D₃×#13049804 #72224524

order	7776
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₃

Aliases

C₃⁵ ⋊ C₂×SD₁₆

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle	C₃×#66472	C₂³
Fitting, socle, hypocenter	C₃⁵	C₂×SD₁₆
prime radical	D₃×#10620206	C₂
LCS	G ▷ C₃×#66472 ▷ C₃×C₃⋊(C₃⋊(C₃⋊D₃)) ▷ C₃⁵
UCS	C₁

1540 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 1540 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.

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

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 3 generators on 15 points:

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

maximal subgroup of	D₃×#72210299, D₃×#72210346
maximal normal subgroup of	D₃×#72210346

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

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