A₄×C₃⋊AGL(1,𝔽₁₃) #72242033

order	5616
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

Fitting, socle, derived	C₂²×C₃×C₁₃ ⋊ C₃²×C₄
Sylow	C₁₃ ⋊ C₃×Dic₃×A₄

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₂²×C₃×C₁₃	C₃²×C₄
prime radical	C₃×A₄×D₁₃⋊C₃	C₂
cosocle	C₂²×C₃×D₁₃	C₂×C₃²
LCS	G ▷ C₂²×C₃×C₁₃
UCS	C₁

272 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.

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

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 156

A minimal faithful permutation representation — 2 generators on 20 points:

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

maximal subgroup of	D₃×A₄×AGL(1,𝔽₁₃), S₄×C₃⋊AGL(1,𝔽₁₃), #72242121
maximal normal subgroup of	D₃×A₄×AGL(1,𝔽₁₃), S₄×C₃⋊AGL(1,𝔽₁₃), #72242121

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

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