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Cayley diagram

C2×C3×D31 #67319

order372
isoclinismisoclinic class of D31
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×C3×D31

Splittings

Fitting, socleC2×C3×C31 ⋊ C2
derived, SylowC31 ⋊ C22×C3
HallC3×C31 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2×C3D31
commutator, cosocle, hypocenterC31C22×C3
FrattiniC1G
Fitting, socleC2×C3×C31C2
LCSG ▷ C31
UCSC1 ◁ C2×C3

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 186

A minimal faithful permutation representation — 2 generators on 36 points:

(1,2)(3,4,5)(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)(33,34)(35,36);(1,2)(6,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,36,34,32,30,28,26,24,22,20,18,16,14,12,10,8)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic3×D31, #10625224, #10625225, …
maximal normal subgroup ofDic3×D31, #10625224, #10625225, …
maximal quotient ofC3×Dic62, C3×C4×D31, C3×D124, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(372, 12);