S₄×C₃⋊(D₁₃⋊C₃) #72242068

order	5616
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×A₄×C₁₃	C₂²×C₃
Fitting, socle	C₂²×C₃×C₁₃	C₂×C₃×D₃
LCS	G ▷ C₃×A₄×C₁₃
UCS	C₁

326 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 34 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:

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

maximal subgroup of	D₃×S₄×D₁₃⋊C₃
maximal normal subgroup of	D₃×S₄×D₁₃⋊C₃

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

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