D₃×#13049813 #72224556

order	7776
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

C₃⁵ ⋊ C₂²×D₄

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle	C₃×C₃⋊(C₃⋊(C₃⋊D₃))	C₂⁴
Fitting, socle, hypocenter	C₃⁵	C₂²×D₄
LCS	G ▷ C₃×C₃⋊(C₃⋊(C₃⋊D₃)) ▷ C₃⁵
UCS	C₁

4560 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 4560 conjugacy classes of subgroups — too many to draw. Only its 114 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 — 4 generators on 15 points:

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

maximal subgroup of	D₃×#72210346, D₃×#72213247
maximal normal subgroup of	D₃×#72210346, D₃×#72213247

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

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