Finite Group Databasebeta

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C5×AGL(1,𝔽5)×H(5,𝔽2) #72275706

order3200
automorphism?
homologyAbelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C29

Aliases

direct productC5×AGL(1,𝔽5)×H(5,𝔽2)

Splittings

FittingC52×H(5,𝔽2) ⋊ C4
SylowC52 ⋊ C4×H(5,𝔽2)

Subgroups

subgroupG/H
centerC2×C5C24×AGL(1,𝔽5)
commutatorC2×C5C24×C4×C5
FrattiniC2C24×C5×AGL(1,𝔽5)
FittingC52×H(5,𝔽2)C4
socleC2×C52C24×C4
prime radicalC5×D5×H(5,𝔽2)C2
cosocleC2×D5C25×C5
hypercenterC5×H(5,𝔽2)AGL(1,𝔽5)
hypocenterC5C4×C5×H(5,𝔽2)
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C5 ◁ C5×H(5,𝔽2)

3403 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 160

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC5×AGL(1,𝔽5)×B(4,𝔽2), C5×AGL(1,𝔽5)×#453, H(5,𝔽2)×AGL(1,ℤ/25)
maximal normal subgroup ofC5×AGL(1,𝔽5)×B(4,𝔽2), C5×AGL(1,𝔽5)×#453
maximal quotient ofH(5,𝔽2)×AGL(1,ℤ/25)

GAP

Past the SmallGroups library coverage — no SmallGroups id.