D₇×C₃₁ #89130

order	434
isoclinism	isoclinic class of D₇
automorphism	Aut(G) = C₂×C₃×C₅×AGL(1,𝔽₇) (order 1260) · Inn(G) = D₇ · Out(G) = C₂×C₃²×C₅
homology	Abelianization (H₁) = C₂×C₃₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₇×C₃₁
unambiguous semidirect product	C₇⋊(C₂×C₃₁)

Fitting, socle, Hall	C₇×C₃₁ ⋊ C₂
derived, Sylow	C₇ ⋊ C₂×C₃₁

	subgroup	G/H
center, hypercenter	C₃₁	D₇
commutator, cosocle, hypocenter	C₇	C₂×C₃₁
Frattini	C₁	G
Fitting, socle	C₇×C₃₁	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₃₁

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 217

A minimal faithful permutation representation — 2 generators on 38 points:

(2,3)(4,5)(6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38);(1,2,4,6,7,5,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(434, 1);