direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C13×C4.D4, C23.C52, C52.58D4, M4(2)⋊3C26, C4.9(D4×C13), (D4×C26).8C2, (C2×D4).2C26, (C22×C26).1C4, C22.3(C2×C52), (C13×M4(2))⋊9C2, (C2×C52).59C22, C26.33(C22⋊C4), (C2×C4).1(C2×C26), (C2×C26).40(C2×C4), C2.4(C13×C22⋊C4), SmallGroup(416,50)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C13×C4.D4
G = < a,b,c,d | a13=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c3 >
Smallest permutation representation of C13×C4.D4
►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 62 49 14)(2 63 50 15)(3 64 51 16)(4 65 52 17)(5 53 40 18)(6 54 41 19)(7 55 42 20)(8 56 43 21)(9 57 44 22)(10 58 45 23)(11 59 46 24)(12 60 47 25)(13 61 48 26)(27 101 90 73)(28 102 91 74)(29 103 79 75)(30 104 80 76)(31 92 81 77)(32 93 82 78)(33 94 83 66)(34 95 84 67)(35 96 85 68)(36 97 86 69)(37 98 87 70)(38 99 88 71)(39 100 89 72)
(1 83 62 94 49 33 14 66)(2 84 63 95 50 34 15 67)(3 85 64 96 51 35 16 68)(4 86 65 97 52 36 17 69)(5 87 53 98 40 37 18 70)(6 88 54 99 41 38 19 71)(7 89 55 100 42 39 20 72)(8 90 56 101 43 27 21 73)(9 91 57 102 44 28 22 74)(10 79 58 103 45 29 23 75)(11 80 59 104 46 30 24 76)(12 81 60 92 47 31 25 77)(13 82 61 93 48 32 26 78)
(1 33 62 94 49 83 14 66)(2 34 63 95 50 84 15 67)(3 35 64 96 51 85 16 68)(4 36 65 97 52 86 17 69)(5 37 53 98 40 87 18 70)(6 38 54 99 41 88 19 71)(7 39 55 100 42 89 20 72)(8 27 56 101 43 90 21 73)(9 28 57 102 44 91 22 74)(10 29 58 103 45 79 23 75)(11 30 59 104 46 80 24 76)(12 31 60 92 47 81 25 77)(13 32 61 93 48 82 26 78)
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), (1,62,49,14)(2,63,50,15)(3,64,51,16)(4,65,52,17)(5,53,40,18)(6,54,41,19)(7,55,42,20)(8,56,43,21)(9,57,44,22)(10,58,45,23)(11,59,46,24)(12,60,47,25)(13,61,48,26)(27,101,90,73)(28,102,91,74)(29,103,79,75)(30,104,80,76)(31,92,81,77)(32,93,82,78)(33,94,83,66)(34,95,84,67)(35,96,85,68)(36,97,86,69)(37,98,87,70)(38,99,88,71)(39,100,89,72), (1,83,62,94,49,33,14,66)(2,84,63,95,50,34,15,67)(3,85,64,96,51,35,16,68)(4,86,65,97,52,36,17,69)(5,87,53,98,40,37,18,70)(6,88,54,99,41,38,19,71)(7,89,55,100,42,39,20,72)(8,90,56,101,43,27,21,73)(9,91,57,102,44,28,22,74)(10,79,58,103,45,29,23,75)(11,80,59,104,46,30,24,76)(12,81,60,92,47,31,25,77)(13,82,61,93,48,32,26,78), (1,33,62,94,49,83,14,66)(2,34,63,95,50,84,15,67)(3,35,64,96,51,85,16,68)(4,36,65,97,52,86,17,69)(5,37,53,98,40,87,18,70)(6,38,54,99,41,88,19,71)(7,39,55,100,42,89,20,72)(8,27,56,101,43,90,21,73)(9,28,57,102,44,91,22,74)(10,29,58,103,45,79,23,75)(11,30,59,104,46,80,24,76)(12,31,60,92,47,81,25,77)(13,32,61,93,48,82,26,78)>;
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), (1,62,49,14)(2,63,50,15)(3,64,51,16)(4,65,52,17)(5,53,40,18)(6,54,41,19)(7,55,42,20)(8,56,43,21)(9,57,44,22)(10,58,45,23)(11,59,46,24)(12,60,47,25)(13,61,48,26)(27,101,90,73)(28,102,91,74)(29,103,79,75)(30,104,80,76)(31,92,81,77)(32,93,82,78)(33,94,83,66)(34,95,84,67)(35,96,85,68)(36,97,86,69)(37,98,87,70)(38,99,88,71)(39,100,89,72), (1,83,62,94,49,33,14,66)(2,84,63,95,50,34,15,67)(3,85,64,96,51,35,16,68)(4,86,65,97,52,36,17,69)(5,87,53,98,40,37,18,70)(6,88,54,99,41,38,19,71)(7,89,55,100,42,39,20,72)(8,90,56,101,43,27,21,73)(9,91,57,102,44,28,22,74)(10,79,58,103,45,29,23,75)(11,80,59,104,46,30,24,76)(12,81,60,92,47,31,25,77)(13,82,61,93,48,32,26,78), (1,33,62,94,49,83,14,66)(2,34,63,95,50,84,15,67)(3,35,64,96,51,85,16,68)(4,36,65,97,52,86,17,69)(5,37,53,98,40,87,18,70)(6,38,54,99,41,88,19,71)(7,39,55,100,42,89,20,72)(8,27,56,101,43,90,21,73)(9,28,57,102,44,91,22,74)(10,29,58,103,45,79,23,75)(11,30,59,104,46,80,24,76)(12,31,60,92,47,81,25,77)(13,32,61,93,48,82,26,78) );
