metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C104⋊1C4, D13.1D8, D26.9D4, D13.1Q16, Dic13.3Q8, C8⋊1(C13⋊C4), C13⋊(C2.D8), C13⋊2C8⋊6C4, C52.9(C2×C4), C26.2(C4⋊C4), (C8×D13).3C2, C52⋊C4.4C2, C2.5(C52⋊C4), (C4×D13).26C22, C4.9(C2×C13⋊C4), SmallGroup(416,69)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D13.D8
G = < a,b,c,d | a13=b2=c8=1, d2=a-1b, bab=a-1, ac=ca, dad-1=a5, bc=cb, dbd-1=a4b, dcd-1=c-1 >
Smallest permutation representation of D13.D8
►On 104 points
Generators in S104
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 20)(15 19)(16 18)(21 26)(22 25)(23 24)(27 29)(30 39)(31 38)(32 37)(33 36)(34 35)(40 47)(41 46)(42 45)(43 44)(48 52)(49 51)(53 65)(54 64)(55 63)(56 62)(57 61)(58 60)(66 76)(67 75)(68 74)(69 73)(70 72)(77 78)(79 83)(80 82)(84 91)(85 90)(86 89)(87 88)(92 98)(93 97)(94 96)(99 104)(100 103)(101 102)
(1 88 35 78 24 102 44 53)(2 89 36 66 25 103 45 54)(3 90 37 67 26 104 46 55)(4 91 38 68 14 92 47 56)(5 79 39 69 15 93 48 57)(6 80 27 70 16 94 49 58)(7 81 28 71 17 95 50 59)(8 82 29 72 18 96 51 60)(9 83 30 73 19 97 52 61)(10 84 31 74 20 98 40 62)(11 85 32 75 21 99 41 63)(12 86 33 76 22 100 42 64)(13 87 34 77 23 101 43 65)
(1 44)(2 52 13 49)(3 47 12 41)(4 42 11 46)(5 50 10 51)(6 45 9 43)(7 40 8 48)(14 33 21 37)(15 28 20 29)(16 36 19 34)(17 31 18 39)(22 32 26 38)(23 27 25 30)(24 35)(54 61 65 58)(55 56 64 63)(57 59 62 60)(66 73 77 70)(67 68 76 75)(69 71 74 72)(79 95 84 96)(80 103 83 101)(81 98 82 93)(85 104 91 100)(86 99 90 92)(87 94 89 97)(88 102)
G:=sub<Sym(104)| (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,20)(15,19)(16,18)(21,26)(22,25)(23,24)(27,29)(30,39)(31,38)(32,37)(33,36)(34,35)(40,47)(41,46)(42,45)(43,44)(48,52)(49,51)(53,65)(54,64)(55,63)(56,62)(57,61)(58,60)(66,76)(67,75)(68,74)(69,73)(70,72)(77,78)(79,83)(80,82)(84,91)(85,90)(86,89)(87,88)(92,98)(93,97)(94,96)(99,104)(100,103)(101,102), (1,88,35,78,24,102,44,53)(2,89,36,66,25,103,45,54)(3,90,37,67,26,104,46,55)(4,91,38,68,14,92,47,56)(5,79,39,69,15,93,48,57)(6,80,27,70,16,94,49,58)(7,81,28,71,17,95,50,59)(8,82,29,72,18,96,51,60)(9,83,30,73,19,97,52,61)(10,84,31,74,20,98,40,62)(11,85,32,75,21,99,41,63)(12,86,33,76,22,100,42,64)(13,87,34,77,23,101,43,65), (1,44)(2,52,13,49)(3,47,12,41)(4,42,11,46)(5,50,10,51)(6,45,9,43)(7,40,8,48)(14,33,21,37)(15,28,20,29)(16,36,19,34)(17,31,18,39)(22,32,26,38)(23,27,25,30)(24,35)(54,61,65,58)(55,56,64,63)(57,59,62,60)(66,73,77,70)(67,68,76,75)(69,71,74,72)(79,95,84,96)(80,103,83,101)(81,98,82,93)(85,104,91,100)(86,99,90,92)(87,94,89,97)(88,102)>;
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)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,20)(15,19)(16,18)(21,26)(22,25)(23,24)(27,29)(30,39)(31,38)(32,37)(33,36)(34,35)(40,47)(41,46)(42,45)(43,44)(48,52)(49,51)(53,65)(54,64)(55,63)(56,62)(57,61)(58,60)(66,76)(67,75)(68,74)(69,73)(70,72)(77,78)(79,83)(80,82)(84,91)(85,90)(86,89)(87,88)(92,98)(93,97)(94,96)(99,104)(100,103)(101,102), (1,88,35,78,24,102,44,53)(2,89,36,66,25,103,45,54)(3,90,37,67,26,104,46,55)(4,91,38,68,14,92,47,56)(5,79,39,69,15,93,48,57)(6,80,27,70,16,94,49,58)(7,81,28,71,17,95,50,59)(8,82,29,72,18,96,51,60)(9,83,30,73,19,97,52,61)(10,84,31,74,20,98,40,62)(11,85,32,75,21,99,41,63)(12,86,33,76,22,100,42,64)(13,87,34,77,23,101,43,65), (1,44)(2,52,13,49)(3,47,12,41)(4,42,11,46)(5,50,10,51)(6,45,9,43)(7,40,8,48)(14,33,21,37)(15,28,20,29)(16,36,19,34)(17,31,18,39)(22,32,26,38)(23,27,25,30)(24,35)(54,61,65,58)(55,56,64,63)(57,59,62,60)(66,73,77,70)(67,68,76,75)(69,71,74,72)(79,95,84,96)(80,103,83,101)(81,98,82,93)(85,104,91,100)(86,99,90,92)(87,94,89,97)(88,102) );
