metabelian, supersoluble, monomial
Aliases: C10.2Dic10, C102.5C22, C22.7D52, (C5×C10).3Q8, C52⋊10(C4⋊C4), C52⋊6C4⋊4C4, (C5×C10).16D4, C10.11(C4×D5), (C2×C10).11D10, (C2×Dic5).2D5, C5⋊3(C10.D4), C10.11(C5⋊D4), (C10×Dic5).1C2, C2.2(C52⋊2D4), C2.2(C52⋊2Q8), C2.5(Dic5⋊2D5), (C5×C10).49(C2×C4), (C2×C52⋊6C4).3C2, SmallGroup(400,75)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.Dic10
G = < a,b,c | a10=b20=1, c2=b10, bab-1=a-1, ac=ca, cbc-1=a5b-1 >
Subgroups: 316 in 68 conjugacy classes, 28 normal (12 characteristic)
C1, C2, C4, C22, C5, C5, C2×C4, C10, C10, C4⋊C4, Dic5, C20, C2×C10, C2×C10, C52, C2×Dic5, C2×Dic5, C2×C20, C5×C10, C10.D4, C5×Dic5, C52⋊6C4, C102, C10×Dic5, C2×C52⋊6C4, C10.Dic10
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C4⋊C4, D10, Dic10, C4×D5, C5⋊D4, C10.D4, D52, Dic5⋊2D5, C52⋊2D4, C52⋊2Q8, C10.Dic10
►On 80 points
Generators in S80
(1 75 9 63 17 71 5 79 13 67)(2 68 14 80 6 72 18 64 10 76)(3 77 11 65 19 73 7 61 15 69)(4 70 16 62 8 74 20 66 12 78)(21 46 29 54 37 42 25 50 33 58)(22 59 34 51 26 43 38 55 30 47)(23 48 31 56 39 44 27 52 35 60)(24 41 36 53 28 45 40 57 32 49)
(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 53 11 43)(2 31 12 21)(3 51 13 41)(4 29 14 39)(5 49 15 59)(6 27 16 37)(7 47 17 57)(8 25 18 35)(9 45 19 55)(10 23 20 33)(22 71 32 61)(24 69 34 79)(26 67 36 77)(28 65 38 75)(30 63 40 73)(42 72 52 62)(44 70 54 80)(46 68 56 78)(48 66 58 76)(50 64 60 74)
G:=sub<Sym(80)| (1,75,9,63,17,71,5,79,13,67)(2,68,14,80,6,72,18,64,10,76)(3,77,11,65,19,73,7,61,15,69)(4,70,16,62,8,74,20,66,12,78)(21,46,29,54,37,42,25,50,33,58)(22,59,34,51,26,43,38,55,30,47)(23,48,31,56,39,44,27,52,35,60)(24,41,36,53,28,45,40,57,32,49), (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), (1,53,11,43)(2,31,12,21)(3,51,13,41)(4,29,14,39)(5,49,15,59)(6,27,16,37)(7,47,17,57)(8,25,18,35)(9,45,19,55)(10,23,20,33)(22,71,32,61)(24,69,34,79)(26,67,36,77)(28,65,38,75)(30,63,40,73)(42,72,52,62)(44,70,54,80)(46,68,56,78)(48,66,58,76)(50,64,60,74)>;
G:=Group( (1,75,9,63,17,71,5,79,13,67)(2,68,14,80,6,72,18,64,10,76)(3,77,11,65,19,73,7,61,15,69)(4,70,16,62,8,74,20,66,12,78)(21,46,29,54,37,42,25,50,33,58)(22,59,34,51,26,43,38,55,30,47)(23,48,31,56,39,44,27,52,35,60)(24,41,36,53,28,45,40,57,32,49), (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), (1,53,11,43)(2,31,12,21)(3,51,13,41)(4,29,14,39)(5,49,15,59)(6,27,16,37)(7,47,17,57)(8,25,18,35)(9,45,19,55)(10,23,20,33)(22,71,32,61)(24,69,34,79)(26,67,36,77)(28,65,38,75)(30,63,40,73)(42,72,52,62)(44,70,54,80)(46,68,56,78)(48,66,58,76)(50,64,60,74) );
G=PermutationGroup([[(1,75,9,63,17,71,5,79,13,67),(2,68,14,80,6,72,18,64,10,76),(3,77,11,65,19,73,7,61,15,69),(4,70,16,62,8,74,20,66,12,78),(21,46,29,54,37,42,25,50,33,58),(22,59,34,51,26,43,38,55,30,47),(23,48,31,56,39,44,27,52,35,60),(24,41,36,53,28,45,40,57,32,49)], [(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)], [(1,53,11,43),(2,31,12,21),(3,51,13,41),(4,29,14,39),(5,49,15,59),(6,27,16,37),(7,47,17,57),(8,25,18,35),(9,45,19,55),(10,23,20,33),(22,71,32,61),(24,69,34,79),(26,67,36,77),(28,65,38,75),(30,63,40,73),(42,72,52,62),(44,70,54,80),(46,68,56,78),(48,66,58,76),(50,64,60,74)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 10A | ··· | 10L | 10M | ··· | 10X | 20A | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 50 | 50 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 10 | ··· | 10 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | - | + | + | - | - | |||
| image | C1 | C2 | C2 | C4 | D4 | Q8 | D5 | D10 | Dic10 | C4×D5 | C5⋊D4 | D52 | Dic5⋊2D5 | C52⋊2D4 | C52⋊2Q8 |
| kernel | C10.Dic10 | C10×Dic5 | C2×C52⋊6C4 | C52⋊6C4 | C5×C10 | C5×C10 | C2×Dic5 | C2×C10 | C10 | C10 | C10 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 4 | 4 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
Matrix representation of C10.Dic10 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 34 |
| 0 | 0 | 0 | 0 | 7 | 7 |
| 35 | 23 | 0 | 0 | 0 | 0 |
| 18 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 2 | 0 | 0 |
| 0 | 0 | 39 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 34 | 40 |
| 16 | 39 | 0 | 0 | 0 | 0 |
| 25 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 18 | 0 | 0 |
| 0 | 0 | 21 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,7,0,0,0,0,34,7],[35,18,0,0,0,0,23,20,0,0,0,0,0,0,28,39,0,0,0,0,2,16,0,0,0,0,0,0,1,34,0,0,0,0,0,40],[16,25,0,0,0,0,39,25,0,0,0,0,0,0,20,21,0,0,0,0,18,21,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;
C10.Dic10 in GAP, Magma, Sage, TeX
C_{10}.{\rm Dic}_{10} % in TeX
G:=Group("C10.Dic10"); // GroupNames label
G:=SmallGroup(400,75);
// by ID
G=gap.SmallGroup(400,75);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,31,970,11525]);
// Polycyclic
G:=Group<a,b,c|a^10=b^20=1,c^2=b^10,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^5*b^-1>;
// generators/relations