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C2×C4⋊(C7⋊AGL(1,𝔽5)) #11765359

order1120
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×C4⋊(C7⋊AGL(1,𝔽5))
unambiguous semidirect productD5⋊(C4⋊Dic7)

Splittings

FittingC2×C4×C5×C7 ⋊ C4
HallC5×C7 ⋊ C2×C4⋊C4
SylowC7 ⋊ C2×C4⋊AGL(1,𝔽5)
SylowC5 ⋊ C2×C4⋊Dic7

Subgroups

subgroupG/H
centerC22C2×C7⋊AGL(1,𝔽5)
commutatorC2×C5×C7C22×C4
FrattiniC2C22×C7⋊AGL(1,𝔽5)
FittingC2×C4×C5×C7C4
socleC22×C5×C7C2×C4
prime radicalC22×C7×D5C22
cosocleC2×C7×D5C23
hypercenterC2×C4C7⋊AGL(1,𝔽5)
hypocenterC5×C7C2×C4⋊C4
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C22 ◁ C2×C4

184 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 81 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) = 280

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C4⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5)), C2×D4×C7⋊AGL(1,𝔽5), C2×AGL(1,𝔽5)×D28, …
maximal normal subgroup ofC2×C4⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5)), C2×D4×C7⋊AGL(1,𝔽5), C2×AGL(1,𝔽5)×D28, …
maximal quotient of#72218126, #72218431

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1120, 965);