C₂×3₋¹⁺²×B(4,𝔽₂) #72268931

order	3456
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁷

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	C₃²×#120
commutator, Frattini, cosocle	C₂³×C₃	C₂⁴×C₃²
prime radical	C₂×C₃²×B(4,𝔽₂)	C₃
LCS	G ▷ C₂³×C₃ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁴×3₋¹⁺² ◁ G

4072 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 4 generators on 19 points:

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

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

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

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