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Cayley diagram

C24×AGL(1,𝔽5) #66300

order320
isoclinismisoclinic class of AGL(1,𝔽5)
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C210

Aliases

direct productC24×AGL(1,𝔽5)
unambiguous semidirect productD5⋊(C23×C4)

Splittings

Fitting, socleC24×C5 ⋊ C4
derived, SylowC5 ⋊ C24×C4

Subgroups

subgroupG/H
center, hypercenterC24AGL(1,𝔽5)
commutator, hypocenterC5C24×C4
FrattiniC1G
Fitting, socleC24×C5C4
prime radicalC24×D5C2
cosocleD5C25
LCSG ▷ C5
UCSC1 ◁ C24

1362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 5 generators on 13 points:

(10,11,12,13);(7,8)(9,10,11,13,12);(5,6);(3,4);(1,2)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10604252, C2×C22⋊C4×AGL(1,𝔽5), C2×#65699, …
maximal normal subgroup of#10604252, C2×C22⋊C4×AGL(1,𝔽5), C2×#65699, …
maximal quotient ofC23×D5⋊C8, C23×#700, C22×D5⋊M16, …
automorphism group of#683, #685, C3×C8×D5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1638);