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AGL(1,𝔽5)×B(4,𝔽2) #13036206

order1280
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productAGL(1,𝔽5)×B(4,𝔽2)

Splittings

FittingC5×B(4,𝔽2) ⋊ C4
SylowC5 ⋊ C4×B(4,𝔽2)

Subgroups

subgroupG/H
centerC2AGL(1,𝔽5)×#120
commutatorC23×C5C23×C4
FrattiniC23C23×AGL(1,𝔽5)
FittingC5×B(4,𝔽2)C4
socleC2×C5C4×#120
prime radicalD5×B(4,𝔽2)C2
cosocleC23×D5C24
hypercenterB(4,𝔽2)AGL(1,𝔽5)
hypocenterC5C4×B(4,𝔽2)
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C23 ◁ B(4,𝔽2)

1910 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 1910 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 40

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72213689, AGL(1,𝔽5)×#1020⋊C2, AGL(1,𝔽5)×#5934, …
maximal normal subgroup ofAGL(1,𝔽5)×#3025, C2×AGL(1,𝔽5)×B(4,𝔽2), AGL(1,𝔽5)×#3027, …
maximal quotient of#72213689, C2×AGL(1,𝔽5)×B(4,𝔽2), C3×AGL(1,𝔽5)×B(4,𝔽2), …
automorphism group ofC2×Dic10, C2×D20, #3966, …
holomorph ofC2×C4×C5

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1106474);