metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C17⋊Dic7, C119⋊1C4, D17.D7, C7⋊(C17⋊C4), (C7×D17).1C2, SmallGroup(476,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C119 — C17⋊Dic7 |
Generators and relations for C17⋊Dic7
G = < a,b,c | a17=b14=1, c2=b7, bab-1=a-1, cac-1=a13, cbc-1=b-1 >
Smallest permutation representation of C17⋊Dic7
►On 119 points
Generators in S119
(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)(103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119)
(1 107 97 82 57 48 25)(2 106 98 81 58 47 26 17 108 96 83 56 49 24)(3 105 99 80 59 46 27 16 109 95 84 55 50 23)(4 104 100 79 60 45 28 15 110 94 85 54 51 22)(5 103 101 78 61 44 29 14 111 93 69 53 35 21)(6 119 102 77 62 43 30 13 112 92 70 52 36 20)(7 118 86 76 63 42 31 12 113 91 71 68 37 19)(8 117 87 75 64 41 32 11 114 90 72 67 38 18)(9 116 88 74 65 40 33 10 115 89 73 66 39 34)
(2 5 17 14)(3 9 16 10)(4 13 15 6)(7 8 12 11)(18 113 32 118)(19 117 31 114)(20 104 30 110)(21 108 29 106)(22 112 28 119)(23 116 27 115)(24 103 26 111)(25 107)(33 105 34 109)(35 96 44 98)(36 100 43 94)(37 87 42 90)(38 91 41 86)(39 95 40 99)(45 102 51 92)(46 89 50 88)(47 93 49 101)(48 97)(52 79 62 85)(53 83 61 81)(54 70 60 77)(55 74 59 73)(56 78 58 69)(57 82)(63 72 68 75)(64 76 67 71)(65 80 66 84)
G:=sub<Sym(119)| (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)(103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119), (1,107,97,82,57,48,25)(2,106,98,81,58,47,26,17,108,96,83,56,49,24)(3,105,99,80,59,46,27,16,109,95,84,55,50,23)(4,104,100,79,60,45,28,15,110,94,85,54,51,22)(5,103,101,78,61,44,29,14,111,93,69,53,35,21)(6,119,102,77,62,43,30,13,112,92,70,52,36,20)(7,118,86,76,63,42,31,12,113,91,71,68,37,19)(8,117,87,75,64,41,32,11,114,90,72,67,38,18)(9,116,88,74,65,40,33,10,115,89,73,66,39,34), (2,5,17,14)(3,9,16,10)(4,13,15,6)(7,8,12,11)(18,113,32,118)(19,117,31,114)(20,104,30,110)(21,108,29,106)(22,112,28,119)(23,116,27,115)(24,103,26,111)(25,107)(33,105,34,109)(35,96,44,98)(36,100,43,94)(37,87,42,90)(38,91,41,86)(39,95,40,99)(45,102,51,92)(46,89,50,88)(47,93,49,101)(48,97)(52,79,62,85)(53,83,61,81)(54,70,60,77)(55,74,59,73)(56,78,58,69)(57,82)(63,72,68,75)(64,76,67,71)(65,80,66,84)>;
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)(103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119), (1,107,97,82,57,48,25)(2,106,98,81,58,47,26,17,108,96,83,56,49,24)(3,105,99,80,59,46,27,16,109,95,84,55,50,23)(4,104,100,79,60,45,28,15,110,94,85,54,51,22)(5,103,101,78,61,44,29,14,111,93,69,53,35,21)(6,119,102,77,62,43,30,13,112,92,70,52,36,20)(7,118,86,76,63,42,31,12,113,91,71,68,37,19)(8,117,87,75,64,41,32,11,114,90,72,67,38,18)(9,116,88,74,65,40,33,10,115,89,73,66,39,34), (2,5,17,14)(3,9,16,10)(4,13,15,6)(7,8,12,11)(18,113,32,118)(19,117,31,114)(20,104,30,110)(21,108,29,106)(22,112,28,119)(23,116,27,115)(24,103,26,111)(25,107)(33,105,34,109)(35,96,44,98)(36,100,43,94)(37,87,42,90)(38,91,41,86)(39,95,40,99)(45,102,51,92)(46,89,50,88)(47,93,49,101)(48,97)(52,79,62,85)(53,83,61,81)(54,70,60,77)(55,74,59,73)(56,78,58,69)(57,82)(63,72,68,75)(64,76,67,71)(65,80,66,84) );
