3₋¹⁺²×#3151 #72244684

order	3456
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

3₋¹⁺²×#3151

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×C₃²×D₄
commutator, Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃²
prime radical	C₃²×#3151	C₃
LCS	G ▷ C₂×C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₄²×3₋¹⁺² ◁ G

2624 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 4 generators on 21 points:

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	AGL(1,ℤ/9)×#3151, #72268954
maximal subgroup of	AGL(1,ℤ/9)×#3151, 3₋¹⁺²×#19745, #72268954
maximal normal subgroup of	AGL(1,ℤ/9)×#3151, 3₋¹⁺²×#19745, #72268954
maximal quotient of	3₋¹⁺²×#19745

GAP

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

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

3−1+2×#3151 #72244684