D₃×AΓL(1,𝔽₁₆) #72241595

order	5760
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

Fitting, socle	C₂⁴×C₃ ⋊ C₂×ΓL(1,𝔽₁₆)
derived	C₃×AGL(1,𝔽₁₆) ⋊ C₂×C₄

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	C₃×AGL(1,𝔽₁₆)	C₂×C₄
Fitting, socle	C₂⁴×C₃	C₂×ΓL(1,𝔽₁₆)
prime radical	D₃×((C₂⁴⋊C₅)⋊C₂)⋊C₃	C₂
cosocle	C₃×((C₂⁴⋊C₅)⋊C₂)⋊C₃	C₂²
LCS	G ▷ C₃×AGL(1,𝔽₁₆)
UCS	C₁

305 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 19 points:

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

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

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