C₂×D₃×((C₃⋊D₃)⋊C₃)⋊D₃ #72228693

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₃

Aliases

Fitting	C₂×C₃²×H(3,𝔽₃) ⋊ C₂³
derived, Sylow	C₃²×H(3,𝔽₃) ⋊ C₂⁴

	subgroup	G/H
center, hypercenter	C₂	D₃×((C₃⋊D₃)⋊C₃)⋊D₃
commutator, cosocle, hypocenter	C₃²×H(3,𝔽₃)	C₂⁴
Frattini	C₃	C₂×D₃²×C₃⋊D₃
Fitting	C₂×C₃²×H(3,𝔽₃)	C₂³
socle	C₂×C₃³	C₂×D₃²
LCS	G ▷ C₃²×H(3,𝔽₃)
UCS	C₁ ◁ C₂

4130 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 4 generators on 17 points:

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

maximal subgroup of	C₂×D₃×C₃⋊AGL(2,𝔽₃), C₂×#72214399
maximal quotient of	C₂×#72214399

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

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