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),(103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119)], [(1,107,97,82,57,48,25),(2,106,98,81,58,47,26,17,108,96,83,56,49,24),(3,105,99,80,59,46,27,16,109,95,84,55,50,23),(4,104,100,79,60,45,28,15,110,94,85,54,51,22),(5,103,101,78,61,44,29,14,111,93,69,53,35,21),(6,119,102,77,62,43,30,13,112,92,70,52,36,20),(7,118,86,76,63,42,31,12,113,91,71,68,37,19),(8,117,87,75,64,41,32,11,114,90,72,67,38,18),(9,116,88,74,65,40,33,10,115,89,73,66,39,34)], [(2,5,17,14),(3,9,16,10),(4,13,15,6),(7,8,12,11),(18,113,32,118),(19,117,31,114),(20,104,30,110),(21,108,29,106),(22,112,28,119),(23,116,27,115),(24,103,26,111),(25,107),(33,105,34,109),(35,96,44,98),(36,100,43,94),(37,87,42,90),(38,91,41,86),(39,95,40,99),(45,102,51,92),(46,89,50,88),(47,93,49,101),(48,97),(52,79,62,85),(53,83,61,81),(54,70,60,77),(55,74,59,73),(56,78,58,69),(57,82),(63,72,68,75),(64,76,67,71),(65,80,66,84)]])
38 conjugacy classes
| class | 1 | 2 | 4A | 4B | 7A | 7B | 7C | 14A | 14B | 14C | 17A | 17B | 17C | 17D | 119A | ··· | 119X |
| order | 1 | 2 | 4 | 4 | 7 | 7 | 7 | 14 | 14 | 14 | 17 | 17 | 17 | 17 | 119 | ··· | 119 |
| size | 1 | 17 | 119 | 119 | 2 | 2 | 2 | 34 | 34 | 34 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | - | + | ||
| image | C1 | C2 | C4 | D7 | Dic7 | C17⋊C4 | C17⋊Dic7 |
| kernel | C17⋊Dic7 | C7×D17 | C119 | D17 | C17 | C7 | C1 |
| # reps | 1 | 1 | 2 | 3 | 3 | 4 | 24 |
Matrix representation of C17⋊Dic7 ►in GL6(𝔽953)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 727 | 1 | 0 | 0 |
| 0 | 0 | 368 | 0 | 1 | 0 |
| 0 | 0 | 728 | 0 | 0 | 1 |
| 0 | 0 | 703 | 826 | 827 | 829 |
| 0 | 952 | 0 | 0 | 0 | 0 |
| 1 | 499 | 0 | 0 | 0 | 0 |
| 0 | 0 | 251 | 478 | 268 | 351 |
| 0 | 0 | 850 | 481 | 756 | 226 |
| 0 | 0 | 261 | 281 | 578 | 70 |
| 0 | 0 | 481 | 624 | 29 | 596 |
| 442 | 0 | 0 | 0 | 0 | 0 |
| 538 | 511 | 0 | 0 | 0 | 0 |
| 0 | 0 | 936 | 294 | 169 | 190 |
| 0 | 0 | 671 | 748 | 730 | 553 |
| 0 | 0 | 394 | 578 | 929 | 198 |
| 0 | 0 | 497 | 494 | 719 | 246 |
G:=sub<GL(6,GF(953))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,727,368,728,703,0,0,1,0,0,826,0,0,0,1,0,827,0,0,0,0,1,829],[0,1,0,0,0,0,952,499,0,0,0,0,0,0,251,850,261,481,0,0,478,481,281,624,0,0,268,756,578,29,0,0,351,226,70,596],[442,538,0,0,0,0,0,511,0,0,0,0,0,0,936,671,394,497,0,0,294,748,578,494,0,0,169,730,929,719,0,0,190,553,198,246] >;
C17⋊Dic7 in GAP, Magma, Sage, TeX
C_{17}\rtimes {\rm Dic}_7 % in TeX
G:=Group("C17:Dic7"); // GroupNames label
G:=SmallGroup(476,6);
// by ID
G=gap.SmallGroup(476,6);
# by ID
G:=PCGroup([4,-2,-2,-7,-17,8,290,1795,3591]);
// Polycyclic
G:=Group<a,b,c|a^17=b^14=1,c^2=b^7,b*a*b^-1=a^-1,c*a*c^-1=a^13,c*b*c^-1=b^-1>;
// generators/relations
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