direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7×D17, C17⋊C14, C119⋊2C2, SmallGroup(238,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C7×D17 |
Generators and relations for C7×D17
G = < a,b,c | a7=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C7×D17
►On 119 points
Generators in S119
(1 115 88 75 60 48 32)(2 116 89 76 61 49 33)(3 117 90 77 62 50 34)(4 118 91 78 63 51 18)(5 119 92 79 64 35 19)(6 103 93 80 65 36 20)(7 104 94 81 66 37 21)(8 105 95 82 67 38 22)(9 106 96 83 68 39 23)(10 107 97 84 52 40 24)(11 108 98 85 53 41 25)(12 109 99 69 54 42 26)(13 110 100 70 55 43 27)(14 111 101 71 56 44 28)(15 112 102 72 57 45 29)(16 113 86 73 58 46 30)(17 114 87 74 59 47 31)
(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 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 28)(19 27)(20 26)(21 25)(22 24)(29 34)(30 33)(31 32)(35 43)(36 42)(37 41)(38 40)(44 51)(45 50)(46 49)(47 48)(52 67)(53 66)(54 65)(55 64)(56 63)(57 62)(58 61)(59 60)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)(81 85)(82 84)(86 89)(87 88)(90 102)(91 101)(92 100)(93 99)(94 98)(95 97)(103 109)(104 108)(105 107)(110 119)(111 118)(112 117)(113 116)(114 115)
G:=sub<Sym(119)| (1,115,88,75,60,48,32)(2,116,89,76,61,49,33)(3,117,90,77,62,50,34)(4,118,91,78,63,51,18)(5,119,92,79,64,35,19)(6,103,93,80,65,36,20)(7,104,94,81,66,37,21)(8,105,95,82,67,38,22)(9,106,96,83,68,39,23)(10,107,97,84,52,40,24)(11,108,98,85,53,41,25)(12,109,99,69,54,42,26)(13,110,100,70,55,43,27)(14,111,101,71,56,44,28)(15,112,102,72,57,45,29)(16,113,86,73,58,46,30)(17,114,87,74,59,47,31), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,28)(19,27)(20,26)(21,25)(22,24)(29,34)(30,33)(31,32)(35,43)(36,42)(37,41)(38,40)(44,51)(45,50)(46,49)(47,48)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)(81,85)(82,84)(86,89)(87,88)(90,102)(91,101)(92,100)(93,99)(94,98)(95,97)(103,109)(104,108)(105,107)(110,119)(111,118)(112,117)(113,116)(114,115)>;
G:=Group( (1,115,88,75,60,48,32)(2,116,89,76,61,49,33)(3,117,90,77,62,50,34)(4,118,91,78,63,51,18)(5,119,92,79,64,35,19)(6,103,93,80,65,36,20)(7,104,94,81,66,37,21)(8,105,95,82,67,38,22)(9,106,96,83,68,39,23)(10,107,97,84,52,40,24)(11,108,98,85,53,41,25)(12,109,99,69,54,42,26)(13,110,100,70,55,43,27)(14,111,101,71,56,44,28)(15,112,102,72,57,45,29)(16,113,86,73,58,46,30)(17,114,87,74,59,47,31), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,28)(19,27)(20,26)(21,25)(22,24)(29,34)(30,33)(31,32)(35,43)(36,42)(37,41)(38,40)(44,51)(45,50)(46,49)(47,48)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)(81,85)(82,84)(86,89)(87,88)(90,102)(91,101)(92,100)(93,99)(94,98)(95,97)(103,109)(104,108)(105,107)(110,119)(111,118)(112,117)(113,116)(114,115) );
G=PermutationGroup([[(1,115,88,75,60,48,32),(2,116,89,76,61,49,33),(3,117,90,77,62,50,34),(4,118,91,78,63,51,18),(5,119,92,79,64,35,19),(6,103,93,80,65,36,20),(7,104,94,81,66,37,21),(8,105,95,82,67,38,22),(9,106,96,83,68,39,23),(10,107,97,84,52,40,24),(11,108,98,85,53,41,25),(12,109,99,69,54,42,26),(13,110,100,70,55,43,27),(14,111,101,71,56,44,28),(15,112,102,72,57,45,29),(16,113,86,73,58,46,30),(17,114,87,74,59,47,31)], [(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)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(18,28),(19,27),(20,26),(21,25),(22,24),(29,34),(30,33),(31,32),(35,43),(36,42),(37,41),(38,40),(44,51),(45,50),(46,49),(47,48),(52,67),(53,66),(54,65),(55,64),(56,63),(57,62),(58,61),(59,60),(69,80),(70,79),(71,78),(72,77),(73,76),(74,75),(81,85),(82,84),(86,89),(87,88),(90,102),(91,101),(92,100),(93,99),(94,98),(95,97),(103,109),(104,108),(105,107),(110,119),(111,118),(112,117),(113,116),(114,115)]])
C7×D17 is a maximal subgroup of
C17⋊Dic7
70 conjugacy classes
| class | 1 | 2 | 7A | ··· | 7F | 14A | ··· | 14F | 17A | ··· | 17H | 119A | ··· | 119AV |
| order | 1 | 2 | 7 | ··· | 7 | 14 | ··· | 14 | 17 | ··· | 17 | 119 | ··· | 119 |
| size | 1 | 17 | 1 | ··· | 1 | 17 | ··· | 17 | 2 | ··· | 2 | 2 | ··· | 2 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C7 | C14 | D17 | C7×D17 |
| kernel | C7×D17 | C119 | D17 | C17 | C7 | C1 |
| # reps | 1 | 1 | 6 | 6 | 8 | 48 |
Matrix representation of C7×D17 ►in GL2(𝔽239) generated by
| 44 | 0 |
| 0 | 44 |
| 129 | 1 |
| 90 | 236 |
| 49 | 138 |
| 232 | 190 |
G:=sub<GL(2,GF(239))| [44,0,0,44],[129,90,1,236],[49,232,138,190] >;
C7×D17 in GAP, Magma, Sage, TeX
C_7\times D_{17} % in TeX
G:=Group("C7xD17"); // GroupNames label
G:=SmallGroup(238,2);
// by ID
G=gap.SmallGroup(238,2);
# by ID
G:=PCGroup([3,-2,-7,-17,2018]);
// Polycyclic
G:=Group<a,b,c|a^7=b^17=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export