Finite Group Databasebeta

← Prev (#66260) Next (#66262) →

Cayley diagram

C2×Q8×AGL(1,𝔽5) #66261

order320
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC2×Q8×AGL(1,𝔽5)
unambiguous semidirect productD5⋊(C4×Q8)

Splittings

FittingC2×C5×Q8 ⋊ C4
SylowC5 ⋊ C2×C4×Q8

Subgroups

subgroupG/H
centerC22C22×AGL(1,𝔽5)
commutatorC2×C5C23×C4
FrattiniC2C23×AGL(1,𝔽5)
FittingC2×C5×Q8C4
socleC22×C5C22×C4
prime radicalC22×D5C23
cosocleC2×D5C24
hypercenterC2×Q8AGL(1,𝔽5)
hypocenterC5C2×C4×Q8
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×Q8

298 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 4 generators on 15 points:

(4,5,6,7);(1,2)(3,4,5,7,6)(8,10,11,15)(9,12,13,14);(8,9,11,13)(10,14,15,12);(1,2)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#101, AGL(1,𝔽5)×#103, #10603984, …
maximal normal subgroup ofAGL(1,𝔽5)×#101, AGL(1,𝔽5)×#103, #10603984, …
maximal quotient ofC2×#65703, C2×#65709, C2×#65707, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1599);