direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C10×D8, C20.41D4, C40⋊12C22, C20.44C23, (C2×C8)⋊3C10, C8⋊2(C2×C10), C4.6(C5×D4), (C2×C40)⋊11C2, D4⋊1(C2×C10), (C2×D4)⋊4C10, (D4×C10)⋊13C2, C10.74(C2×D4), C2.11(D4×C10), (C2×C10).52D4, (C5×D4)⋊10C22, C4.1(C22×C10), C22.14(C5×D4), (C2×C20).129C22, (C2×C4).25(C2×C10), SmallGroup(160,193)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×D8
G = < a,b,c | a10=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 140 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C8, C2×C4, D4, D4, C23, C10, C10, C10, C2×C8, D8, C2×D4, C20, C2×C10, C2×C10, C2×D8, C40, C2×C20, C5×D4, C5×D4, C22×C10, C2×C40, C5×D8, D4×C10, C10×D8
Quotients: C1, C2, C22, C5, D4, C23, C10, D8, C2×D4, C2×C10, C2×D8, C5×D4, C22×C10, C5×D8, D4×C10, C10×D8
►On 80 points
(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 28 50 32 70 20 56 79)(2 29 41 33 61 11 57 80)(3 30 42 34 62 12 58 71)(4 21 43 35 63 13 59 72)(5 22 44 36 64 14 60 73)(6 23 45 37 65 15 51 74)(7 24 46 38 66 16 52 75)(8 25 47 39 67 17 53 76)(9 26 48 40 68 18 54 77)(10 27 49 31 69 19 55 78)
(1 79)(2 80)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 59)(22 60)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 69)(32 70)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)
G:=sub<Sym(80)| (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), (1,28,50,32,70,20,56,79)(2,29,41,33,61,11,57,80)(3,30,42,34,62,12,58,71)(4,21,43,35,63,13,59,72)(5,22,44,36,64,14,60,73)(6,23,45,37,65,15,51,74)(7,24,46,38,66,16,52,75)(8,25,47,39,67,17,53,76)(9,26,48,40,68,18,54,77)(10,27,49,31,69,19,55,78), (1,79)(2,80)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,59)(22,60)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,69)(32,70)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68)>;
G:=Group( (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), (1,28,50,32,70,20,56,79)(2,29,41,33,61,11,57,80)(3,30,42,34,62,12,58,71)(4,21,43,35,63,13,59,72)(5,22,44,36,64,14,60,73)(6,23,45,37,65,15,51,74)(7,24,46,38,66,16,52,75)(8,25,47,39,67,17,53,76)(9,26,48,40,68,18,54,77)(10,27,49,31,69,19,55,78), (1,79)(2,80)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,59)(22,60)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,69)(32,70)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68) );
G=PermutationGroup([[(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)], [(1,28,50,32,70,20,56,79),(2,29,41,33,61,11,57,80),(3,30,42,34,62,12,58,71),(4,21,43,35,63,13,59,72),(5,22,44,36,64,14,60,73),(6,23,45,37,65,15,51,74),(7,24,46,38,66,16,52,75),(8,25,47,39,67,17,53,76),(9,26,48,40,68,18,54,77),(10,27,49,31,69,19,55,78)], [(1,79),(2,80),(3,71),(4,72),(5,73),(6,74),(7,75),(8,76),(9,77),(10,78),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,59),(22,60),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,69),(32,70),(33,61),(34,62),(35,63),(36,64),(37,65),(38,66),(39,67),(40,68)]])
C10×D8 is a maximal subgroup of
C10.D16 D8.Dic5 D8.D10 Dic5⋊D8 C40⋊5D4 D8⋊Dic5 (C2×D8).D5 C40⋊11D4 C40.22D4 D20⋊D4 C40⋊6D4 Dic10⋊D4 C40⋊12D4 C40.23D4 D8⋊13D10
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10AB | 20A | ··· | 20H | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | D4 | D4 | D8 | C5×D4 | C5×D4 | C5×D8 |
| kernel | C10×D8 | C2×C40 | C5×D8 | D4×C10 | C2×D8 | C2×C8 | D8 | C2×D4 | C20 | C2×C10 | C10 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 2 | 4 | 4 | 16 | 8 | 1 | 1 | 4 | 4 | 4 | 16 |
Matrix representation of C10×D8 ►in GL3(𝔽41) generated by
| 40 | 0 | 0 |
| 0 | 25 | 0 |
| 0 | 0 | 25 |
| 40 | 0 | 0 |
| 0 | 0 | 29 |
| 0 | 24 | 17 |
| 1 | 0 | 0 |
| 0 | 17 | 12 |
| 0 | 17 | 24 |
G:=sub<GL(3,GF(41))| [40,0,0,0,25,0,0,0,25],[40,0,0,0,0,24,0,29,17],[1,0,0,0,17,17,0,12,24] >;
C10×D8 in GAP, Magma, Sage, TeX
C_{10}\times D_8 % in TeX
G:=Group("C10xD8"); // GroupNames label
G:=SmallGroup(160,193);
// by ID
G=gap.SmallGroup(160,193);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-2,505,3604,1810,88]);
// Polycyclic
G:=Group<a,b,c|a^10=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations