D₅×C₃₇ #67303

order	370
isoclinism	isoclinic class of D₅
automorphism	Aut(G) = C₄×C₉×AGL(1,𝔽₅) (order 720) · Inn(G) = D₅ · Out(G) = C₂×C₄×C₉
homology	Abelianization (H₁) = C₂×C₃₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₅×C₃₇
unambiguous semidirect product	C₅⋊(C₂×C₃₇)

Fitting, socle, Hall	C₅×C₃₇ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₃₇

	subgroup	G/H
center, hypercenter	C₃₇	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₃₇
Frattini	C₁	G
Fitting, socle	C₅×C₃₇	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₃₇

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 185

A minimal faithful permutation representation — 2 generators on 42 points:

(2,3)(4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42);(1,2,4,5,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(370, 1);