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Cayley diagram

C2×D31 #1245

order124
isoclinismisoclinic class of D31
automorphismAut(G) = C2×AGL(1,𝔽31) (order 1860) · Inn(G) = D31 · Out(G) = C2×C3×C5
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD62
direct productC2×D31
unambiguous semidirect productC31⋊C22
unambiguous semidirect product(C2×C31)⋊C2

Splittings

Fitting, socleC2×C31 ⋊ C2
derived, SylowC31 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2D31
commutator, cosocle, hypocenterC31C22
FrattiniC1G
Fitting, socleC2×C31C2
LCSG ▷ C31
UCSC1 ◁ C2

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 33
transitive degreeμt(G) = 62

A minimal faithful permutation representation — 2 generators on 33 points:

(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33);(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×D31, D124, C22⋊D31, …
maximal normal subgroup ofC4×D31, D124, C22⋊D31, …
maximal quotient ofDic62, C4×D31, D124, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(124, 3);