Copied to
clipboard

G = D5×C19order 190 = 2·5·19

Direct product of C19 and D5

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

Aliases: D5×C19, C5⋊C38, C953C2, SmallGroup(190,1)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C19
C1C5C95 — D5×C19
C5 — D5×C19
C1C19

Generators and relations for D5×C19
 G = < a,b,c | a19=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C38

Smallest permutation representation of D5×C19
On 95 points
Generators in S95
(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 70 45 95 36)(2 71 46 77 37)(3 72 47 78 38)(4 73 48 79 20)(5 74 49 80 21)(6 75 50 81 22)(7 76 51 82 23)(8 58 52 83 24)(9 59 53 84 25)(10 60 54 85 26)(11 61 55 86 27)(12 62 56 87 28)(13 63 57 88 29)(14 64 39 89 30)(15 65 40 90 31)(16 66 41 91 32)(17 67 42 92 33)(18 68 43 93 34)(19 69 44 94 35)
(1 36)(2 37)(3 38)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(58 83)(59 84)(60 85)(61 86)(62 87)(63 88)(64 89)(65 90)(66 91)(67 92)(68 93)(69 94)(70 95)(71 77)(72 78)(73 79)(74 80)(75 81)(76 82)

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

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

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

D5×C19 is a maximal subgroup of   C19⋊F5

76 conjugacy classes

class 1  2 5A5B19A···19R38A···38R95A···95AJ
order125519···1938···3895···95
size15221···15···52···2

76 irreducible representations

dim111122
type+++
imageC1C2C19C38D5D5×C19
kernelD5×C19C95D5C5C19C1
# reps111818236

Matrix representation of D5×C19 in GL2(𝔽191) generated by

1210
0121
,
881
1900
,
01
10
G:=sub<GL(2,GF(191))| [121,0,0,121],[88,190,1,0],[0,1,1,0] >;

D5×C19 in GAP, Magma, Sage, TeX

D_5\times C_{19}
% in TeX

G:=Group("D5xC19");
// GroupNames label

G:=SmallGroup(190,1);
// by ID

G=gap.SmallGroup(190,1);
# by ID

G:=PCGroup([3,-2,-19,-5,1370]);
// Polycyclic

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

Export

Subgroup lattice of D5×C19 in TeX

׿
×
𝔽