C₇⋊#170 #63301

order	280
isoclinism	isoclinic class of C₇⋊AGL(1,𝔽₅)
automorphism	Aut(G) = C₂×AGL(1,𝔽₅)×AGL(1,𝔽₇) (order 1680) · Inn(G) = C₇⋊AGL(1,𝔽₅) · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₇⋊AGL(1,𝔽₅)
commutator, hypocenter	C₅×C₇	C₈
Fitting, socle, prime radical	C₂×C₅×C₇	C₄
cosocle	C₇×Dic₅	C₂
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₂

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)(8,9,12,10,11)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(280, 6);