3₋¹⁺²×#92314 #72243162

order	13122
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃³

Aliases

direct product

3₋¹⁺²×#92314

Splittings

Fitting, Sylow

3₋¹⁺²×#6900 ⋊ C₂

Subgroups

	subgroup	G/H
center	C₃	C₃²×#92314
commutator, cosocle	C₃×C₉²	C₂×C₃³
Frattini, socle	C₃³	C₃³×C₃⋊D₃
Fitting	3₋¹⁺²×#6900	C₂
prime radical	C₃²×#92314	C₃
hypercenter	3₋¹⁺²	#92314
hypocenter	C₉²	C₂×C₃×3₋¹⁺²
LCS	G ▷ C₃×C₉² ▷ C₉²
UCS	C₁ ◁ C₃ ◁ 3₋¹⁺²

6364 conjugacy classes of subgroups; only the 343 normal subgroups are shown.

normal subgroups only

This group has 6364 conjugacy classes of subgroups — too many to draw. Only its 343 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 27 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	AGL(1,ℤ/9)×#92314, 3₋¹⁺²×#92265⋊C₂, #72243381
maximal normal subgroup of	AGL(1,ℤ/9)×#92314, 3₋¹⁺²×#92265⋊C₂, #72243381

GAP

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

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