metabelian, supersoluble, monomial
Aliases: C52:2D8, D20:2D5, C20.9D10, C4.8D52, C5:2(D4:D5), (C5xD20):1C2, (C5xC10).6D4, C52:7C8:1C2, C10.7(C5:D4), (C5xC20).1C22, C2.3(C52:2D4), SmallGroup(400,64)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52:2D8
G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Quotients: C1, C2, C22, D4, D5, D8, D10, C5:D4, D4:D5, D52, C52:2D4, C52:2D8
20C2
20C2
2C5
2C5
10C22
10C22
2C10
2C10
4D5
4D5
20C10
20C10
5D4
5D4
25C8
2D10
2D10
2C20
2C20
10C2xC10
10C2xC10
25D8
10C5:2C8
10C5:2C8
Smallest permutation representation of C52:2D8
►On 80 points
Generators in S80
(1 41 66 75 12)(2 13 76 67 42)(3 43 68 77 14)(4 15 78 69 44)(5 45 70 79 16)(6 9 80 71 46)(7 47 72 73 10)(8 11 74 65 48)(17 60 31 38 54)(18 55 39 32 61)(19 62 25 40 56)(20 49 33 26 63)(21 64 27 34 50)(22 51 35 28 57)(23 58 29 36 52)(24 53 37 30 59)
(1 66 12 41 75)(2 76 42 13 67)(3 68 14 43 77)(4 78 44 15 69)(5 70 16 45 79)(6 80 46 9 71)(7 72 10 47 73)(8 74 48 11 65)(17 38 60 54 31)(18 32 55 61 39)(19 40 62 56 25)(20 26 49 63 33)(21 34 64 50 27)(22 28 51 57 35)(23 36 58 52 29)(24 30 53 59 37)
(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)
(1 59)(2 58)(3 57)(4 64)(5 63)(6 62)(7 61)(8 60)(9 25)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)(33 79)(34 78)(35 77)(36 76)(37 75)(38 74)(39 73)(40 80)(49 70)(50 69)(51 68)(52 67)(53 66)(54 65)(55 72)(56 71)
G:=sub<Sym(80)| (1,41,66,75,12)(2,13,76,67,42)(3,43,68,77,14)(4,15,78,69,44)(5,45,70,79,16)(6,9,80,71,46)(7,47,72,73,10)(8,11,74,65,48)(17,60,31,38,54)(18,55,39,32,61)(19,62,25,40,56)(20,49,33,26,63)(21,64,27,34,50)(22,51,35,28,57)(23,58,29,36,52)(24,53,37,30,59), (1,66,12,41,75)(2,76,42,13,67)(3,68,14,43,77)(4,78,44,15,69)(5,70,16,45,79)(6,80,46,9,71)(7,72,10,47,73)(8,74,48,11,65)(17,38,60,54,31)(18,32,55,61,39)(19,40,62,56,25)(20,26,49,63,33)(21,34,64,50,27)(22,28,51,57,35)(23,36,58,52,29)(24,30,53,59,37), (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), (1,59)(2,58)(3,57)(4,64)(5,63)(6,62)(7,61)(8,60)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(33,79)(34,78)(35,77)(36,76)(37,75)(38,74)(39,73)(40,80)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,72)(56,71)>;
G:=Group( (1,41,66,75,12)(2,13,76,67,42)(3,43,68,77,14)(4,15,78,69,44)(5,45,70,79,16)(6,9,80,71,46)(7,47,72,73,10)(8,11,74,65,48)(17,60,31,38,54)(18,55,39,32,61)(19,62,25,40,56)(20,49,33,26,63)(21,64,27,34,50)(22,51,35,28,57)(23,58,29,36,52)(24,53,37,30,59), (1,66,12,41,75)(2,76,42,13,67)(3,68,14,43,77)(4,78,44,15,69)(5,70,16,45,79)(6,80,46,9,71)(7,72,10,47,73)(8,74,48,11,65)(17,38,60,54,31)(18,32,55,61,39)(19,40,62,56,25)(20,26,49,63,33)(21,34,64,50,27)(22,28,51,57,35)(23,36,58,52,29)(24,30,53,59,37), (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), (1,59)(2,58)(3,57)(4,64)(5,63)(6,62)(7,61)(8,60)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(33,79)(34,78)(35,77)(36,76)(37,75)(38,74)(39,73)(40,80)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,72)(56,71) );
G=PermutationGroup([[(1,41,66,75,12),(2,13,76,67,42),(3,43,68,77,14),(4,15,78,69,44),(5,45,70,79,16),(6,9,80,71,46),(7,47,72,73,10),(8,11,74,65,48),(17,60,31,38,54),(18,55,39,32,61),(19,62,25,40,56),(20,49,33,26,63),(21,64,27,34,50),(22,51,35,28,57),(23,58,29,36,52),(24,53,37,30,59)], [(1,66,12,41,75),(2,76,42,13,67),(3,68,14,43,77),(4,78,44,15,69),(5,70,16,45,79),(6,80,46,9,71),(7,72,10,47,73),(8,74,48,11,65),(17,38,60,54,31),(18,32,55,61,39),(19,40,62,56,25),(20,26,49,63,33),(21,34,64,50,27),(22,28,51,57,35),(23,36,58,52,29),(24,30,53,59,37)], [(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)], [(1,59),(2,58),(3,57),(4,64),(5,63),(6,62),(7,61),(8,60),(9,25),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26),(17,48),(18,47),(19,46),(20,45),(21,44),(22,43),(23,42),(24,41),(33,79),(34,78),(35,77),(36,76),(37,75),(38,74),(39,73),(40,80),(49,70),(50,69),(51,68),(52,67),(53,66),(54,65),(55,72),(56,71)]])
43 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 50 | 50 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 4 | ··· | 4 |
43 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | D4 | D5 | D8 | D10 | C5:D4 | D4:D5 | D52 | C52:2D4 | C52:2D8 |
| kernel | C52:2D8 | C52:7C8 | C5xD20 | C5xC10 | D20 | C52 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 4 | 2 | 4 | 8 | 4 | 4 | 4 | 8 |
Matrix representation of C52:2D8 ►in GL6(F41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 39 | 19 | 0 | 0 | 0 | 0 |
| 26 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 15 | 0 | 0 |
| 0 | 0 | 27 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 6 | 40 |
| 28 | 8 | 0 | 0 | 0 | 0 |
| 20 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 35 | 0 | 0 |
| 0 | 0 | 6 | 23 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 35 | 1 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,1,0,0,0,0,40,0],[39,26,0,0,0,0,19,19,0,0,0,0,0,0,2,27,0,0,0,0,15,39,0,0,0,0,0,0,1,6,0,0,0,0,0,40],[28,20,0,0,0,0,8,13,0,0,0,0,0,0,18,6,0,0,0,0,35,23,0,0,0,0,0,0,40,35,0,0,0,0,0,1] >;
C52:2D8 in GAP, Magma, Sage, TeX
C_5^2\rtimes_2D_8
% in TeX
G:=Group("C5^2:2D8"); // GroupNames label
G:=SmallGroup(400,64);
// by ID
G=gap.SmallGroup(400,64);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,73,218,116,50,970,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^8=d^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
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