Finite Group Databasebeta

← Prev (#66738) Next (#66740) →

Cayley diagram

SO(3,𝔽7) #66739

order336
isoclinismisoclinic class of SO(3,𝔽7)
automorphismAut(G) = SO(3,𝔽7) (order 336) · Inn(G) = G · Out(G) = C1
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C2

Aliases

projective general linearPGL(2,𝔽7)
special orthogonalSO(3,𝔽7)
unambiguous semidirect productGL(3,𝔽2)⋊C2

Splittings

socle, derivedGL(3,𝔽2) ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, socle, cosocle, hypocenterGL(3,𝔽2)C2
LCSG ▷ GL(3,𝔽2)
UCSC1

23 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 8
transitive degreeμt(G) = 8

A minimal faithful permutation representation — 2 generators on 8 points:

(1,4,6,8,5,2,7,3);(1,3,8,6,5,4,7)

Conjugacy classes & character table

C1C2C3C4C5C6C7C8C9
size12128424242485656
order122488736
χ1111111111
χ211-11-1-111-1
χ362002−2-100
χ46200−22-100
χ56-20200-100
χ67-11-1-1-1011
χ77-1-1-11101-1
χ88020001-1-1
χ980-20001-11

Relationships

maximal subgroup ofC2×SO(3,𝔽7), C3⋊SO(3,𝔽7), C3×SO(3,𝔽7), …
maximal normal subgroup ofC2×SO(3,𝔽7), C3×SO(3,𝔽7), C5×SO(3,𝔽7)
maximal quotient ofGL(3,𝔽2)⋊C4, #10621462, #10621463, …
automorphism group ofGL(3,𝔽2), SL(2,𝔽7), SO(3,𝔽7), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 208);