C₇⋊#272 #67390

order	378
isoclinism	isoclinic class of C₇⋊#272
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₃	C₃⋊D₂₁
commutator, Fitting, cosocle, hypocenter	C₇×H(3,𝔽₃)	C₂
socle	C₃×C₇	C₃⋊D₃
LCS	G ▷ C₇×H(3,𝔽₃)
UCS	C₁ ◁ C₃

44 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 63

A minimal faithful permutation representation — 3 generators on 16 points:

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

Polycyclic presentation:

2,3^2,7,3|b^a=b^2,c^a=c^2,d^a=d^6,c^b=ce

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 43);