C₂⁵×C₃
Frattini, socle	C₂²×C₃	C₂⁴×C₃×D₃
Fitting	3₋¹⁺²×#546	C₂
hypercenter	#546	AGL(1,ℤ/9)
hypocenter	C₉	C₂×C₃×#546
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ #546

5867 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 5 generators on 21 points:

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

Conjugacy classes & character table

250 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/9)×#3484
maximal normal subgroup of	AGL(1,ℤ/9)×#3484

GAP

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

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