Finite Group Databasebeta

← Prev (#72226515) Next (#72226517) →

C5×#10615126 #72226516

order3200
automorphism?
homologyAbelianization (H1) = C24×C5 · Schur multiplier (H2) = C26

Aliases

direct productC5×#10615126

Splittings

FittingC52×#535 ⋊ C2
SylowC52 ⋊ #3021

Subgroups

subgroupG/H
centerC2×C5C2×#4083
commutator, cosocleC2×C4×C5C24×C5
FrattiniC2×C4C23×C5×D5
FittingC52×#535C2
socleC2×C52C2×#120
hypercenterC5×#535D5
hypocenterC5C5×#3021
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#535

1806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2,3,4,5)(6,7,9,10,8)(13,20)(16,25)(21,22)(24,26);(12,17)(14,23)(16,24)(18,19)(21,22)(25,26);(11,13)(12,16)(14,22)(15,20)(17,26)(18,25)(19,24)(21,23);(7,8)(9,10)(11,12,15,18)(13,16,20,25)(14,17,23,19)(21,24,22,26)

Conjugacy classes & character table

365 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of#72226522
maximal normal subgroup of#72226522

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,3,4,5)(6,7,9,10,8)(13,20)(16,25)(21,22)(24,26), (12,17)(14,23)(16,24)(18,19)(21,22)(25,26), (11,13)(12,16)(14,22)(15,20)(17,26)(18,25)(19,24)(21,23), (7,8)(9,10)(11,12,15,18)(13,16,20,25)(14,17,23,19)(21,24,22,26));