C₂×#13143663 #72268926

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

Fitting	C₂×3₋¹⁺²×H(5,𝔽₂) ⋊ C₂
Sylow	3₋¹⁺² ⋊ C₂×B(4,𝔽₂)

3503 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:

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

maximal subgroup of	C₂×AGL(1,ℤ/9)×B(4,𝔽₂)
maximal normal subgroup of	C₂×AGL(1,ℤ/9)×B(4,𝔽₂)

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

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