Finite Group Databasebeta

← Prev (#92082) Next (#92084) →

Cayley diagram

C5×#1007 #92083

order480
automorphism?
homologyAbelianization (H1) = C24×C5 · Schur multiplier (H2) = C25

Aliases

direct productC5×#1007

Splittings

FittingC2×C3×C5×Q8 ⋊ C2
HallC3×C5 ⋊ 2−1+4
SylowC3 ⋊ C5×2−1+4

Subgroups

subgroupG/H
centerC2×C5C23×D3
commutator, cosocleC2×C3C24×C5
FrattiniC2C23×C5×D3
FittingC2×C3×C5×Q8C2
socleC2×C3×C5C24
hypercenterC2×C5×Q8D3
hypocenterC3C5×2−1+4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C2×C5×Q8

292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 240

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,3)(5,11)(9,17)(10,19)(16,18)(20,21,22,23,24);(4,7,8,15)(5,10,11,19)(6,12,13,14)(9,16,17,18);(4,6,8,13)(5,9,11,17)(7,14,15,12)(10,18,19,16);(2,3)(4,5)(6,9)(7,10)(8,11)(12,16)(13,17)(14,18)(15,19)

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11749386, #11749387, #11749393, …
maximal normal subgroup of#11749386, #11749387, #11749393, …
maximal quotient ofC5×#5500, C5×#5504, C5×#5507, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1159);