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## G = D5×D19order 380 = 22·5·19

### Direct product of D5 and D19

Aliases: D5×D19, D95⋊C2, C51D38, C95⋊C22, C191D10, (D5×C19)⋊C2, (C5×D19)⋊C2, SmallGroup(380,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C95 — D5×D19
 Chief series C1 — C19 — C95 — C5×D19 — D5×D19
 Lower central C95 — D5×D19
 Upper central C1

Generators and relations for D5×D19
G = < a,b,c,d | a5=b2=c19=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

5C2
19C2
95C2
95C22
19C10
19D5
5C38
5D19
19D10
5D38

Smallest permutation representation of D5×D19
On 95 points
Generators in S95
(1 77 70 45 21)(2 78 71 46 22)(3 79 72 47 23)(4 80 73 48 24)(5 81 74 49 25)(6 82 75 50 26)(7 83 76 51 27)(8 84 58 52 28)(9 85 59 53 29)(10 86 60 54 30)(11 87 61 55 31)(12 88 62 56 32)(13 89 63 57 33)(14 90 64 39 34)(15 91 65 40 35)(16 92 66 41 36)(17 93 67 42 37)(18 94 68 43 38)(19 95 69 44 20)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 20)(39 90)(40 91)(41 92)(42 93)(43 94)(44 95)(45 77)(46 78)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)(53 85)(54 86)(55 87)(56 88)(57 89)
(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 21)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(39 50)(40 49)(41 48)(42 47)(43 46)(44 45)(51 57)(52 56)(53 55)(58 62)(59 61)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)(77 95)(78 94)(79 93)(80 92)(81 91)(82 90)(83 89)(84 88)(85 87)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 5A 5B 10A 10B 19A ··· 19I 38A ··· 38I 95A ··· 95R order 1 2 2 2 5 5 10 10 19 ··· 19 38 ··· 38 95 ··· 95 size 1 5 19 95 2 2 38 38 2 ··· 2 10 ··· 10 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 D5 D10 D19 D38 D5×D19 kernel D5×D19 D5×C19 C5×D19 D95 D19 C19 D5 C5 C1 # reps 1 1 1 1 2 2 9 9 18

Matrix representation of D5×D19 in GL4(𝔽191) generated by

 1 0 0 0 0 1 0 0 0 0 1 123 0 0 21 101
,
 1 0 0 0 0 1 0 0 0 0 1 123 0 0 0 190
,
 74 1 0 0 158 77 0 0 0 0 1 0 0 0 0 1
,
 95 14 0 0 174 96 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(191))| [1,0,0,0,0,1,0,0,0,0,1,21,0,0,123,101],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,123,190],[74,158,0,0,1,77,0,0,0,0,1,0,0,0,0,1],[95,174,0,0,14,96,0,0,0,0,1,0,0,0,0,1] >;

D5×D19 in GAP, Magma, Sage, TeX

D_5\times D_{19}
% in TeX

G:=Group("D5xD19");
// GroupNames label

G:=SmallGroup(380,7);
// by ID

G=gap.SmallGroup(380,7);
# by ID

G:=PCGroup([4,-2,-2,-5,-19,102,5763]);
// Polycyclic

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

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