metabelian, supersoluble, monomial
Aliases: C20.30D10, C52⋊9M4(2), D10.1Dic5, Dic5.1Dic5, C4.15D52, C5⋊2C8⋊4D5, (C4×D5).2D5, C5⋊5(C8⋊D5), C52⋊7C8⋊7C2, (D5×C10).6C4, (D5×C20).1C2, C10.24(C4×D5), C2.3(D5×Dic5), C5⋊3(C4.Dic5), (C5×Dic5).6C4, C10.9(C2×Dic5), (C5×C20).29C22, (C5×C5⋊2C8)⋊6C2, (C5×C10).43(C2×C4), SmallGroup(400,62)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.30D10
G = < a,b,c | a20=1, b10=a5, c2=a15, bab-1=cac-1=a9, cbc-1=a10b9 >
Smallest permutation representation of C20.30D10
►On 80 points
Generators in S80
(1 35 29 23 17 11 5 39 33 27 21 15 9 3 37 31 25 19 13 7)(2 28 14 40 26 12 38 24 10 36 22 8 34 20 6 32 18 4 30 16)(41 75 69 63 57 51 45 79 73 67 61 55 49 43 77 71 65 59 53 47)(42 68 54 80 66 52 78 64 50 76 62 48 74 60 46 72 58 44 70 56)
(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 71 31 61 21 51 11 41)(2 60 32 50 22 80 12 70)(3 49 33 79 23 69 13 59)(4 78 34 68 24 58 14 48)(5 67 35 57 25 47 15 77)(6 56 36 46 26 76 16 66)(7 45 37 75 27 65 17 55)(8 74 38 64 28 54 18 44)(9 63 39 53 29 43 19 73)(10 52 40 42 30 72 20 62)
G:=sub<Sym(80)| (1,35,29,23,17,11,5,39,33,27,21,15,9,3,37,31,25,19,13,7)(2,28,14,40,26,12,38,24,10,36,22,8,34,20,6,32,18,4,30,16)(41,75,69,63,57,51,45,79,73,67,61,55,49,43,77,71,65,59,53,47)(42,68,54,80,66,52,78,64,50,76,62,48,74,60,46,72,58,44,70,56), (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), (1,71,31,61,21,51,11,41)(2,60,32,50,22,80,12,70)(3,49,33,79,23,69,13,59)(4,78,34,68,24,58,14,48)(5,67,35,57,25,47,15,77)(6,56,36,46,26,76,16,66)(7,45,37,75,27,65,17,55)(8,74,38,64,28,54,18,44)(9,63,39,53,29,43,19,73)(10,52,40,42,30,72,20,62)>;
G:=Group( (1,35,29,23,17,11,5,39,33,27,21,15,9,3,37,31,25,19,13,7)(2,28,14,40,26,12,38,24,10,36,22,8,34,20,6,32,18,4,30,16)(41,75,69,63,57,51,45,79,73,67,61,55,49,43,77,71,65,59,53,47)(42,68,54,80,66,52,78,64,50,76,62,48,74,60,46,72,58,44,70,56), (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), (1,71,31,61,21,51,11,41)(2,60,32,50,22,80,12,70)(3,49,33,79,23,69,13,59)(4,78,34,68,24,58,14,48)(5,67,35,57,25,47,15,77)(6,56,36,46,26,76,16,66)(7,45,37,75,27,65,17,55)(8,74,38,64,28,54,18,44)(9,63,39,53,29,43,19,73)(10,52,40,42,30,72,20,62) );
G=PermutationGroup([[(1,35,29,23,17,11,5,39,33,27,21,15,9,3,37,31,25,19,13,7),(2,28,14,40,26,12,38,24,10,36,22,8,34,20,6,32,18,4,30,16),(41,75,69,63,57,51,45,79,73,67,61,55,49,43,77,71,65,59,53,47),(42,68,54,80,66,52,78,64,50,76,62,48,74,60,46,72,58,44,70,56)], [(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)], [(1,71,31,61,21,51,11,41),(2,60,32,50,22,80,12,70),(3,49,33,79,23,69,13,59),(4,78,34,68,24,58,14,48),(5,67,35,57,25,47,15,77),(6,56,36,46,26,76,16,66),(7,45,37,75,27,65,17,55),(8,74,38,64,28,54,18,44),(9,63,39,53,29,43,19,73),(10,52,40,42,30,72,20,62)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 20A | ··· | 20H | 20I | ··· | 20P | 20Q | 20R | 20S | 20T | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 10 | 1 | 1 | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 50 | 50 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 10 | ··· | 10 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D5 | D5 | M4(2) | Dic5 | D10 | Dic5 | C4×D5 | C8⋊D5 | C4.Dic5 | D52 | D5×Dic5 | C20.30D10 |
| kernel | C20.30D10 | C5×C5⋊2C8 | C52⋊7C8 | D5×C20 | C5×Dic5 | D5×C10 | C5⋊2C8 | C4×D5 | C52 | Dic5 | C20 | D10 | C10 | C5 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 8 | 8 | 4 | 4 | 8 |
Matrix representation of C20.30D10 ►in GL4(𝔽41) generated by
| 0 | 9 | 0 | 0 |
| 32 | 13 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 3 | 0 | 0 | 0 |
| 18 | 38 | 0 | 0 |
| 0 | 0 | 32 | 19 |
| 0 | 0 | 22 | 22 |
| 28 | 18 | 0 | 0 |
| 22 | 13 | 0 | 0 |
| 0 | 0 | 9 | 22 |
| 0 | 0 | 0 | 32 |
G:=sub<GL(4,GF(41))| [0,32,0,0,9,13,0,0,0,0,40,0,0,0,0,40],[3,18,0,0,0,38,0,0,0,0,32,22,0,0,19,22],[28,22,0,0,18,13,0,0,0,0,9,0,0,0,22,32] >;
C20.30D10 in GAP, Magma, Sage, TeX
C_{20}._{30}D_{10} % in TeX
G:=Group("C20.30D10"); // GroupNames label
G:=SmallGroup(400,62);
// by ID
G=gap.SmallGroup(400,62);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,121,31,50,970,11525]);
// Polycyclic
G:=Group<a,b,c|a^20=1,b^10=a^5,c^2=a^15,b*a*b^-1=c*a*c^-1=a^9,c*b*c^-1=a^10*b^9>;
// generators/relations
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