Finite Group Databasebeta

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Cayley diagram

C2×#4005 #65760

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

Aliases

direct productC2×#4005

Splittings

SylowC5 ⋊ C2×#101

Subgroups

subgroupG/H
centerC22C22⋊AGL(1,𝔽5)
commutatorC22×C5C22×C4
FrattiniC22C22×AGL(1,𝔽5)
FittingC22×C4×C5C4
socleC22×C5C22⋊C4
prime radicalC23×C5C2×C4
cosocleC2×Dic5C23
hypercenterC22×C4AGL(1,𝔽5)
hypocenterC5C2×#101
LCSG ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4

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:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10598694, #10598724, C2×#64856, …
maximal normal subgroup of#10598694, #10598724, C2×#64856, …
maximal quotient of#10604161, #10604164, C4×#4005, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1098);