metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D40:16C4, Dic20:16C4, M4(2).26D10, C40:C2:8C4, C40:8C4:1C2, C8.11(C4xD5), C5:6(C8.26D4), C40.70(C2xC4), (C2xC8).69D10, C4.212(D4xD5), C10.85(C4xD4), C8.C4:4D5, C5:2C8.51D4, D20:7C4:7C2, D20.24(C2xC4), C20.371(C2xD4), D40:7C2.4C2, (C2xC40).41C22, D20.2C4:10C2, (C2xC20).310C23, C20.113(C22xC4), Dic10.25(C2xC4), C4oD20.17C22, C2.15(D20:8C4), C22.1(Q8:2D5), (C4xDic5).46C22, (C5xM4(2)).20C22, C4.46(C2xC4xD5), (C5xC8.C4):4C2, (C2xC10).1(C4oD4), (C2xC5:2C8).78C22, (C2xC4).413(C22xD5), SmallGroup(320,521)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D40:16C4
G = < a,b,c | a40=b2=c4=1, bab=a-1, cac-1=a29, cbc-1=a18b >
Subgroups: 390 in 104 conjugacy classes, 45 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2xC4, C2xC4, D4, Q8, D5, C10, C10, C42, C2xC8, C2xC8, M4(2), M4(2), D8, SD16, Q16, C4oD4, Dic5, C20, D10, C2xC10, C8:C4, C4wrC2, C8.C4, C8oD4, C4oD8, C5:2C8, C40, C40, Dic10, C4xD5, D20, C2xDic5, C5:D4, C2xC20, C8.26D4, C8xD5, C8:D5, C40:C2, D40, Dic20, C2xC5:2C8, C4xDic5, C2xC40, C5xM4(2), C4oD20, C40:8C4, D20:7C4, C5xC8.C4, D40:7C2, D20.2C4, D40:16C4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, D5, C22xC4, C2xD4, C4oD4, D10, C4xD4, C4xD5, C22xD5, C8.26D4, C2xC4xD5, D4xD5, Q8:2D5, D20:8C4, D40:16C4
►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 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 60)(11 59)(12 58)(13 57)(14 56)(15 55)(16 54)(17 53)(18 52)(19 51)(20 50)(21 49)(22 48)(23 47)(24 46)(25 45)(26 44)(27 43)(28 42)(29 41)(30 80)(31 79)(32 78)(33 77)(34 76)(35 75)(36 74)(37 73)(38 72)(39 71)(40 70)
(1 11 21 31)(2 40 22 20)(3 29 23 9)(4 18 24 38)(5 7 25 27)(6 36 26 16)(8 14 28 34)(10 32 30 12)(13 39 33 19)(15 17 35 37)(41 49)(42 78)(43 67)(44 56)(46 74)(47 63)(48 52)(50 70)(51 59)(53 77)(54 66)(57 73)(58 62)(60 80)(61 69)(64 76)(68 72)(71 79)
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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,60)(11,59)(12,58)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(29,41)(30,80)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,70), (1,11,21,31)(2,40,22,20)(3,29,23,9)(4,18,24,38)(5,7,25,27)(6,36,26,16)(8,14,28,34)(10,32,30,12)(13,39,33,19)(15,17,35,37)(41,49)(42,78)(43,67)(44,56)(46,74)(47,63)(48,52)(50,70)(51,59)(53,77)(54,66)(57,73)(58,62)(60,80)(61,69)(64,76)(68,72)(71,79)>;
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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,60)(11,59)(12,58)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(29,41)(30,80)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,70), (1,11,21,31)(2,40,22,20)(3,29,23,9)(4,18,24,38)(5,7,25,27)(6,36,26,16)(8,14,28,34)(10,32,30,12)(13,39,33,19)(15,17,35,37)(41,49)(42,78)(43,67)(44,56)(46,74)(47,63)(48,52)(50,70)(51,59)(53,77)(54,66)(57,73)(58,62)(60,80)(61,69)(64,76)(68,72)(71,79) );
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)], [(1,69),(2,68),(3,67),(4,66),(5,65),(6,64),(7,63),(8,62),(9,61),(10,60),(11,59),(12,58),(13,57),(14,56),(15,55),(16,54),(17,53),(18,52),(19,51),(20,50),(21,49),(22,48),(23,47),(24,46),(25,45),(26,44),(27,43),(28,42),(29,41),(30,80),(31,79),(32,78),(33,77),(34,76),(35,75),(36,74),(37,73),(38,72),(39,71),(40,70)], [(1,11,21,31),(2,40,22,20),(3,29,23,9),(4,18,24,38),(5,7,25,27),(6,36,26,16),(8,14,28,34),(10,32,30,12),(13,39,33,19),(15,17,35,37),(41,49),(42,78),(43,67),(44,56),(46,74),(47,63),(48,52),(50,70),(51,59),(53,77),(54,66),(57,73),(58,62),(60,80),(61,69),(64,76),(68,72),(71,79)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | ··· | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H | 40I | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 20 | 20 | 1 | 1 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D5 | C4oD4 | D10 | D10 | C4xD5 | C8.26D4 | D4xD5 | Q8:2D5 | D40:16C4 |
| kernel | D40:16C4 | C40:8C4 | D20:7C4 | C5xC8.C4 | D40:7C2 | D20.2C4 | C40:C2 | D40 | Dic20 | C5:2C8 | C8.C4 | C2xC10 | C2xC8 | M4(2) | C8 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 2 | 8 |
Matrix representation of D40:16C4 ►in GL6(F41)
| 6 | 6 | 0 | 0 | 0 | 0 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 1 | 0 |
| 0 | 0 | 17 | 40 | 0 | 1 |
| 0 | 0 | 14 | 0 | 10 | 0 |
| 0 | 0 | 23 | 8 | 24 | 1 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 1 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 30 | 22 | 0 | 9 |
| 0 | 0 | 7 | 17 | 32 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 13 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 25 | 0 | 9 | 0 |
| 0 | 0 | 13 | 2 | 0 | 40 |
G:=sub<GL(6,GF(41))| [6,35,0,0,0,0,6,1,0,0,0,0,0,0,31,17,14,23,0,0,0,40,0,8,0,0,1,0,10,24,0,0,0,1,0,1],[6,1,0,0,0,0,6,35,0,0,0,0,0,0,0,40,30,7,0,0,40,0,22,17,0,0,0,0,0,32,0,0,0,0,9,0],[9,13,0,0,0,0,0,32,0,0,0,0,0,0,32,0,25,13,0,0,0,1,0,2,0,0,0,0,9,0,0,0,0,0,0,40] >;
D40:16C4 in GAP, Magma, Sage, TeX
D_{40}\rtimes_{16}C_4 % in TeX
G:=Group("D40:16C4"); // GroupNames label
G:=SmallGroup(320,521);
// by ID
G=gap.SmallGroup(320,521);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,219,58,136,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^29,c*b*c^-1=a^18*b>;
// generators/relations