metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q16:4D10, SD16:6D10, D40:3C22, C40.5C23, C20.24C24, M4(2):12D10, D20.17C23, Dic10.17C23, D40:C2:3C2, C8:D10:3C2, D4:D5:7C22, (C2xQ8):23D10, C4.192(D4xD5), (C8xD5):5C22, Q8:D5:6C22, C8.C22:6D5, Q8.D10:1C2, C8.5(C22xD5), Q16:D5:2C2, C4oD4.30D10, (C4xD5).101D4, D10.89(C2xD4), C20.245(C2xD4), (D4xD5):10C22, C40:C2:6C22, C8:D5:6C22, (D5xM4(2)):4C2, D4:D10:10C2, (C5xQ16):2C22, (Q8xD5):12C22, C5:Q16:5C22, C4.24(C23xD5), C22.49(D4xD5), SD16:3D5:3C2, (C2xD20):37C22, C5:4(D8:C22), C5:2C8.12C23, (Q8xC10):21C22, (C5xSD16):6C22, (C5xD4).17C23, (C22xD5).52D4, D4.17(C22xD5), (C4xD5).67C23, Q8.17(C22xD5), (C5xQ8).17C23, C20.C23:10C2, D4:2D5:11C22, (C2xC20).115C23, (C2xDic5).252D4, Dic5.102(C2xD4), Q8:2D5:11C22, C4oD20.31C22, C10.125(C22xD4), (C5xM4(2)):6C22, C4.Dic5:15C22, C2.98(C2xD4xD5), (D5xC4oD4):5C2, (C2xC10).70(C2xD4), (C5xC8.C22):2C2, (C2xQ8:2D5):17C2, (C2xC4xD5).172C22, (C2xC4).99(C22xD5), (C5xC4oD4).26C22, SmallGroup(320,1449)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D40:C22
G = < a,b,c,d | a40=b2=c2=d2=1, bab=a-1, cac=a21, dad=a29, cbc=a20b, dbd=a8b, cd=dc >
Subgroups: 1038 in 262 conjugacy classes, 99 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, Q8, C23, D5, C10, C10, C2xC8, M4(2), M4(2), D8, SD16, SD16, Q16, Q16, C22xC4, C2xD4, C2xQ8, C2xQ8, C4oD4, C4oD4, Dic5, Dic5, C20, C20, D10, D10, C2xC10, C2xC10, C2xM4(2), C4oD8, C8:C22, C8.C22, C8.C22, C2xC4oD4, C5:2C8, C40, Dic10, Dic10, C4xD5, C4xD5, D20, D20, D20, C2xDic5, C2xDic5, C5:D4, C2xC20, C2xC20, C5xD4, C5xD4, C5xQ8, C5xQ8, C5xQ8, C22xD5, C22xD5, D8:C22, C8xD5, C8:D5, C40:C2, D40, C4.Dic5, D4:D5, Q8:D5, C5:Q16, C5xM4(2), C5xSD16, C5xQ16, C2xC4xD5, C2xC4xD5, C2xD20, C2xD20, C4oD20, C4oD20, D4xD5, D4xD5, D4:2D5, D4:2D5, Q8xD5, Q8:2D5, Q8:2D5, Q8:2D5, Q8xC10, C5xC4oD4, D5xM4(2), C8:D10, D40:C2, SD16:3D5, Q16:D5, Q8.D10, C20.C23, D4:D10, C5xC8.C22, C2xQ8:2D5, D5xC4oD4, D40:C22
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C24, D10, C22xD4, C22xD5, D8:C22, D4xD5, C23xD5, C2xD4xD5, D40:C22
►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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(41 48)(42 47)(43 46)(44 45)(49 80)(50 79)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 70)(60 69)(61 68)(62 67)(63 66)(64 65)
(1 21)(3 23)(5 25)(7 27)(9 29)(11 31)(13 33)(15 35)(17 37)(19 39)(41 61)(43 63)(45 65)(47 67)(49 69)(51 71)(53 73)(55 75)(57 77)(59 79)
(1 75)(2 64)(3 53)(4 42)(5 71)(6 60)(7 49)(8 78)(9 67)(10 56)(11 45)(12 74)(13 63)(14 52)(15 41)(16 70)(17 59)(18 48)(19 77)(20 66)(21 55)(22 44)(23 73)(24 62)(25 51)(26 80)(27 69)(28 58)(29 47)(30 76)(31 65)(32 54)(33 43)(34 72)(35 61)(36 50)(37 79)(38 68)(39 57)(40 46)
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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,48)(42,47)(43,46)(44,45)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65), (1,21)(3,23)(5,25)(7,27)(9,29)(11,31)(13,33)(15,35)(17,37)(19,39)(41,61)(43,63)(45,65)(47,67)(49,69)(51,71)(53,73)(55,75)(57,77)(59,79), (1,75)(2,64)(3,53)(4,42)(5,71)(6,60)(7,49)(8,78)(9,67)(10,56)(11,45)(12,74)(13,63)(14,52)(15,41)(16,70)(17,59)(18,48)(19,77)(20,66)(21,55)(22,44)(23,73)(24,62)(25,51)(26,80)(27,69)(28,58)(29,47)(30,76)(31,65)(32,54)(33,43)(34,72)(35,61)(36,50)(37,79)(38,68)(39,57)(40,46)>;
