direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×C17⋊C4, C34⋊C12, D17⋊C12, D34.C6, C102⋊2C4, C17⋊(C2×C12), C51⋊3(C2×C4), D17.(C2×C6), (C3×D17)⋊3C4, (C6×D17).3C2, (C3×D17).3C22, SmallGroup(408,39)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C6×C17⋊C4 |
Generators and relations for C6×C17⋊C4
G = < a,b,c | a6=b17=c4=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C6×C17⋊C4
►On 102 points
Generators in S102
(1 69 35 52 18 86)(2 70 36 53 19 87)(3 71 37 54 20 88)(4 72 38 55 21 89)(5 73 39 56 22 90)(6 74 40 57 23 91)(7 75 41 58 24 92)(8 76 42 59 25 93)(9 77 43 60 26 94)(10 78 44 61 27 95)(11 79 45 62 28 96)(12 80 46 63 29 97)(13 81 47 64 30 98)(14 82 48 65 31 99)(15 83 49 66 32 100)(16 84 50 67 33 101)(17 85 51 68 34 102)
(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)
(1 52)(2 65 17 56)(3 61 16 60)(4 57 15 64)(5 53 14 68)(6 66 13 55)(7 62 12 59)(8 58 11 63)(9 54 10 67)(18 69)(19 82 34 73)(20 78 33 77)(21 74 32 81)(22 70 31 85)(23 83 30 72)(24 79 29 76)(25 75 28 80)(26 71 27 84)(35 86)(36 99 51 90)(37 95 50 94)(38 91 49 98)(39 87 48 102)(40 100 47 89)(41 96 46 93)(42 92 45 97)(43 88 44 101)
G:=sub<Sym(102)| (1,69,35,52,18,86)(2,70,36,53,19,87)(3,71,37,54,20,88)(4,72,38,55,21,89)(5,73,39,56,22,90)(6,74,40,57,23,91)(7,75,41,58,24,92)(8,76,42,59,25,93)(9,77,43,60,26,94)(10,78,44,61,27,95)(11,79,45,62,28,96)(12,80,46,63,29,97)(13,81,47,64,30,98)(14,82,48,65,31,99)(15,83,49,66,32,100)(16,84,50,67,33,101)(17,85,51,68,34,102), (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), (1,52)(2,65,17,56)(3,61,16,60)(4,57,15,64)(5,53,14,68)(6,66,13,55)(7,62,12,59)(8,58,11,63)(9,54,10,67)(18,69)(19,82,34,73)(20,78,33,77)(21,74,32,81)(22,70,31,85)(23,83,30,72)(24,79,29,76)(25,75,28,80)(26,71,27,84)(35,86)(36,99,51,90)(37,95,50,94)(38,91,49,98)(39,87,48,102)(40,100,47,89)(41,96,46,93)(42,92,45,97)(43,88,44,101)>;
G:=Group( (1,69,35,52,18,86)(2,70,36,53,19,87)(3,71,37,54,20,88)(4,72,38,55,21,89)(5,73,39,56,22,90)(6,74,40,57,23,91)(7,75,41,58,24,92)(8,76,42,59,25,93)(9,77,43,60,26,94)(10,78,44,61,27,95)(11,79,45,62,28,96)(12,80,46,63,29,97)(13,81,47,64,30,98)(14,82,48,65,31,99)(15,83,49,66,32,100)(16,84,50,67,33,101)(17,85,51,68,34,102), (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), (1,52)(2,65,17,56)(3,61,16,60)(4,57,15,64)(5,53,14,68)(6,66,13,55)(7,62,12,59)(8,58,11,63)(9,54,10,67)(18,69)(19,82,34,73)(20,78,33,77)(21,74,32,81)(22,70,31,85)(23,83,30,72)(24,79,29,76)(25,75,28,80)(26,71,27,84)(35,86)(36,99,51,90)(37,95,50,94)(38,91,49,98)(39,87,48,102)(40,100,47,89)(41,96,46,93)(42,92,45,97)(43,88,44,101) );
G=PermutationGroup([[(1,69,35,52,18,86),(2,70,36,53,19,87),(3,71,37,54,20,88),(4,72,38,55,21,89),(5,73,39,56,22,90),(6,74,40,57,23,91),(7,75,41,58,24,92),(8,76,42,59,25,93),(9,77,43,60,26,94),(10,78,44,61,27,95),(11,79,45,62,28,96),(12,80,46,63,29,97),(13,81,47,64,30,98),(14,82,48,65,31,99),(15,83,49,66,32,100),(16,84,50,67,33,101),(17,85,51,68,34,102)], [(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)], [(1,52),(2,65,17,56),(3,61,16,60),(4,57,15,64),(5,53,14,68),(6,66,13,55),(7,62,12,59),(8,58,11,63),(9,54,10,67),(18,69),(19,82,34,73),(20,78,33,77),(21,74,32,81),(22,70,31,85),(23,83,30,72),(24,79,29,76),(25,75,28,80),(26,71,27,84),(35,86),(36,99,51,90),(37,95,50,94),(38,91,49,98),(39,87,48,102),(40,100,47,89),(41,96,46,93),(42,92,45,97),(43,88,44,101)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 12A | ··· | 12H | 17A | 17B | 17C | 17D | 34A | 34B | 34C | 34D | 51A | ··· | 51H | 102A | ··· | 102H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 17 | 17 | 17 | 17 | 34 | 34 | 34 | 34 | 51 | ··· | 51 | 102 | ··· | 102 |
| size | 1 | 1 | 17 | 17 | 1 | 1 | 17 | 17 | 17 | 17 | 1 | 1 | 17 | 17 | 17 | 17 | 17 | ··· | 17 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | C17⋊C4 | C2×C17⋊C4 | C3×C17⋊C4 | C6×C17⋊C4 |
| kernel | C6×C17⋊C4 | C3×C17⋊C4 | C6×D17 | C2×C17⋊C4 | C3×D17 | C102 | C17⋊C4 | D34 | D17 | C34 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 4 | 8 | 8 |
Matrix representation of C6×C17⋊C4 ►in GL5(𝔽409)
| 408 | 0 | 0 | 0 | 0 |
| 0 | 53 | 0 | 0 | 0 |
| 0 | 0 | 53 | 0 | 0 |
| 0 | 0 | 0 | 53 | 0 |
| 0 | 0 | 0 | 0 | 53 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 342 | 38 | 253 | 104 |
| 0 | 1 | 0 | 0 | 51 |
| 0 | 0 | 1 | 0 | 341 |
| 0 | 0 | 0 | 1 | 50 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 52 | 345 | 336 | 296 |
| 0 | 13 | 369 | 365 | 272 |
| 0 | 185 | 59 | 361 | 353 |
| 0 | 210 | 363 | 107 | 36 |
G:=sub<GL(5,GF(409))| [408,0,0,0,0,0,53,0,0,0,0,0,53,0,0,0,0,0,53,0,0,0,0,0,53],[1,0,0,0,0,0,342,1,0,0,0,38,0,1,0,0,253,0,0,1,0,104,51,341,50],[1,0,0,0,0,0,52,13,185,210,0,345,369,59,363,0,336,365,361,107,0,296,272,353,36] >;
C6×C17⋊C4 in GAP, Magma, Sage, TeX
C_6\times C_{17}\rtimes C_4 % in TeX
G:=Group("C6xC17:C4"); // GroupNames label
G:=SmallGroup(408,39);
// by ID
G=gap.SmallGroup(408,39);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-17,60,7804,819]);
// Polycyclic
G:=Group<a,b,c|a^6=b^17=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
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