Finite Group Databasebeta

← Prev (#91979) Next (#91981) →

Cayley diagram

C3×Q8×AGL(1,𝔽5) #91980

order480
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC3×Q8×AGL(1,𝔽5)

Splittings

FittingC3×C5×Q8 ⋊ C4
HallC3×C5 ⋊ C4×Q8
SylowC5 ⋊ C3×C4×Q8

Subgroups

subgroupG/H
centerC2×C3C22×AGL(1,𝔽5)
commutatorC2×C5C22×C3×C4
FrattiniC2C22×C3×AGL(1,𝔽5)
FittingC3×C5×Q8C4
socleC2×C3×C5C22×C4
prime radicalC2×C3×D5C23
cosocleC2×D5C23×C3
hypercenterC3×Q8AGL(1,𝔽5)
hypocenterC5C3×C4×Q8
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C3×Q8

140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 120

A minimal faithful permutation representation — 3 generators on 16 points:

(2,5,4,3)(14,15,16);(1,2,3,5,4)(6,13,9,8)(7,12,11,10);(6,11,9,7)(8,10,13,12)

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#215, #11751964, AGL(1,𝔽5)×#216, …
maximal normal subgroup ofAGL(1,𝔽5)×#215, #11751964, AGL(1,𝔽5)×#216, …
maximal quotient ofC3×#65703, C3×#65707, C3×#65709, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1056);