direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×D19, C19⋊C10, C95⋊2C2, SmallGroup(190,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C5×D19 |
Generators and relations for C5×D19
G = < a,b,c | a5=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D19
►On 95 points
Generators in S95
(1 92 72 48 26)(2 93 73 49 27)(3 94 74 50 28)(4 95 75 51 29)(5 77 76 52 30)(6 78 58 53 31)(7 79 59 54 32)(8 80 60 55 33)(9 81 61 56 34)(10 82 62 57 35)(11 83 63 39 36)(12 84 64 40 37)(13 85 65 41 38)(14 86 66 42 20)(15 87 67 43 21)(16 88 68 44 22)(17 89 69 45 23)(18 90 70 46 24)(19 91 71 47 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 56)(40 55)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)(58 66)(59 65)(60 64)(61 63)(67 76)(68 75)(69 74)(70 73)(71 72)(77 87)(78 86)(79 85)(80 84)(81 83)(88 95)(89 94)(90 93)(91 92)
G:=sub<Sym(95)| (1,92,72,48,26)(2,93,73,49,27)(3,94,74,50,28)(4,95,75,51,29)(5,77,76,52,30)(6,78,58,53,31)(7,79,59,54,32)(8,80,60,55,33)(9,81,61,56,34)(10,82,62,57,35)(11,83,63,39,36)(12,84,64,40,37)(13,85,65,41,38)(14,86,66,42,20)(15,87,67,43,21)(16,88,68,44,22)(17,89,69,45,23)(18,90,70,46,24)(19,91,71,47,25), (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)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(58,66)(59,65)(60,64)(61,63)(67,76)(68,75)(69,74)(70,73)(71,72)(77,87)(78,86)(79,85)(80,84)(81,83)(88,95)(89,94)(90,93)(91,92)>;
G:=Group( (1,92,72,48,26)(2,93,73,49,27)(3,94,74,50,28)(4,95,75,51,29)(5,77,76,52,30)(6,78,58,53,31)(7,79,59,54,32)(8,80,60,55,33)(9,81,61,56,34)(10,82,62,57,35)(11,83,63,39,36)(12,84,64,40,37)(13,85,65,41,38)(14,86,66,42,20)(15,87,67,43,21)(16,88,68,44,22)(17,89,69,45,23)(18,90,70,46,24)(19,91,71,47,25), (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)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(58,66)(59,65)(60,64)(61,63)(67,76)(68,75)(69,74)(70,73)(71,72)(77,87)(78,86)(79,85)(80,84)(81,83)(88,95)(89,94)(90,93)(91,92) );
G=PermutationGroup([[(1,92,72,48,26),(2,93,73,49,27),(3,94,74,50,28),(4,95,75,51,29),(5,77,76,52,30),(6,78,58,53,31),(7,79,59,54,32),(8,80,60,55,33),(9,81,61,56,34),(10,82,62,57,35),(11,83,63,39,36),(12,84,64,40,37),(13,85,65,41,38),(14,86,66,42,20),(15,87,67,43,21),(16,88,68,44,22),(17,89,69,45,23),(18,90,70,46,24),(19,91,71,47,25)], [(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),(39,56),(40,55),(41,54),(42,53),(43,52),(44,51),(45,50),(46,49),(47,48),(58,66),(59,65),(60,64),(61,63),(67,76),(68,75),(69,74),(70,73),(71,72),(77,87),(78,86),(79,85),(80,84),(81,83),(88,95),(89,94),(90,93),(91,92)]])
55 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 19A | ··· | 19I | 95A | ··· | 95AJ |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 19 | ··· | 19 | 95 | ··· | 95 |
| size | 1 | 19 | 1 | 1 | 1 | 1 | 19 | 19 | 19 | 19 | 2 | ··· | 2 | 2 | ··· | 2 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C5 | C10 | D19 | C5×D19 |
| kernel | C5×D19 | C95 | D19 | C19 | C5 | C1 |
| # reps | 1 | 1 | 4 | 4 | 9 | 36 |
Matrix representation of C5×D19 ►in GL2(𝔽191) generated by
| 184 | 0 |
| 0 | 184 |
| 0 | 1 |
| 190 | 132 |
| 0 | 1 |
| 1 | 0 |
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
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