direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7×C17⋊C4, C17⋊C28, C119⋊2C4, D17.C14, (C7×D17).2C2, SmallGroup(476,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C7×C17⋊C4 |
Generators and relations for C7×C17⋊C4
G = < a,b,c | a7=b17=c4=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C7×C17⋊C4
►On 119 points
Generators in S119
(1 103 86 69 52 35 18)(2 104 87 70 53 36 19)(3 105 88 71 54 37 20)(4 106 89 72 55 38 21)(5 107 90 73 56 39 22)(6 108 91 74 57 40 23)(7 109 92 75 58 41 24)(8 110 93 76 59 42 25)(9 111 94 77 60 43 26)(10 112 95 78 61 44 27)(11 113 96 79 62 45 28)(12 114 97 80 63 46 29)(13 115 98 81 64 47 30)(14 116 99 82 65 48 31)(15 117 100 83 66 49 32)(16 118 101 84 67 50 33)(17 119 102 85 68 51 34)
(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)
(2 14 17 5)(3 10 16 9)(4 6 15 13)(7 11 12 8)(19 31 34 22)(20 27 33 26)(21 23 32 30)(24 28 29 25)(36 48 51 39)(37 44 50 43)(38 40 49 47)(41 45 46 42)(53 65 68 56)(54 61 67 60)(55 57 66 64)(58 62 63 59)(70 82 85 73)(71 78 84 77)(72 74 83 81)(75 79 80 76)(87 99 102 90)(88 95 101 94)(89 91 100 98)(92 96 97 93)(104 116 119 107)(105 112 118 111)(106 108 117 115)(109 113 114 110)
G:=sub<Sym(119)| (1,103,86,69,52,35,18)(2,104,87,70,53,36,19)(3,105,88,71,54,37,20)(4,106,89,72,55,38,21)(5,107,90,73,56,39,22)(6,108,91,74,57,40,23)(7,109,92,75,58,41,24)(8,110,93,76,59,42,25)(9,111,94,77,60,43,26)(10,112,95,78,61,44,27)(11,113,96,79,62,45,28)(12,114,97,80,63,46,29)(13,115,98,81,64,47,30)(14,116,99,82,65,48,31)(15,117,100,83,66,49,32)(16,118,101,84,67,50,33)(17,119,102,85,68,51,34), (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), (2,14,17,5)(3,10,16,9)(4,6,15,13)(7,11,12,8)(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)(36,48,51,39)(37,44,50,43)(38,40,49,47)(41,45,46,42)(53,65,68,56)(54,61,67,60)(55,57,66,64)(58,62,63,59)(70,82,85,73)(71,78,84,77)(72,74,83,81)(75,79,80,76)(87,99,102,90)(88,95,101,94)(89,91,100,98)(92,96,97,93)(104,116,119,107)(105,112,118,111)(106,108,117,115)(109,113,114,110)>;
G:=Group( (1,103,86,69,52,35,18)(2,104,87,70,53,36,19)(3,105,88,71,54,37,20)(4,106,89,72,55,38,21)(5,107,90,73,56,39,22)(6,108,91,74,57,40,23)(7,109,92,75,58,41,24)(8,110,93,76,59,42,25)(9,111,94,77,60,43,26)(10,112,95,78,61,44,27)(11,113,96,79,62,45,28)(12,114,97,80,63,46,29)(13,115,98,81,64,47,30)(14,116,99,82,65,48,31)(15,117,100,83,66,49,32)(16,118,101,84,67,50,33)(17,119,102,85,68,51,34), (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), (2,14,17,5)(3,10,16,9)(4,6,15,13)(7,11,12,8)(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)(36,48,51,39)(37,44,50,43)(38,40,49,47)(41,45,46,42)(53,65,68,56)(54,61,67,60)(55,57,66,64)(58,62,63,59)(70,82,85,73)(71,78,84,77)(72,74,83,81)(75,79,80,76)(87,99,102,90)(88,95,101,94)(89,91,100,98)(92,96,97,93)(104,116,119,107)(105,112,118,111)(106,108,117,115)(109,113,114,110) );
G=PermutationGroup([[(1,103,86,69,52,35,18),(2,104,87,70,53,36,19),(3,105,88,71,54,37,20),(4,106,89,72,55,38,21),(5,107,90,73,56,39,22),(6,108,91,74,57,40,23),(7,109,92,75,58,41,24),(8,110,93,76,59,42,25),(9,111,94,77,60,43,26),(10,112,95,78,61,44,27),(11,113,96,79,62,45,28),(12,114,97,80,63,46,29),(13,115,98,81,64,47,30),(14,116,99,82,65,48,31),(15,117,100,83,66,49,32),(16,118,101,84,67,50,33),(17,119,102,85,68,51,34)], [(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)], [(2,14,17,5),(3,10,16,9),(4,6,15,13),(7,11,12,8),(19,31,34,22),(20,27,33,26),(21,23,32,30),(24,28,29,25),(36,48,51,39),(37,44,50,43),(38,40,49,47),(41,45,46,42),(53,65,68,56),(54,61,67,60),(55,57,66,64),(58,62,63,59),(70,82,85,73),(71,78,84,77),(72,74,83,81),(75,79,80,76),(87,99,102,90),(88,95,101,94),(89,91,100,98),(92,96,97,93),(104,116,119,107),(105,112,118,111),(106,108,117,115),(109,113,114,110)]])
56 conjugacy classes
| class | 1 | 2 | 4A | 4B | 7A | ··· | 7F | 14A | ··· | 14F | 17A | 17B | 17C | 17D | 28A | ··· | 28L | 119A | ··· | 119X |
| order | 1 | 2 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 17 | 17 | 17 | 17 | 28 | ··· | 28 | 119 | ··· | 119 |
| size | 1 | 17 | 17 | 17 | 1 | ··· | 1 | 17 | ··· | 17 | 4 | 4 | 4 | 4 | 17 | ··· | 17 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | |||||
| image | C1 | C2 | C4 | C7 | C14 | C28 | C17⋊C4 | C7×C17⋊C4 |
| kernel | C7×C17⋊C4 | C7×D17 | C119 | C17⋊C4 | D17 | C17 | C7 | C1 |
| # reps | 1 | 1 | 2 | 6 | 6 | 12 | 4 | 24 |
Matrix representation of C7×C17⋊C4 ►in GL4(𝔽953) generated by
| 879 | 0 | 0 | 0 |
| 0 | 879 | 0 | 0 |
| 0 | 0 | 879 | 0 |
| 0 | 0 | 0 | 879 |
| 933 | 348 | 933 | 952 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 952 | 933 | 348 | 933 |
| 952 | 1 | 20 | 604 |
| 604 | 20 | 1 | 952 |
G:=sub<GL(4,GF(953))| [879,0,0,0,0,879,0,0,0,0,879,0,0,0,0,879],[933,1,0,0,348,0,1,0,933,0,0,1,952,0,0,0],[1,952,952,604,0,933,1,20,0,348,20,1,0,933,604,952] >;
C7×C17⋊C4 in GAP, Magma, Sage, TeX
C_7\times C_{17}\rtimes C_4 % in TeX
G:=Group("C7xC17:C4"); // GroupNames label
G:=SmallGroup(476,5);
// by ID
G=gap.SmallGroup(476,5);
# by ID
G:=PCGroup([4,-2,-7,-2,-17,56,5827,523]);
// Polycyclic
G:=Group<a,b,c|a^7=b^17=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
Export