Finite Group Databasebeta

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C5×#10614894 #72226583

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

Aliases

direct productC5×#10614894

Splittings

FittingC2×C52×#127 ⋊ C2
SylowC52 ⋊ #3151

Subgroups

subgroupG/H
centerC22×C5C22×C22⋊D5
commutator, cosocleC2×C4×C5C24×C5
FrattiniC2×C4C23×C5×D5
FittingC2×C52×#127C2
socleC22×C52C22×D4
hypercenterC2×C5×#127D5
hypocenterC5C5×#3151
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C5×#127

1810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 160

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of#72226589
maximal normal subgroup of#72226589

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