direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D7×D17, C7⋊1D34, D119⋊C2, C17⋊1D14, C119⋊C22, (C7×D17)⋊C2, (D7×C17)⋊C2, SmallGroup(476,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C119 — D7×D17 |
Generators and relations for D7×D17
G = < a,b,c,d | a7=b2=c17=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of D7×D17
►On 119 points
Generators in S119
(1 113 96 74 58 42 28)(2 114 97 75 59 43 29)(3 115 98 76 60 44 30)(4 116 99 77 61 45 31)(5 117 100 78 62 46 32)(6 118 101 79 63 47 33)(7 119 102 80 64 48 34)(8 103 86 81 65 49 18)(9 104 87 82 66 50 19)(10 105 88 83 67 51 20)(11 106 89 84 68 35 21)(12 107 90 85 52 36 22)(13 108 91 69 53 37 23)(14 109 92 70 54 38 24)(15 110 93 71 55 39 25)(16 111 94 72 56 40 26)(17 112 95 73 57 41 27)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 18)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(35 106)(36 107)(37 108)(38 109)(39 110)(40 111)(41 112)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 103)(50 104)(51 105)(52 90)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 97)(60 98)(61 99)(62 100)(63 101)(64 102)(65 86)(66 87)(67 88)(68 89)
(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 20)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)(49 51)(52 63)(53 62)(54 61)(55 60)(56 59)(57 58)(64 68)(65 67)(69 78)(70 77)(71 76)(72 75)(73 74)(79 85)(80 84)(81 83)(86 88)(89 102)(90 101)(91 100)(92 99)(93 98)(94 97)(95 96)(103 105)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)
G:=sub<Sym(119)| (1,113,96,74,58,42,28)(2,114,97,75,59,43,29)(3,115,98,76,60,44,30)(4,116,99,77,61,45,31)(5,117,100,78,62,46,32)(6,118,101,79,63,47,33)(7,119,102,80,64,48,34)(8,103,86,81,65,49,18)(9,104,87,82,66,50,19)(10,105,88,83,67,51,20)(11,106,89,84,68,35,21)(12,107,90,85,52,36,22)(13,108,91,69,53,37,23)(14,109,92,70,54,38,24)(15,110,93,71,55,39,25)(16,111,94,72,56,40,26)(17,112,95,73,57,41,27), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,18)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(41,112)(42,113)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(49,103)(50,104)(51,105)(52,90)(53,91)(54,92)(55,93)(56,94)(57,95)(58,96)(59,97)(60,98)(61,99)(62,100)(63,101)(64,102)(65,86)(66,87)(67,88)(68,89), (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)(18,20)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(49,51)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(64,68)(65,67)(69,78)(70,77)(71,76)(72,75)(73,74)(79,85)(80,84)(81,83)(86,88)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)(103,105)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)>;
G:=Group( (1,113,96,74,58,42,28)(2,114,97,75,59,43,29)(3,115,98,76,60,44,30)(4,116,99,77,61,45,31)(5,117,100,78,62,46,32)(6,118,101,79,63,47,33)(7,119,102,80,64,48,34)(8,103,86,81,65,49,18)(9,104,87,82,66,50,19)(10,105,88,83,67,51,20)(11,106,89,84,68,35,21)(12,107,90,85,52,36,22)(13,108,91,69,53,37,23)(14,109,92,70,54,38,24)(15,110,93,71,55,39,25)(16,111,94,72,56,40,26)(17,112,95,73,57,41,27), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,18)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(41,112)(42,113)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(49,103)(50,104)(51,105)(52,90)(53,91)(54,92)(55,93)(56,94)(57,95)(58,96)(59,97)(60,98)(61,99)(62,100)(63,101)(64,102)(65,86)(66,87)(67,88)(68,89), (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)(18,20)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(49,51)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)(64,68)(65,67)(69,78)(70,77)(71,76)(72,75)(73,74)(79,85)(80,84)(81,83)(86,88)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)(103,105)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113) );
G=PermutationGroup([[(1,113,96,74,58,42,28),(2,114,97,75,59,43,29),(3,115,98,76,60,44,30),(4,116,99,77,61,45,31),(5,117,100,78,62,46,32),(6,118,101,79,63,47,33),(7,119,102,80,64,48,34),(8,103,86,81,65,49,18),(9,104,87,82,66,50,19),(10,105,88,83,67,51,20),(11,106,89,84,68,35,21),(12,107,90,85,52,36,22),(13,108,91,69,53,37,23),(14,109,92,70,54,38,24),(15,110,93,71,55,39,25),(16,111,94,72,56,40,26),(17,112,95,73,57,41,27)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,18),(9,19),(10,20),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(35,106),(36,107),(37,108),(38,109),(39,110),(40,111),(41,112),(42,113),(43,114),(44,115),(45,116),(46,117),(47,118),(48,119),(49,103),(50,104),(51,105),(52,90),(53,91),(54,92),(55,93),(56,94),(57,95),(58,96),(59,97),(60,98),(61,99),(62,100),(63,101),(64,102),(65,86),(66,87),(67,88),(68,89)], [(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),(18,20),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43),(41,42),(49,51),(52,63),(53,62),(54,61),(55,60),(56,59),(57,58),(64,68),(65,67),(69,78),(70,77),(71,76),(72,75),(73,74),(79,85),(80,84),(81,83),(86,88),(89,102),(90,101),(91,100),(92,99),(93,98),(94,97),(95,96),(103,105),(106,119),(107,118),(108,117),(109,116),(110,115),(111,114),(112,113)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 7A | 7B | 7C | 14A | 14B | 14C | 17A | ··· | 17H | 34A | ··· | 34H | 119A | ··· | 119X |
| order | 1 | 2 | 2 | 2 | 7 | 7 | 7 | 14 | 14 | 14 | 17 | ··· | 17 | 34 | ··· | 34 | 119 | ··· | 119 |
| size | 1 | 7 | 17 | 119 | 2 | 2 | 2 | 34 | 34 | 34 | 2 | ··· | 2 | 14 | ··· | 14 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | D7 | D14 | D17 | D34 | D7×D17 |
| kernel | D7×D17 | D7×C17 | C7×D17 | D119 | D17 | C17 | D7 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 3 | 3 | 8 | 8 | 24 |
Matrix representation of D7×D17 ►in GL4(𝔽239) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 238 | 34 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 238 | 100 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(239))| [1,0,0,0,0,1,0,0,0,0,0,238,0,0,1,34],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[0,238,0,0,1,100,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;
D7×D17 in GAP, Magma, Sage, TeX
D_7\times D_{17} % in TeX
G:=Group("D7xD17"); // GroupNames label
G:=SmallGroup(476,7);
// by ID
G=gap.SmallGroup(476,7);
# by ID
G:=PCGroup([4,-2,-2,-7,-17,150,7171]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^2=c^17=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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