direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8xD13, C104:3C2, D26.4C4, C4.12D26, C52.12C22, Dic13.4C4, C13:3(C2xC8), C13:2C8:6C2, C26.8(C2xC4), C2.1(C4xD13), (C4xD13).7C2, SmallGroup(208,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C8xD13 |
Generators and relations for C8xD13
G = < a,b,c | a8=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >
Quotients: C1, C2, C4, C22, C8, C2xC4, C2xC8, D13, D26, C4xD13, C8xD13
Smallest permutation representation of C8xD13
►On 104 points
Generators in S104
(1 97 50 66 25 84 31 59)(2 98 51 67 26 85 32 60)(3 99 52 68 14 86 33 61)(4 100 40 69 15 87 34 62)(5 101 41 70 16 88 35 63)(6 102 42 71 17 89 36 64)(7 103 43 72 18 90 37 65)(8 104 44 73 19 91 38 53)(9 92 45 74 20 79 39 54)(10 93 46 75 21 80 27 55)(11 94 47 76 22 81 28 56)(12 95 48 77 23 82 29 57)(13 96 49 78 24 83 30 58)
(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 96 97 98 99 100 101 102 103 104)
(1 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 26)(13 25)(27 40)(28 52)(29 51)(30 50)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(53 71)(54 70)(55 69)(56 68)(57 67)(58 66)(59 78)(60 77)(61 76)(62 75)(63 74)(64 73)(65 72)(79 101)(80 100)(81 99)(82 98)(83 97)(84 96)(85 95)(86 94)(87 93)(88 92)(89 104)(90 103)(91 102)
G:=sub<Sym(104)| (1,97,50,66,25,84,31,59)(2,98,51,67,26,85,32,60)(3,99,52,68,14,86,33,61)(4,100,40,69,15,87,34,62)(5,101,41,70,16,88,35,63)(6,102,42,71,17,89,36,64)(7,103,43,72,18,90,37,65)(8,104,44,73,19,91,38,53)(9,92,45,74,20,79,39,54)(10,93,46,75,21,80,27,55)(11,94,47,76,22,81,28,56)(12,95,48,77,23,82,29,57)(13,96,49,78,24,83,30,58), (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)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,26)(13,25)(27,40)(28,52)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(79,101)(80,100)(81,99)(82,98)(83,97)(84,96)(85,95)(86,94)(87,93)(88,92)(89,104)(90,103)(91,102)>;
G:=Group( (1,97,50,66,25,84,31,59)(2,98,51,67,26,85,32,60)(3,99,52,68,14,86,33,61)(4,100,40,69,15,87,34,62)(5,101,41,70,16,88,35,63)(6,102,42,71,17,89,36,64)(7,103,43,72,18,90,37,65)(8,104,44,73,19,91,38,53)(9,92,45,74,20,79,39,54)(10,93,46,75,21,80,27,55)(11,94,47,76,22,81,28,56)(12,95,48,77,23,82,29,57)(13,96,49,78,24,83,30,58), (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)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,26)(13,25)(27,40)(28,52)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(79,101)(80,100)(81,99)(82,98)(83,97)(84,96)(85,95)(86,94)(87,93)(88,92)(89,104)(90,103)(91,102) );
G=PermutationGroup([[(1,97,50,66,25,84,31,59),(2,98,51,67,26,85,32,60),(3,99,52,68,14,86,33,61),(4,100,40,69,15,87,34,62),(5,101,41,70,16,88,35,63),(6,102,42,71,17,89,36,64),(7,103,43,72,18,90,37,65),(8,104,44,73,19,91,38,53),(9,92,45,74,20,79,39,54),(10,93,46,75,21,80,27,55),(11,94,47,76,22,81,28,56),(12,95,48,77,23,82,29,57),(13,96,49,78,24,83,30,58)], [(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),(92,93,94,95,96,97,98,99,100,101,102,103,104)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,26),(13,25),(27,40),(28,52),(29,51),(30,50),(31,49),(32,48),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(53,71),(54,70),(55,69),(56,68),(57,67),(58,66),(59,78),(60,77),(61,76),(62,75),(63,74),(64,73),(65,72),(79,101),(80,100),(81,99),(82,98),(83,97),(84,96),(85,95),(86,94),(87,93),(88,92),(89,104),(90,103),(91,102)]])
C8xD13 is a maximal subgroup of
C208:C2 D13:C16 D26.C8 C104:C4 D26.8D4 D13.D8 C104.C4 C104.1C4 D52.3C4 D52.2C4 D8:3D13 D26.6D4 D104:C2
C8xD13 is a maximal quotient of
C208:C2 C52.8Q8 D26:1C8
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 13A | ··· | 13F | 26A | ··· | 26F | 52A | ··· | 52L | 104A | ··· | 104X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 13 | 13 | 1 | 1 | 13 | 13 | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D13 | D26 | C4xD13 | C8xD13 |
| kernel | C8xD13 | C13:2C8 | C104 | C4xD13 | Dic13 | D26 | D13 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 6 | 6 | 12 | 24 |
Matrix representation of C8xD13 ►in GL2(F313) generated by
| 188 | 0 |
| 0 | 188 |
| 142 | 270 |
| 312 | 256 |
| 291 | 181 |
| 25 | 22 |
G:=sub<GL(2,GF(313))| [188,0,0,188],[142,312,270,256],[291,25,181,22] >;
C8xD13 in GAP, Magma, Sage, TeX
C_8\times D_{13} % in TeX
G:=Group("C8xD13"); // GroupNames label
G:=SmallGroup(208,4);
// by ID
G=gap.SmallGroup(208,4);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,26,42,4804]);
// Polycyclic
G:=Group<a,b,c|a^8=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export