AGL(1,ℤ/9)×#6900 #72243195

order	13122
automorphism	?
homology	Abelianization (H₁) = C₂×C₃⁴ · Schur multiplier (H₂) = C₃⁴

Aliases

direct product

AGL(1,ℤ/9)×#6900

Splittings

Fitting, Sylow

3₋¹⁺²×#6900 ⋊ C₂

Subgroups

	subgroup	G/H
center	C₃²	C₃³×AGL(1,ℤ/9)
commutator, cosocle	C₃²×C₉	C₂×C₃⁴
Frattini, socle	C₃³	C₃⁴×D₃
Fitting	3₋¹⁺²×#6900	C₂
prime radical	C₃³×AGL(1,ℤ/9)	C₃²
hypercenter	#6900	AGL(1,ℤ/9)
hypocenter	C₉	C₂×C₃×#6900
LCS	G ▷ C₃²×C₉ ▷ C₉
UCS	C₁ ◁ C₃² ◁ #6900

7474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 486

A minimal faithful permutation representation — 4 generators on 27 points:

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

Conjugacy classes & character table

510 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	AGL(1,ℤ/9)×#92314, AGL(1,ℤ/9)×#92265
maximal normal subgroup of	AGL(1,ℤ/9)×#92314, AGL(1,ℤ/9)×#92265

GAP

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

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