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Cayley diagram

C2×C61⋊C3 #67253

order366
isoclinismisoclinic class of C61⋊C3
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC2×C61⋊C3
unambiguous semidirect product(C2×C61)⋊C3

Splittings

Fitting, socle, HallC2×C61 ⋊ C3
derived, SylowC61 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC2C61⋊C3
commutator, cosocle, hypocenterC61C2×C3
FrattiniC1G
Fitting, socleC2×C61C3
LCSG ▷ C61
UCSC1 ◁ C2

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 63
transitive degreeμt(G) = 122

A minimal faithful permutation representation — 2 generators on 63 points:

(1,2)(4,5,7)(6,9,14)(8,12,11)(10,16,24)(13,19,21)(15,22,18)(17,26,36)(20,28,33)(23,31,30)(25,34,27)(29,39,46)(32,41,43)(35,44,40)(37,47,38)(42,50,54)(45,52,51)(48,55,49)(53,58,59)(56,60,57)(61,63,62);(3,4,6,10,17,26,37,48,56,61,60,44,22,7,11,18,27,38,39,16,25,35,45,53,55,34,14,21,30,40,49,50,28,9,15,23,32,42,47,24,33,43,51,57,58,41,19,5,8,13,20,29,36,46,54,59,62,63,52,31,12)

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic61⋊C3, C4×C61⋊C3, C2×D61⋊C3, …
maximal normal subgroup ofDic61⋊C3, C4×C61⋊C3, C2×D61⋊C3, …
maximal quotient ofC4×C61⋊C3, C22×C61⋊C3, C2×C61⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(366, 2);