Finite Group Databasebeta

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Cayley diagram

C2×#3939 #65251

order320
isoclinismisoclinic class of #686
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×#3939

Splittings

SylowC5 ⋊ C2×#107

Subgroups

subgroupG/H
centerC23C22⋊D5
commutatorC4×C5C22×C4
FrattiniC2×C4C22×D5
FittingC2×C5×C4⋊C4C2
socle, prime radicalC23×C5D4
cosocleC2×C4×C5C23
hypercenterC2×C4⋊C4D5
hypocenterC5C2×#107
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊C4

130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 320

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10599578, #10599579, #10599580, …
maximal normal subgroup of#10599578, #10599579, #10599580, …
maximal quotient of#10600259, C4×#3939, #10600297, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 589);