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),(92,93,94,95,96,97,98,99,100,101,102,103,104)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,20),(15,19),(16,18),(21,26),(22,25),(23,24),(27,29),(30,39),(31,38),(32,37),(33,36),(34,35),(40,47),(41,46),(42,45),(43,44),(48,52),(49,51),(53,65),(54,64),(55,63),(56,62),(57,61),(58,60),(66,76),(67,75),(68,74),(69,73),(70,72),(77,78),(79,83),(80,82),(84,91),(85,90),(86,89),(87,88),(92,98),(93,97),(94,96),(99,104),(100,103),(101,102)], [(1,88,35,78,24,102,44,53),(2,89,36,66,25,103,45,54),(3,90,37,67,26,104,46,55),(4,91,38,68,14,92,47,56),(5,79,39,69,15,93,48,57),(6,80,27,70,16,94,49,58),(7,81,28,71,17,95,50,59),(8,82,29,72,18,96,51,60),(9,83,30,73,19,97,52,61),(10,84,31,74,20,98,40,62),(11,85,32,75,21,99,41,63),(12,86,33,76,22,100,42,64),(13,87,34,77,23,101,43,65)], [(1,44),(2,52,13,49),(3,47,12,41),(4,42,11,46),(5,50,10,51),(6,45,9,43),(7,40,8,48),(14,33,21,37),(15,28,20,29),(16,36,19,34),(17,31,18,39),(22,32,26,38),(23,27,25,30),(24,35),(54,61,65,58),(55,56,64,63),(57,59,62,60),(66,73,77,70),(67,68,76,75),(69,71,74,72),(79,95,84,96),(80,103,83,101),(81,98,82,93),(85,104,91,100),(86,99,90,92),(87,94,89,97),(88,102)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 13A | 13B | 13C | 26A | 26B | 26C | 52A | ··· | 52F | 104A | ··· | 104L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | 13 | 13 | 26 | 26 | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 13 | 13 | 2 | 26 | 52 | 52 | 52 | 52 | 2 | 2 | 26 | 26 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | + | - | + | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | Q8 | D4 | D8 | Q16 | C13⋊C4 | C2×C13⋊C4 | C52⋊C4 | D13.D8 |
| kernel | D13.D8 | C8×D13 | C52⋊C4 | C13⋊2C8 | C104 | Dic13 | D26 | D13 | D13 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 3 | 3 | 6 | 12 |
Matrix representation of D13.D8 ►in GL6(𝔽313)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 32 | 31 | 31 |
| 0 | 0 | 282 | 282 | 283 | 282 |
| 0 | 0 | 242 | 242 | 242 | 243 |
| 0 | 0 | 312 | 312 | 312 | 312 |
| 312 | 0 | 0 | 0 | 0 | 0 |
| 0 | 312 | 0 | 0 | 0 | 0 |
| 0 | 0 | 312 | 281 | 40 | 311 |
| 0 | 0 | 0 | 31 | 61 | 30 |
| 0 | 0 | 0 | 71 | 281 | 71 |
| 0 | 0 | 0 | 1 | 242 | 2 |
| 253 | 60 | 0 | 0 | 0 | 0 |
| 253 | 253 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 68 | 86 | 60 |
| 0 | 0 | 227 | 223 | 227 | 0 |
| 0 | 0 | 26 | 8 | 4 | 253 |
| 0 | 0 | 18 | 18 | 0 | 22 |
| 0 | 25 | 0 | 0 | 0 | 0 |
| 25 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 283 | 0 | 272 | 32 |
| 0 | 0 | 243 | 0 | 252 | 252 |
| 0 | 0 | 282 | 312 | 32 | 272 |
| 0 | 0 | 29 | 0 | 71 | 311 |
G:=sub<GL(6,GF(313))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,282,242,312,0,0,32,282,242,312,0,0,31,283,242,312,0,0,31,282,243,312],[312,0,0,0,0,0,0,312,0,0,0,0,0,0,312,0,0,0,0,0,281,31,71,1,0,0,40,61,281,242,0,0,311,30,71,2],[253,253,0,0,0,0,60,253,0,0,0,0,0,0,64,227,26,18,0,0,68,223,8,18,0,0,86,227,4,0,0,0,60,0,253,22],[0,25,0,0,0,0,25,0,0,0,0,0,0,0,283,243,282,29,0,0,0,0,312,0,0,0,272,252,32,71,0,0,32,252,272,311] >;
D13.D8 in GAP, Magma, Sage, TeX
D_{13}.D_8 % in TeX
G:=Group("D13.D8"); // GroupNames label
G:=SmallGroup(416,69);
// by ID
G=gap.SmallGroup(416,69);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,151,579,69,9221,3473]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^2=c^8=1,d^2=a^-1*b,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations
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