metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D40⋊8C4, C40.86D4, Dic20⋊8C4, M5(2)⋊6D5, C20.7SD16, C22.3D40, C8.6(C4×D5), C40⋊6C4⋊1C2, (C2×C10).2D8, C5⋊4(D8⋊2C4), C40.44(C2×C4), (C2×C8).48D10, (C2×C4).11D20, (C2×C20).101D4, C8.43(C5⋊D4), D40⋊7C2.7C2, C4.12(C40⋊C2), (C5×M5(2))⋊10C2, (C2×C40).52C22, C20.91(C22⋊C4), C10.34(D4⋊C4), C4.20(D10⋊C4), C2.11(D20⋊5C4), SmallGroup(320,76)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D40⋊8C4
G = < a,b,c | a40=b2=c4=1, bab=a-1, cac-1=a19, cbc-1=a23b >
Subgroups: 334 in 58 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C16, C4⋊C4, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, D10, C2×C10, C4.Q8, M5(2), C4○D8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, D8⋊2C4, C80, C40⋊C2, D40, Dic20, C4⋊Dic5, C2×C40, C4○D20, C40⋊6C4, C5×M5(2), D40⋊7C2, D40⋊8C4
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D8, SD16, D10, D4⋊C4, C4×D5, D20, C5⋊D4, D8⋊2C4, C40⋊C2, D40, D10⋊C4, D20⋊5C4, D40⋊8C4
►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 77)(2 76)(3 75)(4 74)(5 73)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 63)(16 62)(17 61)(18 60)(19 59)(20 58)(21 57)(22 56)(23 55)(24 54)(25 53)(26 52)(27 51)(28 50)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 80)(39 79)(40 78)
(2 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 70 61 50)(42 49 62 69)(43 68 63 48)(44 47 64 67)(45 66 65 46)(51 60 71 80)(52 79 72 59)(53 58 73 78)(54 77 74 57)(55 56 75 76)
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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,60)(19,59)(20,58)(21,57)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,80)(39,79)(40,78), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,70,61,50)(42,49,62,69)(43,68,63,48)(44,47,64,67)(45,66,65,46)(51,60,71,80)(52,79,72,59)(53,58,73,78)(54,77,74,57)(55,56,75,76)>;
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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,60)(19,59)(20,58)(21,57)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,80)(39,79)(40,78), (2,20)(3,39)(4,18)(5,37)(6,16)(7,35)(8,14)(9,33)(10,12)(11,31)(13,29)(15,27)(17,25)(19,23)(22,40)(24,38)(26,36)(28,34)(30,32)(41,70,61,50)(42,49,62,69)(43,68,63,48)(44,47,64,67)(45,66,65,46)(51,60,71,80)(52,79,72,59)(53,58,73,78)(54,77,74,57)(55,56,75,76) );
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)], [(1,77),(2,76),(3,75),(4,74),(5,73),(6,72),(7,71),(8,70),(9,69),(10,68),(11,67),(12,66),(13,65),(14,64),(15,63),(16,62),(17,61),(18,60),(19,59),(20,58),(21,57),(22,56),(23,55),(24,54),(25,53),(26,52),(27,51),(28,50),(29,49),(30,48),(31,47),(32,46),(33,45),(34,44),(35,43),(36,42),(37,41),(38,80),(39,79),(40,78)], [(2,20),(3,39),(4,18),(5,37),(6,16),(7,35),(8,14),(9,33),(10,12),(11,31),(13,29),(15,27),(17,25),(19,23),(22,40),(24,38),(26,36),(28,34),(30,32),(41,70,61,50),(42,49,62,69),(43,68,63,48),(44,47,64,67),(45,66,65,46),(51,60,71,80),(52,79,72,59),(53,58,73,78),(54,77,74,57),(55,56,75,76)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 10A | 10B | 10C | 10D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H | 40I | 40J | 40K | 40L | 80A | ··· | 80P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 40 | 2 | 2 | 40 | 40 | 40 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D5 | SD16 | D8 | D10 | C4×D5 | C5⋊D4 | D20 | C40⋊C2 | D40 | D8⋊2C4 | D40⋊8C4 |
| kernel | D40⋊8C4 | C40⋊6C4 | C5×M5(2) | D40⋊7C2 | D40 | Dic20 | C40 | C2×C20 | M5(2) | C20 | C2×C10 | C2×C8 | C8 | C8 | C2×C4 | C4 | C22 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 2 | 8 |
Matrix representation of D40⋊8C4 ►in GL4(𝔽241) generated by
| 31 | 200 | 108 | 187 |
| 41 | 109 | 197 | 227 |
| 0 | 0 | 169 | 4 |
| 0 | 0 | 200 | 173 |
| 159 | 110 | 16 | 1 |
| 159 | 110 | 110 | 117 |
| 153 | 188 | 146 | 25 |
| 90 | 178 | 131 | 67 |
| 190 | 1 | 57 | 0 |
| 51 | 51 | 9 | 38 |
| 0 | 0 | 173 | 234 |
| 0 | 0 | 41 | 68 |
G:=sub<GL(4,GF(241))| [31,41,0,0,200,109,0,0,108,197,169,200,187,227,4,173],[159,159,153,90,110,110,188,178,16,110,146,131,1,117,25,67],[190,51,0,0,1,51,0,0,57,9,173,41,0,38,234,68] >;
D40⋊8C4 in GAP, Magma, Sage, TeX
D_{40}\rtimes_8C_4 % in TeX
G:=Group("D40:8C4"); // GroupNames label
G:=SmallGroup(320,76);
// by ID
G=gap.SmallGroup(320,76);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,85,92,422,387,268,570,136,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^23*b>;
// generators/relations