3₋¹⁺²×#1712 #72245261

order	3456
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₈

Aliases

direct product

3₋¹⁺²×#1712

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₃²×#127
commutator, Frattini, cosocle	C₃×C₄²	C₂³×C₃²
prime radical	C₃²×#1712	C₃
LCS	G ▷ C₃×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₄²×3₋¹⁺² ◁ G

1120 conjugacy classes of subgroups; only the 336 normal subgroups are shown.

normal subgroups only

This group has 1120 conjugacy classes of subgroups — too many to draw. Only its 336 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) = 25
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 25 points:

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	AGL(1,ℤ/9)×#1712, #72268948
maximal subgroup of	AGL(1,ℤ/9)×#1712, 3₋¹⁺²×#19745, #72268948
maximal normal subgroup of	AGL(1,ℤ/9)×#1712, 3₋¹⁺²×#19745, #72268948

GAP

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

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