ΓL(1,𝔽₈)×#672 #13131982

order	1680
isoclinism	isoclinic class of D₅×ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₃×C₁₆ · Schur multiplier (H₂) = C₁

Aliases

direct product	ΓL(1,𝔽₈)×#672
unambiguous semidirect product	(C₇×#672)⋊C₃

	subgroup	G/H
center, Frattini, hypercenter	C₈	D₅×ΓL(1,𝔽₈)
commutator, hypocenter	C₅×C₇	C₃×C₁₆
Fitting, cosocle	C₅×C₇×C₈	C₂×C₃
socle	C₂×C₅×C₇	C₃×C₈
prime radical	C₂×C₅×ΓL(1,𝔽₈)	C₈
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₈

40 conjugacy classes of subgroups.

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 560

A minimal faithful permutation representation — 2 generators on 28 points:

(2,3,4)(5,7,6)(9,10)(11,12)(13,14,15,18,16,19,21,24,17,20,22,25,23,26,27,28);(1,2,3,5,4,6,7)(8,9,11,12,10)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 2);