Finite Group Databasebeta

← Prev (#67361) Next (#67363) →

Cayley diagram

D9×ΓL(1,𝔽8) #67362

order378
isoclinismisoclinic class of D9×ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productD9×ΓL(1,𝔽8)

Splittings

Fitting, derivedC7×C9 ⋊ C2×C3
HallC9×ΓL(1,𝔽8) ⋊ C2
SylowC7 ⋊ C3×D9

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC7×C9C2×C3
FrattiniC3D3×ΓL(1,𝔽8)
socleC3×C7C3×D3
LCSG ▷ C7×C9
UCSC1

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 63

A minimal faithful permutation representation — 2 generators on 16 points:

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD9×AGL(1,𝔽7), C2×D9×ΓL(1,𝔽8), ΓL(1,𝔽8)×D27, …
maximal normal subgroup ofD9×AGL(1,𝔽7), C2×D9×ΓL(1,𝔽8), C3×D9×ΓL(1,𝔽8), …
maximal quotient ofΓL(1,𝔽8)×Dic9, C2×D9×ΓL(1,𝔽8), ΓL(1,𝔽8)×D27, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 15);