metabelian, supersoluble, monomial
Aliases: C52⋊2Q16, C20.14D10, Dic10.2D5, C4.10D52, C5⋊2(C5⋊Q16), (C5×C10).11D4, C10.9(C5⋊D4), (C5×C20).6C22, C52⋊7C8.1C2, (C5×Dic10).1C2, C2.5(C52⋊2D4), SmallGroup(400,69)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52⋊2Q16
G = < a,b,c,d | a5=b5=c8=1, d2=c4, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=c-1 >
Smallest permutation representation of C52⋊2Q16
►On 80 points
Generators in S80
(1 37 66 10 77)(2 78 11 67 38)(3 39 68 12 79)(4 80 13 69 40)(5 33 70 14 73)(6 74 15 71 34)(7 35 72 16 75)(8 76 9 65 36)(17 60 42 29 50)(18 51 30 43 61)(19 62 44 31 52)(20 53 32 45 63)(21 64 46 25 54)(22 55 26 47 57)(23 58 48 27 56)(24 49 28 41 59)
(1 66 77 37 10)(2 11 38 78 67)(3 68 79 39 12)(4 13 40 80 69)(5 70 73 33 14)(6 15 34 74 71)(7 72 75 35 16)(8 9 36 76 65)(17 29 60 50 42)(18 43 51 61 30)(19 31 62 52 44)(20 45 53 63 32)(21 25 64 54 46)(22 47 55 57 26)(23 27 58 56 48)(24 41 49 59 28)
(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 60 5 64)(2 59 6 63)(3 58 7 62)(4 57 8 61)(9 30 13 26)(10 29 14 25)(11 28 15 32)(12 27 16 31)(17 33 21 37)(18 40 22 36)(19 39 23 35)(20 38 24 34)(41 74 45 78)(42 73 46 77)(43 80 47 76)(44 79 48 75)(49 71 53 67)(50 70 54 66)(51 69 55 65)(52 68 56 72)
G:=sub<Sym(80)| (1,37,66,10,77)(2,78,11,67,38)(3,39,68,12,79)(4,80,13,69,40)(5,33,70,14,73)(6,74,15,71,34)(7,35,72,16,75)(8,76,9,65,36)(17,60,42,29,50)(18,51,30,43,61)(19,62,44,31,52)(20,53,32,45,63)(21,64,46,25,54)(22,55,26,47,57)(23,58,48,27,56)(24,49,28,41,59), (1,66,77,37,10)(2,11,38,78,67)(3,68,79,39,12)(4,13,40,80,69)(5,70,73,33,14)(6,15,34,74,71)(7,72,75,35,16)(8,9,36,76,65)(17,29,60,50,42)(18,43,51,61,30)(19,31,62,52,44)(20,45,53,63,32)(21,25,64,54,46)(22,47,55,57,26)(23,27,58,56,48)(24,41,49,59,28), (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), (1,60,5,64)(2,59,6,63)(3,58,7,62)(4,57,8,61)(9,30,13,26)(10,29,14,25)(11,28,15,32)(12,27,16,31)(17,33,21,37)(18,40,22,36)(19,39,23,35)(20,38,24,34)(41,74,45,78)(42,73,46,77)(43,80,47,76)(44,79,48,75)(49,71,53,67)(50,70,54,66)(51,69,55,65)(52,68,56,72)>;
G:=Group( (1,37,66,10,77)(2,78,11,67,38)(3,39,68,12,79)(4,80,13,69,40)(5,33,70,14,73)(6,74,15,71,34)(7,35,72,16,75)(8,76,9,65,36)(17,60,42,29,50)(18,51,30,43,61)(19,62,44,31,52)(20,53,32,45,63)(21,64,46,25,54)(22,55,26,47,57)(23,58,48,27,56)(24,49,28,41,59), (1,66,77,37,10)(2,11,38,78,67)(3,68,79,39,12)(4,13,40,80,69)(5,70,73,33,14)(6,15,34,74,71)(7,72,75,35,16)(8,9,36,76,65)(17,29,60,50,42)(18,43,51,61,30)(19,31,62,52,44)(20,45,53,63,32)(21,25,64,54,46)(22,47,55,57,26)(23,27,58,56,48)(24,41,49,59,28), (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), (1,60,5,64)(2,59,6,63)(3,58,7,62)(4,57,8,61)(9,30,13,26)(10,29,14,25)(11,28,15,32)(12,27,16,31)(17,33,21,37)(18,40,22,36)(19,39,23,35)(20,38,24,34)(41,74,45,78)(42,73,46,77)(43,80,47,76)(44,79,48,75)(49,71,53,67)(50,70,54,66)(51,69,55,65)(52,68,56,72) );