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)], [(1,62,49,14),(2,63,50,15),(3,64,51,16),(4,65,52,17),(5,53,40,18),(6,54,41,19),(7,55,42,20),(8,56,43,21),(9,57,44,22),(10,58,45,23),(11,59,46,24),(12,60,47,25),(13,61,48,26),(27,101,90,73),(28,102,91,74),(29,103,79,75),(30,104,80,76),(31,92,81,77),(32,93,82,78),(33,94,83,66),(34,95,84,67),(35,96,85,68),(36,97,86,69),(37,98,87,70),(38,99,88,71),(39,100,89,72)], [(1,83,62,94,49,33,14,66),(2,84,63,95,50,34,15,67),(3,85,64,96,51,35,16,68),(4,86,65,97,52,36,17,69),(5,87,53,98,40,37,18,70),(6,88,54,99,41,38,19,71),(7,89,55,100,42,39,20,72),(8,90,56,101,43,27,21,73),(9,91,57,102,44,28,22,74),(10,79,58,103,45,29,23,75),(11,80,59,104,46,30,24,76),(12,81,60,92,47,31,25,77),(13,82,61,93,48,32,26,78)], [(1,33,62,94,49,83,14,66),(2,34,63,95,50,84,15,67),(3,35,64,96,51,85,16,68),(4,36,65,97,52,86,17,69),(5,37,53,98,40,87,18,70),(6,38,54,99,41,88,19,71),(7,39,55,100,42,89,20,72),(8,27,56,101,43,90,21,73),(9,28,57,102,44,91,22,74),(10,29,58,103,45,79,23,75),(11,30,59,104,46,80,24,76),(12,31,60,92,47,81,25,77),(13,32,61,93,48,82,26,78)]])
143 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 8A | 8B | 8C | 8D | 13A | ··· | 13L | 26A | ··· | 26L | 26M | ··· | 26X | 26Y | ··· | 26AV | 52A | ··· | 52X | 104A | ··· | 104AV |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
143 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C4 | C13 | C26 | C26 | C52 | D4 | D4×C13 | C4.D4 | C13×C4.D4 |
| kernel | C13×C4.D4 | C13×M4(2) | D4×C26 | C22×C26 | C4.D4 | M4(2) | C2×D4 | C23 | C52 | C4 | C13 | C1 |
| # reps | 1 | 2 | 1 | 4 | 12 | 24 | 12 | 48 | 2 | 24 | 1 | 12 |
Matrix representation of C13×C4.D4 ►in GL4(𝔽313) generated by
| 48 | 0 | 0 | 0 |
| 0 | 48 | 0 | 0 |
| 0 | 0 | 48 | 0 |
| 0 | 0 | 0 | 48 |
| 0 | 1 | 0 | 0 |
| 312 | 0 | 0 | 0 |
| 45 | 168 | 0 | 1 |
| 168 | 268 | 312 | 0 |
| 148 | 123 | 311 | 0 |
| 213 | 188 | 0 | 2 |
| 107 | 162 | 165 | 123 |
| 54 | 59 | 213 | 125 |
| 148 | 123 | 311 | 0 |
| 100 | 125 | 0 | 311 |
| 200 | 200 | 165 | 190 |
| 35 | 34 | 213 | 188 |
G:=sub<GL(4,GF(313))| [48,0,0,0,0,48,0,0,0,0,48,0,0,0,0,48],[0,312,45,168,1,0,168,268,0,0,0,312,0,0,1,0],[148,213,107,54,123,188,162,59,311,0,165,213,0,2,123,125],[148,100,200,35,123,125,200,34,311,0,165,213,0,311,190,188] >;
C13×C4.D4 in GAP, Magma, Sage, TeX
C_{13}\times C_4.D_4 % in TeX
G:=Group("C13xC4.D4"); // GroupNames label
G:=SmallGroup(416,50);
// by ID
G=gap.SmallGroup(416,50);
# by ID
G:=PCGroup([6,-2,-2,-13,-2,-2,-2,624,649,6243,4690,88]);
// Polycyclic
G:=Group<a,b,c,d|a^13=b^4=1,c^4=b^2,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations
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