Finite Group Databasebeta

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Cayley diagram

C2×#6122 #88243

order416
isoclinismisoclinic class of #6118
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22

Aliases

direct productC2×#6122

Splittings

SylowC13 ⋊ C2×M16

Subgroups

subgroupG/H
centerC22C2×#261
commutatorC2×C13C22×C4
FrattiniC2C22×#261
Fitting, prime radicalC23×C13C4
socleC22×C13C2×C4
cosocleDic13C23
hypercenterC23#261
hypocenterC13C2×M16
LCSG ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 208

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

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11718239, #11718291, #11719062, …
maximal normal subgroup of#11718239, #11718291, #11719062, …
maximal quotient ofC2×#88110, C4×#6122, C2×#88112, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 210);