C₂×3₋¹⁺²×#453 #72244668

order	3456
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	C₃²×#120
commutator, Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃²
prime radical	C₂×C₃²×#453	C₃
LCS	G ▷ C₂×C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₄×3₋¹⁺² ◁ G

3376 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);(5,10)(7,9)(11,12,15,16,17,19,13,14,18);(4,7)(5,10)(8,9);(1,2)

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