C₃²×SO(3,𝔽₇) #72275484

order	3024
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

socle	C₃²×GL(3,𝔽₂) ⋊ C₂
derived	GL(3,𝔽₂) ⋊ C₂×C₃²

	subgroup	G/H
center, Fitting, hypercenter	C₃²	SO(3,𝔽₇)
commutator, cosocle, hypocenter	GL(3,𝔽₂)	C₂×C₃²
Frattini	C₁	G
socle	C₃²×GL(3,𝔽₂)	C₂
LCS	G ▷ GL(3,𝔽₂)
UCS	C₁ ◁ C₃²

194 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 2 generators on 14 points:

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

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

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