metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C13⋊Dic9, C117⋊1C4, D13.D9, C39.Dic3, C9⋊(C13⋊C4), C3.(C39⋊C4), (C9×D13).1C2, (C3×D13).2S3, SmallGroup(468,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C117 — C13⋊Dic9 |
Generators and relations for C13⋊Dic9
G = < a,b,c | a13=b18=1, c2=b9, bab-1=a-1, cac-1=a8, cbc-1=b-1 >
Smallest permutation representation of C13⋊Dic9
►On 117 points
Generators in S117
(1 67 55 109 15 37 88 97 28 24 100 46 76)(2 77 47 101 25 29 98 89 38 16 110 56 68)(3 69 57 111 17 39 90 99 30 26 102 48 78)(4 79 49 103 27 31 82 91 40 18 112 58 70)(5 71 59 113 19 41 92 83 32 10 104 50 80)(6 81 51 105 11 33 84 93 42 20 114 60 72)(7 73 61 115 21 43 94 85 34 12 106 52 64)(8 65 53 107 13 35 86 95 44 22 116 62 74)(9 75 63 117 23 45 96 87 36 14 108 54 66)
(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)(100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117)
(2 9)(3 8)(4 7)(5 6)(10 84 19 93)(11 83 20 92)(12 82 21 91)(13 99 22 90)(14 98 23 89)(15 97 24 88)(16 96 25 87)(17 95 26 86)(18 94 27 85)(28 67 37 76)(29 66 38 75)(30 65 39 74)(31 64 40 73)(32 81 41 72)(33 80 42 71)(34 79 43 70)(35 78 44 69)(36 77 45 68)(46 109 55 100)(47 108 56 117)(48 107 57 116)(49 106 58 115)(50 105 59 114)(51 104 60 113)(52 103 61 112)(53 102 62 111)(54 101 63 110)
G:=sub<Sym(117)| (1,67,55,109,15,37,88,97,28,24,100,46,76)(2,77,47,101,25,29,98,89,38,16,110,56,68)(3,69,57,111,17,39,90,99,30,26,102,48,78)(4,79,49,103,27,31,82,91,40,18,112,58,70)(5,71,59,113,19,41,92,83,32,10,104,50,80)(6,81,51,105,11,33,84,93,42,20,114,60,72)(7,73,61,115,21,43,94,85,34,12,106,52,64)(8,65,53,107,13,35,86,95,44,22,116,62,74)(9,75,63,117,23,45,96,87,36,14,108,54,66), (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)(100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117), (2,9)(3,8)(4,7)(5,6)(10,84,19,93)(11,83,20,92)(12,82,21,91)(13,99,22,90)(14,98,23,89)(15,97,24,88)(16,96,25,87)(17,95,26,86)(18,94,27,85)(28,67,37,76)(29,66,38,75)(30,65,39,74)(31,64,40,73)(32,81,41,72)(33,80,42,71)(34,79,43,70)(35,78,44,69)(36,77,45,68)(46,109,55,100)(47,108,56,117)(48,107,57,116)(49,106,58,115)(50,105,59,114)(51,104,60,113)(52,103,61,112)(53,102,62,111)(54,101,63,110)>;
G:=Group( (1,67,55,109,15,37,88,97,28,24,100,46,76)(2,77,47,101,25,29,98,89,38,16,110,56,68)(3,69,57,111,17,39,90,99,30,26,102,48,78)(4,79,49,103,27,31,82,91,40,18,112,58,70)(5,71,59,113,19,41,92,83,32,10,104,50,80)(6,81,51,105,11,33,84,93,42,20,114,60,72)(7,73,61,115,21,43,94,85,34,12,106,52,64)(8,65,53,107,13,35,86,95,44,22,116,62,74)(9,75,63,117,23,45,96,87,36,14,108,54,66), (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)(100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117), (2,9)(3,8)(4,7)(5,6)(10,84,19,93)(11,83,20,92)(12,82,21,91)(13,99,22,90)(14,98,23,89)(15,97,24,88)(16,96,25,87)(17,95,26,86)(18,94,27,85)(28,67,37,76)(29,66,38,75)(30,65,39,74)(31,64,40,73)(32,81,41,72)(33,80,42,71)(34,79,43,70)(35,78,44,69)(36,77,45,68)(46,109,55,100)(47,108,56,117)(48,107,57,116)(49,106,58,115)(50,105,59,114)(51,104,60,113)(52,103,61,112)(53,102,62,111)(54,101,63,110) );
