C₅⋊SO(3,𝔽₇) #13132904

order	1680
isoclinism	isoclinic class of C₅⋊SO(3,𝔽₇)
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	C₅⋊SO(3,𝔽₇)
unambiguous semidirect product	GL(3,𝔽₂)⋊D₅

Fitting, Sylow	C₅ ⋊ SO(3,𝔽₇)
socle, derived	C₅×GL(3,𝔽₂) ⋊ C₂

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, socle, cosocle, hypocenter	C₅×GL(3,𝔽₂)	C₂
Fitting	C₅	SO(3,𝔽₇)
LCS	G ▷ C₅×GL(3,𝔽₂)
UCS	C₁

46 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 40

A minimal faithful permutation representation — 2 generators on 13 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 924);