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Cayley diagram

C5×Q8×A4 #92053

order480
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×Q8×A4

Splittings

Fitting, HallC22×C5×Q8 ⋊ C3
SylowC22×Q8 ⋊ C3×C5

Subgroups

subgroupG/H
centerC2×C5C22×A4
commutator, cosocleC23C22×C3×C5
FrattiniC2C22×C5×A4
FittingC22×C5×Q8C3
socleC23×C5C22×C3
prime radicalC2×C5×A4C22
hypercenterC5×Q8A4
hypocenterC22C3×C5×Q8
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C2×C5 ◁ C5×Q8

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 120

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

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×#5416, C5×#5417, #11752135, …
maximal normal subgroup ofC5×#5416, C5×#5417, #11752135, …
maximal quotient ofC5×A4×C4⋊C4, C5×#557⋊C3, C5×Q8×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1129);