G=PermutationGroup([[(1,67,55,109,15,37,88,97,28,24,100,46,76),(2,77,47,101,25,29,98,89,38,16,110,56,68),(3,69,57,111,17,39,90,99,30,26,102,48,78),(4,79,49,103,27,31,82,91,40,18,112,58,70),(5,71,59,113,19,41,92,83,32,10,104,50,80),(6,81,51,105,11,33,84,93,42,20,114,60,72),(7,73,61,115,21,43,94,85,34,12,106,52,64),(8,65,53,107,13,35,86,95,44,22,116,62,74),(9,75,63,117,23,45,96,87,36,14,108,54,66)], [(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),(100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117)], [(2,9),(3,8),(4,7),(5,6),(10,84,19,93),(11,83,20,92),(12,82,21,91),(13,99,22,90),(14,98,23,89),(15,97,24,88),(16,96,25,87),(17,95,26,86),(18,94,27,85),(28,67,37,76),(29,66,38,75),(30,65,39,74),(31,64,40,73),(32,81,41,72),(33,80,42,71),(34,79,43,70),(35,78,44,69),(36,77,45,68),(46,109,55,100),(47,108,56,117),(48,107,57,116),(49,106,58,115),(50,105,59,114),(51,104,60,113),(52,103,61,112),(53,102,62,111),(54,101,63,110)]])
39 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 9A | 9B | 9C | 13A | 13B | 13C | 18A | 18B | 18C | 39A | ··· | 39F | 117A | ··· | 117R |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 9 | 9 | 9 | 13 | 13 | 13 | 18 | 18 | 18 | 39 | ··· | 39 | 117 | ··· | 117 |
| size | 1 | 13 | 2 | 117 | 117 | 26 | 2 | 2 | 2 | 4 | 4 | 4 | 26 | 26 | 26 | 4 | ··· | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | - | + | - | + | |||
| image | C1 | C2 | C4 | S3 | Dic3 | D9 | Dic9 | C13⋊C4 | C39⋊C4 | C13⋊Dic9 |
| kernel | C13⋊Dic9 | C9×D13 | C117 | C3×D13 | C39 | D13 | C13 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 3 | 3 | 3 | 6 | 18 |
Matrix representation of C13⋊Dic9 ►in GL4(𝔽937) generated by
| 928 | 496 | 1 | 0 |
| 441 | 432 | 0 | 1 |
| 936 | 0 | 0 | 0 |
| 0 | 936 | 0 | 0 |
| 675 | 203 | 55 | 846 |
| 734 | 472 | 91 | 146 |
| 0 | 0 | 262 | 734 |
| 0 | 0 | 203 | 465 |
| 0 | 1 | 361 | 372 |
| 1 | 0 | 11 | 576 |
| 0 | 0 | 684 | 131 |
| 0 | 0 | 384 | 253 |
G:=sub<GL(4,GF(937))| [928,441,936,0,496,432,0,936,1,0,0,0,0,1,0,0],[675,734,0,0,203,472,0,0,55,91,262,203,846,146,734,465],[0,1,0,0,1,0,0,0,361,11,684,384,372,576,131,253] >;
C13⋊Dic9 in GAP, Magma, Sage, TeX
C_{13}\rtimes {\rm Dic}_9 % in TeX
G:=Group("C13:Dic9"); // GroupNames label
G:=SmallGroup(468,10);
// by ID
G=gap.SmallGroup(468,10);
# by ID
G:=PCGroup([5,-2,-2,-3,-13,-3,10,2462,1182,1203,1448,7804]);
// Polycyclic
G:=Group<a,b,c|a^13=b^18=1,c^2=b^9,b*a*b^-1=a^-1,c*a*c^-1=a^8,c*b*c^-1=b^-1>;
// generators/relations
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