Finite Group Databasebeta

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C2×#13104918 #72226391

order3200
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4×C5

Aliases

direct productC2×#13104918

Splittings

FittingC22×C52×D4 ⋊ C4
SylowC52 ⋊ C2×#390

Subgroups

subgroupG/H
centerC23C22×#1044
commutatorC22×C52C23×C4
FrattiniC22C23×#1044
FittingC22×C52×D4C4
socleC23×C52C22×C4
prime radicalC22×D4×C5⋊D5C2
cosocleC22×C5⋊D5C24
hypercenterC22×D4#1044
hypocenterC52C2×#390
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C23 ◁ C22×D4

4082 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 160

A minimal faithful permutation representation — 4 generators on 20 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72226392, #72226600

GAP

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

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