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C22×C5×ΓL(1,𝔽8)⋊C4 #13132563

order1680
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C22×C3×C4×C5 · Schur multiplier (H2) = C23

Aliases

direct productC22×C5×ΓL(1,𝔽8)⋊C4

Splittings

derived, SylowC7 ⋊ C22×C3×C4×C5
HallC22×C5×Dic7 ⋊ C3
HallC22×Dic7 ⋊ C3×C5
HallC5×ΓL(1,𝔽8) ⋊ C22×C4
HallC5×C7 ⋊ C22×C3×C4
HallΓL(1,𝔽8) ⋊ C22×C4×C5

Subgroups

subgroupG/H
center, hypercenterC23×C5AGL(1,𝔽7)
commutator, hypocenterC7C22×C3×C4×C5
FrattiniC2C22×C5×AGL(1,𝔽7)
Fitting, socleC23×C5×C7C2×C3
prime radicalC23×C5×ΓL(1,𝔽8)C2
cosocleC2×C7C23×C3×C5
LCSG ▷ C7
UCSC1 ◁ C23×C5

216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 560

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 583);