C₇⋊#168 #63298

order	280
isoclinism	isoclinic class of D₃₅
automorphism	?
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₃₅
commutator, hypocenter	C₅×C₇	C₈
Fitting, cosocle	C₄×C₅×C₇	C₂
socle, prime radical	C₂×C₅×C₇	C₄
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₄

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 280

A minimal faithful permutation representation — 2 generators on 20 points:

(2,3)(4,5)(6,7)(9,10)(11,12)(13,14,15,17,16,18,19,20);(1,2,4,6,7,5,3)(8,9,11,12,10)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(280, 3);