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Cayley diagram

Dic5×SL(2,𝔽3) #91190

order480
isoclinismisoclinic class of C5⋊(#41⋊C3)
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productDic5×SL(2,𝔽3)
unambiguous semidirect product(Q8×Dic5)⋊C3

Splittings

derivedC5×Q8 ⋊ C3×C4
HallQ8×Dic5 ⋊ C3
SylowC5 ⋊ C4×SL(2,𝔽3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC22D5×A4
commutator, hypocenterC5×Q8C3×C4
Fitting, cosocleC2×C5×Q8C2×C3
socleC22×C5C2×A4
prime radicalC2×C5×SL(2,𝔽3)C2
LCSG ▷ C5×Q8
UCSC1 ◁ C22

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 160

A minimal faithful permutation representation — 2 generators on 17 points:

(1,2,3,4)(6,7,9)(10,12,11)(14,15)(16,17);(5,9,8,11)(6,12,10,7)(13,14,16,17,15)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofSL(2,𝔽3)⋊#170, SL(2,𝔽3)×#170, Dic5×#226, …
maximal normal subgroup ofSL(2,𝔽3)⋊#170, SL(2,𝔽3)×#170, Dic5×#226, …
maximal quotient ofSL(2,𝔽3)×#168, C2×Dic5×SL(2,𝔽3), Dic5×Q8⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 266);