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G = C5×D19order 190 = 2·5·19

Direct product of C5 and D19

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C5×D19, C19⋊C10, C952C2, SmallGroup(190,2)

Series: Derived Chief Lower central Upper central

C1C19 — C5×D19
C1C19C95 — C5×D19
C19 — C5×D19
C1C5

Generators and relations for C5×D19
 G = < a,b,c | a5=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >

19C2
19C10

Smallest permutation representation of C5×D19
On 95 points
Generators in S95
(1 80 61 50 26)(2 81 62 51 27)(3 82 63 52 28)(4 83 64 53 29)(5 84 65 54 30)(6 85 66 55 31)(7 86 67 56 32)(8 87 68 57 33)(9 88 69 39 34)(10 89 70 40 35)(11 90 71 41 36)(12 91 72 42 37)(13 92 73 43 38)(14 93 74 44 20)(15 94 75 45 21)(16 95 76 46 22)(17 77 58 47 23)(18 78 59 48 24)(19 79 60 49 25)
(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)
(1 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)(32 38)(33 37)(34 36)(39 41)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(58 63)(59 62)(60 61)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(77 82)(78 81)(79 80)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)

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

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

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

55 conjugacy classes

class 1  2 5A5B5C5D10A10B10C10D19A···19I95A···95AJ
order1255551010101019···1995···95
size1191111191919192···22···2

55 irreducible representations

dim111122
type+++
imageC1C2C5C10D19C5×D19
kernelC5×D19C95D19C19C5C1
# reps1144936

Matrix representation of C5×D19 in GL2(𝔽191) generated by

1840
0184
,
01
190132
,
01
10
G:=sub<GL(2,GF(191))| [184,0,0,184],[0,190,1,132],[0,1,1,0] >;

C5×D19 in GAP, Magma, Sage, TeX

C_5\times D_{19}
% in TeX

G:=Group("C5xD19");
// GroupNames label

G:=SmallGroup(190,2);
// by ID

G=gap.SmallGroup(190,2);
# by ID

G:=PCGroup([3,-2,-5,-19,1622]);
// Polycyclic

G:=Group<a,b,c|a^5=b^19=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×D19 in TeX

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