C₂⁴×C₅×ΓL(1,𝔽₈) #13132973

order	1680
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃×C₅ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center, hypercenter	C₂⁴×C₅	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂⁴×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂⁴×C₅×C₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂⁴×C₅

536 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 560

A minimal faithful permutation representation — 4 generators on 20 points:

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

400 conjugacy classes. The character table is too large to display.

maximal subgroup of	C₂⁴×D₅×ΓL(1,𝔽₈), C₂⁴×C₅×AGL(1,𝔽₇), C₂⁴×C₅⋊AGL(1,𝔽₇), …
maximal normal subgroup of	C₂⁴×D₅×ΓL(1,𝔽₈), C₂⁴×C₅×AGL(1,𝔽₇), C₂⁴×C₅⋊AGL(1,𝔽₇), …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 993);