metacyclic, supersoluble, monomial, Z-group
Aliases: C13⋊2C24, C52.2C6, C26.2C12, C13⋊2C8⋊C3, C13⋊C3⋊2C8, C4.2(C13⋊C6), C2.(C26.C6), (C4×C13⋊C3).2C2, (C2×C13⋊C3).2C4, SmallGroup(312,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C13⋊2C24 |
Generators and relations for C13⋊2C24
G = < a,b | a13=b24=1, bab-1=a10 >
Smallest permutation representation of C13⋊2C24
►On 104 points
Generators in S104
(1 79 22 63 42 14 30 97 81 71 34 89 50)(2 35 98 43 80 90 82 15 23 51 72 31 64)(3 73 16 57 36 32 24 91 99 65 52 83 44)(4 53 92 37 74 84 100 9 17 45 66 25 58)(5 67 10 75 54 26 18 85 93 59 46 101 38)(6 47 86 55 68 102 94 27 11 39 60 19 76)(7 61 28 69 48 20 12 103 87 77 40 95 56)(8 41 104 49 62 96 88 21 29 33 78 13 70)
(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)
G:=sub<Sym(104)| (1,79,22,63,42,14,30,97,81,71,34,89,50)(2,35,98,43,80,90,82,15,23,51,72,31,64)(3,73,16,57,36,32,24,91,99,65,52,83,44)(4,53,92,37,74,84,100,9,17,45,66,25,58)(5,67,10,75,54,26,18,85,93,59,46,101,38)(6,47,86,55,68,102,94,27,11,39,60,19,76)(7,61,28,69,48,20,12,103,87,77,40,95,56)(8,41,104,49,62,96,88,21,29,33,78,13,70), (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)>;
G:=Group( (1,79,22,63,42,14,30,97,81,71,34,89,50)(2,35,98,43,80,90,82,15,23,51,72,31,64)(3,73,16,57,36,32,24,91,99,65,52,83,44)(4,53,92,37,74,84,100,9,17,45,66,25,58)(5,67,10,75,54,26,18,85,93,59,46,101,38)(6,47,86,55,68,102,94,27,11,39,60,19,76)(7,61,28,69,48,20,12,103,87,77,40,95,56)(8,41,104,49,62,96,88,21,29,33,78,13,70), (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) );
G=PermutationGroup([[(1,79,22,63,42,14,30,97,81,71,34,89,50),(2,35,98,43,80,90,82,15,23,51,72,31,64),(3,73,16,57,36,32,24,91,99,65,52,83,44),(4,53,92,37,74,84,100,9,17,45,66,25,58),(5,67,10,75,54,26,18,85,93,59,46,101,38),(6,47,86,55,68,102,94,27,11,39,60,19,76),(7,61,28,69,48,20,12,103,87,77,40,95,56),(8,41,104,49,62,96,88,21,29,33,78,13,70)], [(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)]])
32 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 13A | 13B | 24A | ··· | 24H | 26A | 26B | 52A | 52B | 52C | 52D |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 13 | 13 | 24 | ··· | 24 | 26 | 26 | 52 | 52 | 52 | 52 |
| size | 1 | 1 | 13 | 13 | 1 | 1 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 13 | 6 | 6 | 13 | ··· | 13 | 6 | 6 | 6 | 6 | 6 | 6 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 |
| type | + | + | + | - | |||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 | C13⋊C6 | C26.C6 | C13⋊2C24 |
| kernel | C13⋊2C24 | C4×C13⋊C3 | C13⋊2C8 | C2×C13⋊C3 | C52 | C13⋊C3 | C26 | C13 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 2 | 2 | 4 |
Matrix representation of C13⋊2C24 ►in GL6(𝔽313)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 312 | 110 | 311 | 111 | 311 | 110 |
| 163 | 72 | 153 | 232 | 306 | 232 |
| 213 | 305 | 59 | 1 | 203 | 304 |
| 6 | 28 | 159 | 91 | 97 | 250 |
| 153 | 232 | 74 | 79 | 84 | 232 |
| 81 | 17 | 234 | 239 | 81 | 160 |
| 9 | 162 | 10 | 312 | 19 | 152 |
G:=sub<GL(6,GF(313))| [0,0,0,0,0,312,1,0,0,0,0,110,0,1,0,0,0,311,0,0,1,0,0,111,0,0,0,1,0,311,0,0,0,0,1,110],[163,213,6,153,81,9,72,305,28,232,17,162,153,59,159,74,234,10,232,1,91,79,239,312,306,203,97,84,81,19,232,304,250,232,160,152] >;
C13⋊2C24 in GAP, Magma, Sage, TeX
C_{13}\rtimes_2C_{24} % in TeX
G:=Group("C13:2C24"); // GroupNames label
G:=SmallGroup(312,1);
// by ID
G=gap.SmallGroup(312,1);
# by ID
G:=PCGroup([5,-2,-3,-2,-2,-13,30,42,7204,909]);
// Polycyclic
G:=Group<a,b|a^13=b^24=1,b*a*b^-1=a^10>;
// generators/relations
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