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C3×C22⋊(C7⋊AGL(1,𝔽5)) #13132616

order1680
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC3×C22⋊(C7⋊AGL(1,𝔽5))

Splittings

FittingC22×C3×C5×C7 ⋊ C4
HallC3×C5×C7 ⋊ C22⋊C4
HallC5×C7 ⋊ C3×C22⋊C4
HallC3×C7 ⋊ C22⋊AGL(1,𝔽5)
HallC3×C5 ⋊ C22⋊Dic7
SylowC7 ⋊ C3×C22⋊AGL(1,𝔽5)
SylowC5 ⋊ C3×C22⋊Dic7

Subgroups

subgroupG/H
centerC2×C3C2×C7⋊AGL(1,𝔽5)
commutatorC2×C5×C7C2×C3×C4
FrattiniC2C2×C3×C7⋊AGL(1,𝔽5)
FittingC22×C3×C5×C7C4
socleC2×C3×C5×C7C2×C4
prime radicalC22×C3×C7×D5C2
cosocleC2×C7×D5C22×C3
hypercenterC22×C3C7⋊AGL(1,𝔽5)
hypocenterC5×C7C3×C22⋊C4
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2×C3 ◁ C22×C3

136 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 420

Conjugacy classes & character table

186 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 636);