metabelian, supersoluble, monomial, A-group
Aliases: C52:5C16, C20.9F5, C5:2(C5:C16), C10.4(C5:C8), (C5xC10).5C8, (C5xC20).8C4, C2.(C52:5C8), C52:7C8.4C2, C4.2(C52:C4), SmallGroup(400,59)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C52:5C16 |
Generators and relations for C52:5C16
G = < a,b,c | a5=b5=c16=1, ab=ba, cac-1=a2, cbc-1=b3 >
Quotients: C1, C2, C4, C8, C16, F5, C5:C8, C5:C16, C52:C4, C52:5C8, C52:5C16
Smallest permutation representation of C52:5C16
►On 80 points
Generators in S80
(1 20 62 69 44)(2 63 45 21 70)(3 46 71 64 22)(4 72 23 47 49)(5 24 50 73 48)(6 51 33 25 74)(7 34 75 52 26)(8 76 27 35 53)(9 28 54 77 36)(10 55 37 29 78)(11 38 79 56 30)(12 80 31 39 57)(13 32 58 65 40)(14 59 41 17 66)(15 42 67 60 18)(16 68 19 43 61)
(1 44 69 62 20)(2 63 45 21 70)(3 22 64 71 46)(4 72 23 47 49)(5 48 73 50 24)(6 51 33 25 74)(7 26 52 75 34)(8 76 27 35 53)(9 36 77 54 28)(10 55 37 29 78)(11 30 56 79 38)(12 80 31 39 57)(13 40 65 58 32)(14 59 41 17 66)(15 18 60 67 42)(16 68 19 43 61)
(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)| (1,20,62,69,44)(2,63,45,21,70)(3,46,71,64,22)(4,72,23,47,49)(5,24,50,73,48)(6,51,33,25,74)(7,34,75,52,26)(8,76,27,35,53)(9,28,54,77,36)(10,55,37,29,78)(11,38,79,56,30)(12,80,31,39,57)(13,32,58,65,40)(14,59,41,17,66)(15,42,67,60,18)(16,68,19,43,61), (1,44,69,62,20)(2,63,45,21,70)(3,22,64,71,46)(4,72,23,47,49)(5,48,73,50,24)(6,51,33,25,74)(7,26,52,75,34)(8,76,27,35,53)(9,36,77,54,28)(10,55,37,29,78)(11,30,56,79,38)(12,80,31,39,57)(13,40,65,58,32)(14,59,41,17,66)(15,18,60,67,42)(16,68,19,43,61), (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( (1,20,62,69,44)(2,63,45,21,70)(3,46,71,64,22)(4,72,23,47,49)(5,24,50,73,48)(6,51,33,25,74)(7,34,75,52,26)(8,76,27,35,53)(9,28,54,77,36)(10,55,37,29,78)(11,38,79,56,30)(12,80,31,39,57)(13,32,58,65,40)(14,59,41,17,66)(15,42,67,60,18)(16,68,19,43,61), (1,44,69,62,20)(2,63,45,21,70)(3,22,64,71,46)(4,72,23,47,49)(5,48,73,50,24)(6,51,33,25,74)(7,26,52,75,34)(8,76,27,35,53)(9,36,77,54,28)(10,55,37,29,78)(11,30,56,79,38)(12,80,31,39,57)(13,40,65,58,32)(14,59,41,17,66)(15,18,60,67,42)(16,68,19,43,61), (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([[(1,20,62,69,44),(2,63,45,21,70),(3,46,71,64,22),(4,72,23,47,49),(5,24,50,73,48),(6,51,33,25,74),(7,34,75,52,26),(8,76,27,35,53),(9,28,54,77,36),(10,55,37,29,78),(11,38,79,56,30),(12,80,31,39,57),(13,32,58,65,40),(14,59,41,17,66),(15,42,67,60,18),(16,68,19,43,61)], [(1,44,69,62,20),(2,63,45,21,70),(3,22,64,71,46),(4,72,23,47,49),(5,48,73,50,24),(6,51,33,25,74),(7,26,52,75,34),(8,76,27,35,53),(9,36,77,54,28),(10,55,37,29,78),(11,30,56,79,38),(12,80,31,39,57),(13,40,65,58,32),(14,59,41,17,66),(15,18,60,67,42),(16,68,19,43,61)], [(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)]])
40 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | ··· | 5F | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 16A | ··· | 16H | 20A | ··· | 20L |
| order | 1 | 2 | 4 | 4 | 5 | ··· | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 25 | 25 | 25 | 25 | 4 | ··· | 4 | 25 | ··· | 25 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | - | |||||
| image | C1 | C2 | C4 | C8 | C16 | F5 | C5:C8 | C5:C16 | C52:C4 | C52:5C8 | C52:5C16 |
| kernel | C52:5C16 | C52:7C8 | C5xC20 | C5xC10 | C52 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C52:5C16 ►in GL5(F241)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 51 | 240 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 51 |
| 0 | 0 | 0 | 190 | 190 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 240 | 51 | 0 | 0 |
| 0 | 0 | 0 | 240 | 51 |
| 0 | 0 | 0 | 190 | 190 |
| 197 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 207 | 240 | 0 | 0 |
| 0 | 193 | 34 | 0 | 0 |
G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,51,1,0,0,0,240,0,0,0,0,0,0,240,190,0,0,0,51,190],[1,0,0,0,0,0,0,240,0,0,0,1,51,0,0,0,0,0,240,190,0,0,0,51,190],[197,0,0,0,0,0,0,0,207,193,0,0,0,240,34,0,1,0,0,0,0,0,1,0,0] >;
C52:5C16 in GAP, Magma, Sage, TeX
C_5^2\rtimes_5C_{16} % in TeX
G:=Group("C5^2:5C16"); // GroupNames label
G:=SmallGroup(400,59);
// by ID
G=gap.SmallGroup(400,59);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,12,31,50,1444,970,5765,5771]);
// Polycyclic
G:=Group<a,b,c|a^5=b^5=c^16=1,a*b=b*a,c*a*c^-1=a^2,c*b*c^-1=b^3>;
// generators/relations
Export