direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D7×C17, C7⋊C34, C119⋊3C2, SmallGroup(238,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C17 |
Generators and relations for D7×C17
G = < a,b,c | a17=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D7×C17
►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)
(1 117 50 64 99 82 27)(2 118 51 65 100 83 28)(3 119 35 66 101 84 29)(4 103 36 67 102 85 30)(5 104 37 68 86 69 31)(6 105 38 52 87 70 32)(7 106 39 53 88 71 33)(8 107 40 54 89 72 34)(9 108 41 55 90 73 18)(10 109 42 56 91 74 19)(11 110 43 57 92 75 20)(12 111 44 58 93 76 21)(13 112 45 59 94 77 22)(14 113 46 60 95 78 23)(15 114 47 61 96 79 24)(16 115 48 62 97 80 25)(17 116 49 63 98 81 26)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(35 101)(36 102)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(50 99)(51 100)(69 104)(70 105)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 113)(79 114)(80 115)(81 116)(82 117)(83 118)(84 119)(85 103)
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), (1,117,50,64,99,82,27)(2,118,51,65,100,83,28)(3,119,35,66,101,84,29)(4,103,36,67,102,85,30)(5,104,37,68,86,69,31)(6,105,38,52,87,70,32)(7,106,39,53,88,71,33)(8,107,40,54,89,72,34)(9,108,41,55,90,73,18)(10,109,42,56,91,74,19)(11,110,43,57,92,75,20)(12,111,44,58,93,76,21)(13,112,45,59,94,77,22)(14,113,46,60,95,78,23)(15,114,47,61,96,79,24)(16,115,48,62,97,80,25)(17,116,49,63,98,81,26), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(35,101)(36,102)(37,86)(38,87)(39,88)(40,89)(41,90)(42,91)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,99)(51,100)(69,104)(70,105)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,113)(79,114)(80,115)(81,116)(82,117)(83,118)(84,119)(85,103)>;
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), (1,117,50,64,99,82,27)(2,118,51,65,100,83,28)(3,119,35,66,101,84,29)(4,103,36,67,102,85,30)(5,104,37,68,86,69,31)(6,105,38,52,87,70,32)(7,106,39,53,88,71,33)(8,107,40,54,89,72,34)(9,108,41,55,90,73,18)(10,109,42,56,91,74,19)(11,110,43,57,92,75,20)(12,111,44,58,93,76,21)(13,112,45,59,94,77,22)(14,113,46,60,95,78,23)(15,114,47,61,96,79,24)(16,115,48,62,97,80,25)(17,116,49,63,98,81,26), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(35,101)(36,102)(37,86)(38,87)(39,88)(40,89)(41,90)(42,91)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,99)(51,100)(69,104)(70,105)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,113)(79,114)(80,115)(81,116)(82,117)(83,118)(84,119)(85,103) );
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)], [(1,117,50,64,99,82,27),(2,118,51,65,100,83,28),(3,119,35,66,101,84,29),(4,103,36,67,102,85,30),(5,104,37,68,86,69,31),(6,105,38,52,87,70,32),(7,106,39,53,88,71,33),(8,107,40,54,89,72,34),(9,108,41,55,90,73,18),(10,109,42,56,91,74,19),(11,110,43,57,92,75,20),(12,111,44,58,93,76,21),(13,112,45,59,94,77,22),(14,113,46,60,95,78,23),(15,114,47,61,96,79,24),(16,115,48,62,97,80,25),(17,116,49,63,98,81,26)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,18),(10,19),(11,20),(12,21),(13,22),(14,23),(15,24),(16,25),(17,26),(35,101),(36,102),(37,86),(38,87),(39,88),(40,89),(41,90),(42,91),(43,92),(44,93),(45,94),(46,95),(47,96),(48,97),(49,98),(50,99),(51,100),(69,104),(70,105),(71,106),(72,107),(73,108),(74,109),(75,110),(76,111),(77,112),(78,113),(79,114),(80,115),(81,116),(82,117),(83,118),(84,119),(85,103)]])
85 conjugacy classes
| class | 1 | 2 | 7A | 7B | 7C | 17A | ··· | 17P | 34A | ··· | 34P | 119A | ··· | 119AV |
| order | 1 | 2 | 7 | 7 | 7 | 17 | ··· | 17 | 34 | ··· | 34 | 119 | ··· | 119 |
| size | 1 | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C17 | C34 | D7 | D7×C17 |
| kernel | D7×C17 | C119 | D7 | C7 | C17 | C1 |
| # reps | 1 | 1 | 16 | 16 | 3 | 48 |
Matrix representation of D7×C17 ►in GL2(𝔽239) generated by
| 75 | 0 |
| 0 | 75 |
| 232 | 49 |
| 238 | 41 |
| 232 | 48 |
| 238 | 7 |
G:=sub<GL(2,GF(239))| [75,0,0,75],[232,238,49,41],[232,238,48,7] >;
D7×C17 in GAP, Magma, Sage, TeX
D_7\times C_{17} % in TeX
G:=Group("D7xC17"); // GroupNames label
G:=SmallGroup(238,1);
// by ID
G=gap.SmallGroup(238,1);
# by ID
G:=PCGroup([3,-2,-17,-7,1838]);
// Polycyclic
G:=Group<a,b,c|a^17=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export