D₇×C₂₇ #67351

order	378
isoclinism	isoclinic class of D₇
automorphism	Aut(G) = C₂×C₉×AGL(1,𝔽₇) (order 756) · Inn(G) = D₇ · Out(G) = C₂×C₃×C₉
homology	Abelianization (H₁) = C₂×C₂₇ · Schur multiplier (H₂) = C₁

Aliases

D₇×C₂₇

Fitting, Hall	C₇×C₂₇ ⋊ C₂
derived, Sylow	C₇ ⋊ C₂×C₂₇

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 189

A minimal faithful permutation representation — 2 generators on 34 points:

(2,3)(4,5)(6,7)(8,9,12,10,13,18,15,20,25,11,14,19,16,21,26,23,28,31,17,22,27,24,29,32,30,33,34);(1,2,4,6,7,5,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 4);