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Cayley diagram

D5×SL(2,𝔽3) #6748

order240
isoclinismisoclinic class of C5⋊(#41⋊C3)
automorphismAut(G) = C2×AGL(1,𝔽5)×S4 (order 960) · Inn(G) = D5×A4 · Out(G) = C23
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productD5×SL(2,𝔽3)
unambiguous semidirect productC5⋊(C2×SL(2,𝔽3))
unambiguous semidirect product(Q8×D5)⋊C3

Splittings

Fitting, derivedC5×Q8 ⋊ C2×C3
HallQ8×D5 ⋊ C3
SylowC5 ⋊ C2×SL(2,𝔽3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D5×A4
commutator, Fitting, cosocle, hypocenterC5×Q8C2×C3
socleC2×C5C2×A4
LCSG ▷ C5×Q8
UCSC1 ◁ C2

36 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)(10,11)(12,13);(1,5,4,7)(2,8,6,3)(9,10,12,13,11)

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#91886, AGL(1,𝔽5)×SL(2,𝔽3), D5×#226, …
maximal normal subgroup of#91886, AGL(1,𝔽5)×SL(2,𝔽3), D5×#226, …
maximal quotient ofDic5×SL(2,𝔽3), C2×D5×SL(2,𝔽3), D5×Q8⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 109);