Finite Group Databasebeta

← Prev (#1204) Next (#1206) →

Cayley diagram

C5×SL(2,𝔽3) #1205

order120
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C4×S4 (order 96) · Inn(G) = A4 · Out(G) = C2×C4
holomorphG ⋊ Aut(G) = AGL(1,𝔽5)×SO+(4,𝔽3)
homologyAbelianization (H1) = C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×SL(2,𝔽3)
unambiguous semidirect productQ8⋊(C3×C5)
unambiguous semidirect product(C5×Q8)⋊C3

Splittings

Fitting, HallC5×Q8 ⋊ C3
derived, SylowQ8 ⋊ C3×C5

Subgroups

subgroupG/H
center, socle, hypercenterC2×C5A4
commutator, cosocle, hypocenterQ8C3×C5
FrattiniC2C5×A4
FittingC5×Q8C3
LCSG ▷ Q8
UCSC1 ◁ C2×C5

14 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 40

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC5⋊#226, C5⋊GL(2,𝔽3), #91184, …
maximal subgroup ofC5×#226, C5×GL(2,𝔽3), C5⋊#226, …
maximal normal subgroup ofC5×#226, C5×GL(2,𝔽3), C5⋊#226, …
maximal quotient ofC2×C5×SL(2,𝔽3), C5×Q8⋊C9, C3×C5×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(120, 15);