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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,48)(42,47)(43,46)(44,45)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65), (1,21)(3,23)(5,25)(7,27)(9,29)(11,31)(13,33)(15,35)(17,37)(19,39)(41,61)(43,63)(45,65)(47,67)(49,69)(51,71)(53,73)(55,75)(57,77)(59,79), (1,75)(2,64)(3,53)(4,42)(5,71)(6,60)(7,49)(8,78)(9,67)(10,56)(11,45)(12,74)(13,63)(14,52)(15,41)(16,70)(17,59)(18,48)(19,77)(20,66)(21,55)(22,44)(23,73)(24,62)(25,51)(26,80)(27,69)(28,58)(29,47)(30,76)(31,65)(32,54)(33,43)(34,72)(35,61)(36,50)(37,79)(38,68)(39,57)(40,46) );
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)], [(1,40),(2,39),(3,38),(4,37),(5,36),(6,35),(7,34),(8,33),(9,32),(10,31),(11,30),(12,29),(13,28),(14,27),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(41,48),(42,47),(43,46),(44,45),(49,80),(50,79),(51,78),(52,77),(53,76),(54,75),(55,74),(56,73),(57,72),(58,71),(59,70),(60,69),(61,68),(62,67),(63,66),(64,65)], [(1,21),(3,23),(5,25),(7,27),(9,29),(11,31),(13,33),(15,35),(17,37),(19,39),(41,61),(43,63),(45,65),(47,67),(49,69),(51,71),(53,73),(55,75),(57,77),(59,79)], [(1,75),(2,64),(3,53),(4,42),(5,71),(6,60),(7,49),(8,78),(9,67),(10,56),(11,45),(12,74),(13,63),(14,52),(15,41),(16,70),(17,59),(18,48),(19,77),(20,66),(21,55),(22,44),(23,73),(24,62),(25,51),(26,80),(27,69),(28,58),(29,47),(30,76),(31,65),(32,54),(33,43),(34,72),(35,61),(36,50),(37,79),(38,68),(39,57),(40,46)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 2 | 4 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 10 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | D10 | D8:C22 | D4xD5 | D4xD5 | D40:C22 |
| kernel | D40:C22 | D5xM4(2) | C8:D10 | D40:C2 | SD16:3D5 | Q16:D5 | Q8.D10 | C20.C23 | D4:D10 | C5xC8.C22 | C2xQ8:2D5 | D5xC4oD4 | C4xD5 | C2xDic5 | C22xD5 | C8.C22 | M4(2) | SD16 | Q16 | C2xQ8 | C4oD4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 |
Matrix representation of D40:C22 ►in GL8(F41)
| 28 | 13 | 0 | 27 | 0 | 0 | 0 | 0 |
| 28 | 19 | 14 | 0 | 0 | 0 | 0 | 0 |
| 32 | 19 | 22 | 28 | 0 | 0 | 0 | 0 |
| 0 | 32 | 13 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 | 25 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 33 | 9 |
| 0 | 0 | 0 | 0 | 1 | 1 | 36 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 23 | 0 |
| 28 | 13 | 0 | 27 | 0 | 0 | 0 | 0 |
| 22 | 13 | 27 | 16 | 0 | 0 | 0 | 0 |
| 22 | 9 | 19 | 18 | 0 | 0 | 0 | 0 |
| 9 | 0 | 28 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 38 | 38 | 16 | 9 |
| 0 | 0 | 0 | 0 | 39 | 39 | 8 | 32 |
| 0 | 0 | 0 | 0 | 40 | 40 | 5 | 0 |
| 0 | 0 | 0 | 0 | 23 | 32 | 31 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 30 | 0 | 28 | 22 | 0 | 0 | 0 | 0 |
| 36 | 11 | 22 | 23 | 0 | 0 | 0 | 0 |
| 34 | 37 | 5 | 36 | 0 | 0 | 0 | 0 |
| 37 | 20 | 30 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 9 | 23 | 38 |
| 0 | 0 | 0 | 0 | 23 | 32 | 8 | 39 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 39 | 32 |
G:=sub<GL(8,GF(41))| [28,28,32,0,0,0,0,0,13,19,19,32,0,0,0,0,0,14,22,13,0,0,0,0,27,0,28,13,0,0,0,0,0,0,0,0,3,2,1,0,0,0,0,0,3,2,1,9,0,0,0,0,25,33,36,23,0,0,0,0,0,9,0,0],[28,22,22,9,0,0,0,0,13,13,9,0,0,0,0,0,0,27,19,28,0,0,0,0,27,16,18,22,0,0,0,0,0,0,0,0,38,39,40,23,0,0,0,0,38,39,40,32,0,0,0,0,16,8,5,31,0,0,0,0,9,32,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,6,4,1,0,0,0,0,0,0,0,0,1],[30,36,34,37,0,0,0,0,0,11,37,20,0,0,0,0,28,22,5,30,0,0,0,0,22,23,36,36,0,0,0,0,0,0,0,0,9,23,0,0,0,0,0,0,9,32,0,0,0,0,0,0,23,8,9,39,0,0,0,0,38,39,40,32] >;
D40:C22 in GAP, Magma, Sage, TeX
D_{40}\rtimes C_2^2 % in TeX
G:=Group("D40:C2^2"); // GroupNames label
G:=SmallGroup(320,1449);
// by ID
G=gap.SmallGroup(320,1449);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,1123,185,136,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^40=b^2=c^2=d^2=1,b*a*b=a^-1,c*a*c=a^21,d*a*d=a^29,c*b*c=a^20*b,d*b*d=a^8*b,c*d=d*c>;
// generators/relations