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G = C5×C19⋊C3order 285 = 3·5·19

Direct product of C5 and C19⋊C3

direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C5×C19⋊C3, C95⋊C3, C19⋊C15, SmallGroup(285,1)

Series: Derived Chief Lower central Upper central

C1C19 — C5×C19⋊C3
C1C19C95 — C5×C19⋊C3
C19 — C5×C19⋊C3
C1C5

Generators and relations for C5×C19⋊C3
 G = < a,b,c | a5=b19=c3=1, ab=ba, ac=ca, cbc-1=b11 >

19C3
19C15

Smallest permutation representation of C5×C19⋊C3
On 95 points
Generators in S95
(1 77 58 39 20)(2 78 59 40 21)(3 79 60 41 22)(4 80 61 42 23)(5 81 62 43 24)(6 82 63 44 25)(7 83 64 45 26)(8 84 65 46 27)(9 85 66 47 28)(10 86 67 48 29)(11 87 68 49 30)(12 88 69 50 31)(13 89 70 51 32)(14 90 71 52 33)(15 91 72 53 34)(16 92 73 54 35)(17 93 74 55 36)(18 94 75 56 37)(19 95 76 57 38)
(1 2 3 4 5 6 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 37 38)(39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57)(58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76)(77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95)
(2 8 12)(3 15 4)(5 10 7)(6 17 18)(9 19 13)(11 14 16)(21 27 31)(22 34 23)(24 29 26)(25 36 37)(28 38 32)(30 33 35)(40 46 50)(41 53 42)(43 48 45)(44 55 56)(47 57 51)(49 52 54)(59 65 69)(60 72 61)(62 67 64)(63 74 75)(66 76 70)(68 71 73)(78 84 88)(79 91 80)(81 86 83)(82 93 94)(85 95 89)(87 90 92)

G:=sub<Sym(95)| (1,77,58,39,20)(2,78,59,40,21)(3,79,60,41,22)(4,80,61,42,23)(5,81,62,43,24)(6,82,63,44,25)(7,83,64,45,26)(8,84,65,46,27)(9,85,66,47,28)(10,86,67,48,29)(11,87,68,49,30)(12,88,69,50,31)(13,89,70,51,32)(14,90,71,52,33)(15,91,72,53,34)(16,92,73,54,35)(17,93,74,55,36)(18,94,75,56,37)(19,95,76,57,38), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57)(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76)(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)(40,46,50)(41,53,42)(43,48,45)(44,55,56)(47,57,51)(49,52,54)(59,65,69)(60,72,61)(62,67,64)(63,74,75)(66,76,70)(68,71,73)(78,84,88)(79,91,80)(81,86,83)(82,93,94)(85,95,89)(87,90,92)>;

G:=Group( (1,77,58,39,20)(2,78,59,40,21)(3,79,60,41,22)(4,80,61,42,23)(5,81,62,43,24)(6,82,63,44,25)(7,83,64,45,26)(8,84,65,46,27)(9,85,66,47,28)(10,86,67,48,29)(11,87,68,49,30)(12,88,69,50,31)(13,89,70,51,32)(14,90,71,52,33)(15,91,72,53,34)(16,92,73,54,35)(17,93,74,55,36)(18,94,75,56,37)(19,95,76,57,38), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57)(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76)(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)(40,46,50)(41,53,42)(43,48,45)(44,55,56)(47,57,51)(49,52,54)(59,65,69)(60,72,61)(62,67,64)(63,74,75)(66,76,70)(68,71,73)(78,84,88)(79,91,80)(81,86,83)(82,93,94)(85,95,89)(87,90,92) );

G=PermutationGroup([(1,77,58,39,20),(2,78,59,40,21),(3,79,60,41,22),(4,80,61,42,23),(5,81,62,43,24),(6,82,63,44,25),(7,83,64,45,26),(8,84,65,46,27),(9,85,66,47,28),(10,86,67,48,29),(11,87,68,49,30),(12,88,69,50,31),(13,89,70,51,32),(14,90,71,52,33),(15,91,72,53,34),(16,92,73,54,35),(17,93,74,55,36),(18,94,75,56,37),(19,95,76,57,38)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19),(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38),(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57),(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76),(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95)], [(2,8,12),(3,15,4),(5,10,7),(6,17,18),(9,19,13),(11,14,16),(21,27,31),(22,34,23),(24,29,26),(25,36,37),(28,38,32),(30,33,35),(40,46,50),(41,53,42),(43,48,45),(44,55,56),(47,57,51),(49,52,54),(59,65,69),(60,72,61),(62,67,64),(63,74,75),(66,76,70),(68,71,73),(78,84,88),(79,91,80),(81,86,83),(82,93,94),(85,95,89),(87,90,92)])

45 conjugacy classes

class 1 3A3B5A5B5C5D15A···15H19A···19F95A···95X
order133555515···1519···1995···95
size11919111119···193···33···3

45 irreducible representations

dim111133
type+
imageC1C3C5C15C19⋊C3C5×C19⋊C3
kernelC5×C19⋊C3C95C19⋊C3C19C5C1
# reps1248624

Matrix representation of C5×C19⋊C3 in GL3(𝔽11) generated by

400
040
004
,
039
087
325
,
138
065
094
G:=sub<GL(3,GF(11))| [4,0,0,0,4,0,0,0,4],[0,0,3,3,8,2,9,7,5],[1,0,0,3,6,9,8,5,4] >;

C5×C19⋊C3 in GAP, Magma, Sage, TeX

C_5\times C_{19}\rtimes C_3
% in TeX

G:=Group("C5xC19:C3");
// GroupNames label

G:=SmallGroup(285,1);
// by ID

G=gap.SmallGroup(285,1);
# by ID

G:=PCGroup([3,-3,-5,-19,947]);
// Polycyclic

G:=Group<a,b,c|a^5=b^19=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^11>;
// generators/relations

Export

Subgroup lattice of C5×C19⋊C3 in TeX

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