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Cayley diagram

D61⋊C3 #67252

order366
isoclinismisoclinic class of D61⋊C3
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productD61⋊C3
unambiguous semidirect product(C61⋊C3)⋊C2

Splittings

Fitting, socle, derived, SylowC61 ⋊ C2×C3
HallC61⋊C3 ⋊ C2
HallD61 ⋊ C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC61C2×C3
LCSG ▷ C61
UCSC1

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 61
transitive degreeμt(G) = 61

A minimal faithful permutation representation — 2 generators on 61 points:

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

Polycyclic presentation:

2,3,61|c^a=c^60,c^b=c^47

Conjugacy classes & character table

16 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#6923⋊C3, C2×D61⋊C3, C3×D61⋊C3, …
maximal normal subgroup of#6923⋊C3, C2×D61⋊C3, C3×D61⋊C3, …
maximal quotient ofDic61⋊C3, C2×D61⋊C3, D61⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(366, 1);