direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C5xC5:C16, C5:C80, C10.C40, C52:2C16, C20.2C20, C20.16F5, C4.2(C5xF5), C10.6(C5:C8), (C5xC10).2C8, (C5xC20).5C4, C5:2C8.2C10, C2.(C5xC5:C8), (C5xC5:2C8).1C2, SmallGroup(400,56)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C5xC5:C16 |
Generators and relations for C5xC5:C16
G = < a,b,c | a5=b5=c16=1, ab=ba, ac=ca, cbc-1=b3 >
Quotients: C1, C2, C4, C5, C8, C10, C16, C20, F5, C40, C5:C8, C80, C5:C16, C5xF5, C5xC5:C8, C5xC5:C16
Smallest permutation representation of C5xC5:C16
►On 80 points
Generators in S80
(1 39 69 28 58)(2 40 70 29 59)(3 41 71 30 60)(4 42 72 31 61)(5 43 73 32 62)(6 44 74 17 63)(7 45 75 18 64)(8 46 76 19 49)(9 47 77 20 50)(10 48 78 21 51)(11 33 79 22 52)(12 34 80 23 53)(13 35 65 24 54)(14 36 66 25 55)(15 37 67 26 56)(16 38 68 27 57)
(1 39 69 28 58)(2 29 40 59 70)(3 60 30 71 41)(4 72 61 42 31)(5 43 73 32 62)(6 17 44 63 74)(7 64 18 75 45)(8 76 49 46 19)(9 47 77 20 50)(10 21 48 51 78)(11 52 22 79 33)(12 80 53 34 23)(13 35 65 24 54)(14 25 36 55 66)(15 56 26 67 37)(16 68 57 38 27)
(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)
G:=sub<Sym(80)| (1,39,69,28,58)(2,40,70,29,59)(3,41,71,30,60)(4,42,72,31,61)(5,43,73,32,62)(6,44,74,17,63)(7,45,75,18,64)(8,46,76,19,49)(9,47,77,20,50)(10,48,78,21,51)(11,33,79,22,52)(12,34,80,23,53)(13,35,65,24,54)(14,36,66,25,55)(15,37,67,26,56)(16,38,68,27,57), (1,39,69,28,58)(2,29,40,59,70)(3,60,30,71,41)(4,72,61,42,31)(5,43,73,32,62)(6,17,44,63,74)(7,64,18,75,45)(8,76,49,46,19)(9,47,77,20,50)(10,21,48,51,78)(11,52,22,79,33)(12,80,53,34,23)(13,35,65,24,54)(14,25,36,55,66)(15,56,26,67,37)(16,68,57,38,27), (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)>;
G:=Group( (1,39,69,28,58)(2,40,70,29,59)(3,41,71,30,60)(4,42,72,31,61)(5,43,73,32,62)(6,44,74,17,63)(7,45,75,18,64)(8,46,76,19,49)(9,47,77,20,50)(10,48,78,21,51)(11,33,79,22,52)(12,34,80,23,53)(13,35,65,24,54)(14,36,66,25,55)(15,37,67,26,56)(16,38,68,27,57), (1,39,69,28,58)(2,29,40,59,70)(3,60,30,71,41)(4,72,61,42,31)(5,43,73,32,62)(6,17,44,63,74)(7,64,18,75,45)(8,76,49,46,19)(9,47,77,20,50)(10,21,48,51,78)(11,52,22,79,33)(12,80,53,34,23)(13,35,65,24,54)(14,25,36,55,66)(15,56,26,67,37)(16,68,57,38,27), (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) );
G=PermutationGroup([[(1,39,69,28,58),(2,40,70,29,59),(3,41,71,30,60),(4,42,72,31,61),(5,43,73,32,62),(6,44,74,17,63),(7,45,75,18,64),(8,46,76,19,49),(9,47,77,20,50),(10,48,78,21,51),(11,33,79,22,52),(12,34,80,23,53),(13,35,65,24,54),(14,36,66,25,55),(15,37,67,26,56),(16,38,68,27,57)], [(1,39,69,28,58),(2,29,40,59,70),(3,60,30,71,41),(4,72,61,42,31),(5,43,73,32,62),(6,17,44,63,74),(7,64,18,75,45),(8,76,49,46,19),(9,47,77,20,50),(10,21,48,51,78),(11,52,22,79,33),(12,80,53,34,23),(13,35,65,24,54),(14,25,36,55,66),(15,56,26,67,37),(16,68,57,38,27)], [(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)]])
100 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | ··· | 5I | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | ··· | 10I | 16A | ··· | 16H | 20A | ··· | 20H | 20I | ··· | 20R | 40A | ··· | 40P | 80A | ··· | 80AF |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 5 | ··· | 5 | 1 | ··· | 1 | 4 | ··· | 4 | 5 | ··· | 5 | 5 | ··· | 5 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | ||||||||||||
| image | C1 | C2 | C4 | C5 | C8 | C10 | C16 | C20 | C40 | C80 | F5 | C5:C8 | C5:C16 | C5xF5 | C5xC5:C8 | C5xC5:C16 |
| kernel | C5xC5:C16 | C5xC5:2C8 | C5xC20 | C5:C16 | C5xC10 | C5:2C8 | C52 | C20 | C10 | C5 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 16 | 32 | 1 | 1 | 2 | 4 | 4 | 8 |
Matrix representation of C5xC5:C16 ►in GL5(F241)
| 87 | 0 | 0 | 0 | 0 |
| 0 | 87 | 0 | 0 | 0 |
| 0 | 0 | 87 | 0 | 0 |
| 0 | 0 | 0 | 87 | 0 |
| 0 | 0 | 0 | 0 | 87 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 87 | 0 | 0 | 0 |
| 0 | 88 | 205 | 0 | 0 |
| 0 | 98 | 0 | 98 | 0 |
| 0 | 189 | 0 | 0 | 91 |
| 76 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 204 | 0 |
| 0 | 0 | 0 | 240 | 1 |
| 0 | 0 | 1 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(241))| [87,0,0,0,0,0,87,0,0,0,0,0,87,0,0,0,0,0,87,0,0,0,0,0,87],[1,0,0,0,0,0,87,88,98,189,0,0,205,0,0,0,0,0,98,0,0,0,0,0,91],[76,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,204,240,1,1,0,0,1,0,0] >;
C5xC5:C16 in GAP, Magma, Sage, TeX
C_5\times C_5\rtimes C_{16} % in TeX
G:=Group("C5xC5:C16"); // GroupNames label
G:=SmallGroup(400,56);
// by ID
G=gap.SmallGroup(400,56);
# by ID
G:=PCGroup([6,-2,-5,-2,-2,-2,-5,60,50,69,5765,1169]);
// Polycyclic
G:=Group<a,b,c|a^5=b^5=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
Export