Finite Group Databasebeta

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C5×#10602444 #72226409

order3200
automorphism?
homologyAbelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24

Aliases

direct productC5×#10602444

Splittings

FittingC52×#518 ⋊ C2
SylowC52 ⋊ #1904

Subgroups

subgroupG/H
centerC2×C5#65502
commutatorC2×C4×C5C22×C4×C5
FrattiniC22×C4C22×C5×D5
FittingC52×#518C2
socleC2×C52#386
prime radicalC2×C5×#4078C2
cosocleC22×C4×C5C23×C5
hypercenterC5×#518D5
hypocenterC5C5×#1904
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#518

1030 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of#72226413
maximal normal subgroup of#72226413

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