Finite Group Databasebeta

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Cayley diagram

C2×#3956 #65425

order320
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×#3956

Splittings

SylowC5 ⋊ C2×#101

Subgroups

subgroupG/H
centerC22#685
commutatorC22×C5C22×C4
FrattiniC2×C4C22×D5
FittingC2×C5×M16C2
socleC22×C5C22⋊C4
prime radicalC23×C5C2×C4
cosocleC2×C4×C5C23
hypercenterC2×M16D5
hypocenterC5C2×#101
LCSG ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 160

A minimal faithful permutation representation — 3 generators on 23 points:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10599501, #10599502, #10599516, …
maximal normal subgroup of#10599501, #10599502, #10599516, …
maximal quotient ofC2×#64685, #10600594, #10600655, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 763);