G=PermutationGroup([[(1,37,66,10,77),(2,78,11,67,38),(3,39,68,12,79),(4,80,13,69,40),(5,33,70,14,73),(6,74,15,71,34),(7,35,72,16,75),(8,76,9,65,36),(17,60,42,29,50),(18,51,30,43,61),(19,62,44,31,52),(20,53,32,45,63),(21,64,46,25,54),(22,55,26,47,57),(23,58,48,27,56),(24,49,28,41,59)], [(1,66,77,37,10),(2,11,38,78,67),(3,68,79,39,12),(4,13,40,80,69),(5,70,73,33,14),(6,15,34,74,71),(7,72,75,35,16),(8,9,36,76,65),(17,29,60,50,42),(18,43,51,61,30),(19,31,62,52,44),(20,45,53,63,32),(21,25,64,54,46),(22,47,55,57,26),(23,27,58,56,48),(24,41,49,59,28)], [(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)], [(1,60,5,64),(2,59,6,63),(3,58,7,62),(4,57,8,61),(9,30,13,26),(10,29,14,25),(11,28,15,32),(12,27,16,31),(17,33,21,37),(18,40,22,36),(19,39,23,35),(20,38,24,34),(41,74,45,78),(42,73,46,77),(43,80,47,76),(44,79,48,75),(49,71,53,67),(50,70,54,66),(51,69,55,65),(52,68,56,72)]])
43 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 50 | 50 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 20 | ··· | 20 |
43 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | - | ||
| image | C1 | C2 | C2 | D4 | D5 | Q16 | D10 | C5⋊D4 | C5⋊Q16 | D52 | C52⋊2D4 | C52⋊2Q16 |
| kernel | C52⋊2Q16 | C52⋊7C8 | C5×Dic10 | C5×C10 | Dic10 | C52 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 4 | 2 | 4 | 8 | 4 | 4 | 4 | 8 |
Matrix representation of C52⋊2Q16 ►in GL6(𝔽41)
| 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 29 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 39 | 0 | 0 |
| 0 | 0 | 31 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 23 |
| 0 | 0 | 0 | 0 | 20 | 21 |
| 21 | 38 | 0 | 0 | 0 | 0 |
| 38 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 6 | 0 | 0 |
| 0 | 0 | 35 | 23 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 40 |
| 0 | 0 | 0 | 0 | 35 | 35 |
G:=sub<GL(6,GF(41))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,29,12,0,0,0,0,0,0,15,31,0,0,0,0,39,26,0,0,0,0,0,0,20,20,0,0,0,0,23,21],[21,38,0,0,0,0,38,20,0,0,0,0,0,0,18,35,0,0,0,0,6,23,0,0,0,0,0,0,6,35,0,0,0,0,40,35] >;
C52⋊2Q16 in GAP, Magma, Sage, TeX
C_5^2\rtimes_2Q_{16} % in TeX
G:=Group("C5^2:2Q16"); // GroupNames label
G:=SmallGroup(400,69);
// by ID
G=gap.SmallGroup(400,69);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,73,55,218,116,50,970,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^8=1,d^2=c^4,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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