metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D40⋊10C4, C8.27D20, C40.49D4, Dic20⋊10C4, C42.23D10, C8.8(C4×D5), C8⋊C4⋊5D5, C40⋊C2⋊6C4, C5⋊4(C8.26D4), C40.46(C2×C4), C4.78(C2×D20), C2.17(C4×D20), C10.44(C4×D4), D20⋊4C4⋊1C2, D20.29(C2×C4), C20.298(C2×D4), (C2×C8).161D10, C40.6C4⋊12C2, D40⋊7C2.8C2, (C4×C20).17C22, D20.3C4⋊12C2, C20.167(C22×C4), (C2×C20).792C23, (C2×C40).229C22, Dic10.30(C2×C4), C4○D20.36C22, C22.21(C4○D20), C4.Dic5.34C22, C4.66(C2×C4×D5), (C5×C8⋊C4)⋊1C2, (C2×C10).63(C4○D4), (C2×C4).682(C22×D5), SmallGroup(320,344)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D40⋊10C4
G = < a,b,c | a40=b2=c4=1, bab=a-1, cac-1=a21, cbc-1=a30b >
Subgroups: 374 in 104 conjugacy classes, 47 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C42, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C4○D4, Dic5, C20, C20, D10, C2×C10, C8⋊C4, C4≀C2, C8.C4, C8○D4, C4○D8, C5⋊2C8, C40, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C8.26D4, C8×D5, C8⋊D5, C40⋊C2, D40, Dic20, C4.Dic5, C4×C20, C2×C40, C4○D20, D20⋊4C4, C40.6C4, C5×C8⋊C4, D20.3C4, D40⋊7C2, D40⋊10C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C4×D4, C4×D5, D20, C22×D5, C8.26D4, C2×C4×D5, C2×D20, C4○D20, C4×D20, D40⋊10C4
►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 51)(2 50)(3 49)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 80)(13 79)(14 78)(15 77)(16 76)(17 75)(18 74)(19 73)(20 72)(21 71)(22 70)(23 69)(24 68)(25 67)(26 66)(27 65)(28 64)(29 63)(30 62)(31 61)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)
(1 31 21 11)(2 12 22 32)(3 33 23 13)(4 14 24 34)(5 35 25 15)(6 16 26 36)(7 37 27 17)(8 18 28 38)(9 39 29 19)(10 20 30 40)(42 62)(44 64)(46 66)(48 68)(50 70)(52 72)(54 74)(56 76)(58 78)(60 80)
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), (1,51)(2,50)(3,49)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,73)(20,72)(21,71)(22,70)(23,69)(24,68)(25,67)(26,66)(27,65)(28,64)(29,63)(30,62)(31,61)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52), (1,31,21,11)(2,12,22,32)(3,33,23,13)(4,14,24,34)(5,35,25,15)(6,16,26,36)(7,37,27,17)(8,18,28,38)(9,39,29,19)(10,20,30,40)(42,62)(44,64)(46,66)(48,68)(50,70)(52,72)(54,74)(56,76)(58,78)(60,80)>;
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), (1,51)(2,50)(3,49)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,73)(20,72)(21,71)(22,70)(23,69)(24,68)(25,67)(26,66)(27,65)(28,64)(29,63)(30,62)(31,61)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52), (1,31,21,11)(2,12,22,32)(3,33,23,13)(4,14,24,34)(5,35,25,15)(6,16,26,36)(7,37,27,17)(8,18,28,38)(9,39,29,19)(10,20,30,40)(42,62)(44,64)(46,66)(48,68)(50,70)(52,72)(54,74)(56,76)(58,78)(60,80) );
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)], [(1,51),(2,50),(3,49),(4,48),(5,47),(6,46),(7,45),(8,44),(9,43),(10,42),(11,41),(12,80),(13,79),(14,78),(15,77),(16,76),(17,75),(18,74),(19,73),(20,72),(21,71),(22,70),(23,69),(24,68),(25,67),(26,66),(27,65),(28,64),(29,63),(30,62),(31,61),(32,60),(33,59),(34,58),(35,57),(36,56),(37,55),(38,54),(39,53),(40,52)], [(1,31,21,11),(2,12,22,32),(3,33,23,13),(4,14,24,34),(5,35,25,15),(6,16,26,36),(7,37,27,17),(8,18,28,38),(9,39,29,19),(10,20,30,40),(42,62),(44,64),(46,66),(48,68),(50,70),(52,72),(54,74),(56,76),(58,78),(60,80)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 20 | 20 | 1 | 1 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D5 | C4○D4 | D10 | D10 | C4×D5 | D20 | C4○D20 | C8.26D4 | D40⋊10C4 |
| kernel | D40⋊10C4 | D20⋊4C4 | C40.6C4 | C5×C8⋊C4 | D20.3C4 | D40⋊7C2 | C40⋊C2 | D40 | Dic20 | C40 | C8⋊C4 | C2×C10 | C42 | C2×C8 | C8 | C8 | C22 | C5 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 8 | 2 | 8 |
Matrix representation of D40⋊10C4 ►in GL6(𝔽41)
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 21 | 9 | 21 |
| 0 | 0 | 23 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 9 | 5 |
| 0 | 0 | 0 | 0 | 2 | 32 |
| 0 | 33 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 32 |
| 0 | 0 | 1 | 21 | 0 | 16 |
| 0 | 0 | 23 | 0 | 0 | 40 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 5 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 9 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [36,0,0,0,0,0,0,8,0,0,0,0,0,0,0,23,0,0,0,0,21,0,0,0,0,0,9,0,9,2,0,0,21,40,5,32],[0,5,0,0,0,0,33,0,0,0,0,0,0,0,1,0,1,23,0,0,0,0,21,0,0,0,0,2,0,0,0,0,0,32,16,40],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,5,0,9,1] >;
D40⋊10C4 in GAP, Magma, Sage, TeX
D_{40}\rtimes_{10}C_4 % in TeX
G:=Group("D40:10C4"); // GroupNames label
G:=SmallGroup(320,344);
// by ID
G=gap.SmallGroup(320,344);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,387,58,136,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^21,c*b*c^-1=a^30*b>;
// generators/relations