D₁₅⋊D₇ #88305

order	420
isoclinism	isoclinic class of D₁₅⋊D₇
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	D₁₅⋊D₇
unambiguous semidirect product	D₂₁⋊D₅
unambiguous semidirect product	D₃₅⋊D₃

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₇	C₂²
LCS	G ▷ C₃×C₅×C₇
UCS	C₁

40 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 105

A minimal faithful permutation representation — 2 generators on 15 points:

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

Polycyclic presentation:

2^2,3,5,7|c^a=c^2,e^a=e^6,c^b=c^2,d^b=d^4

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(420, 30);