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Cayley diagram

D67⋊C3 #87929

order402
isoclinismisoclinic class of D67⋊C3
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productD67⋊C3
unambiguous semidirect product(C67⋊C3)⋊C2

Splittings

Fitting, socle, derived, SylowC67 ⋊ C2×C3
HallC67⋊C3 ⋊ C2
HallD67 ⋊ C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC67C2×C3
LCSG ▷ C67
UCSC1

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 67
transitive degreeμt(G) = 67

A minimal faithful permutation representation — 2 generators on 67 points:

(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,59)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,55,54,40,49,65)(34,62,63,50,48,56)(37,42,61,51,52,57)(53,60,66,58,64,67);(1,2,5,11,19,32,36,10,16,24,38,54,66,64,31,48,55,57,59,63,28,43,44,27,42,58,51,35,22,7,6,12,21,34,50,33,20,4,8,13,23,37,53,52,18,29,39,41,56,47,61,49,62,46,60,67,65,45,15,26,14,25,40,30,17,9,3)

Polycyclic presentation:

2,3,67|c^a=c^66,c^b=c^37

Conjugacy classes & character table

17 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D67⋊C3, C3×D67⋊C3, C3⋊(D67⋊C3), …
maximal normal subgroup ofC2×D67⋊C3, C3×D67⋊C3, C5×D67⋊C3, …
maximal quotient of(C67⋊C3)⋊C4, C2×D67⋊C3, D67⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(402, 1);