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C5×D67⋊C3 #14523054

order2010
isoclinismisoclinic class of D67⋊C3
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D67⋊C3
unambiguous semidirect productD67⋊(C3×C5)
unambiguous semidirect product(C67⋊C3)⋊(C2×C5)
unambiguous semidirect product(C5×D67)⋊C3

Splittings

Fitting, socle, HallC5×C67 ⋊ C2×C3
derived, SylowC67 ⋊ C2×C3×C5
HallC5×C67⋊C3 ⋊ C2
HallC5×D67 ⋊ C3
HallC67⋊C3 ⋊ C2×C5
HallD67 ⋊ C3×C5

Subgroups

subgroupG/H
center, hypercenterC5D67⋊C3
commutator, cosocle, hypocenterC67C2×C3×C5
FrattiniC1G
Fitting, socleC5×C67C2×C3
LCSG ▷ C67
UCSC1 ◁ C5

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 72
transitive degreeμt(G) = 335

A minimal faithful permutation representation — 2 generators on 72 points:

(1,2,3,4,5)(7,11,12,8,9,13)(10,19,20,14,15,29)(16,32,33,22,23,46)(17,31,27,25,21,18)(24,51,52,35,36,64)(26,49,50,38,34,43)(28,30,48,40,41,44)(37,60,59,45,54,70)(39,67,68,55,53,61)(42,47,66,56,57,62)(58,65,71,63,69,72);(6,7,10,16,24,37,41,15,21,29,43,59,71,69,36,53,60,62,64,68,33,48,49,32,47,63,56,40,27,12,11,17,26,39,55,38,25,9,13,18,28,42,58,57,23,34,44,46,61,52,66,54,67,51,65,72,70,50,20,31,19,30,45,35,22,14,8)

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×D67⋊C3, C2×C5×D67⋊C3, C5×C7×D67⋊C3, …
maximal normal subgroup ofD5×D67⋊C3, C2×C5×D67⋊C3, C5×C7×D67⋊C3, …
maximal quotient ofC5×(C67⋊C3)⋊C4, C2×C5×D67⋊C3, C5×C7×D67⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2010, 10);