metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D40⋊16C4, Dic20⋊16C4, M4(2).26D10, C40⋊C2⋊8C4, C40⋊8C4⋊1C2, C8.11(C4×D5), C5⋊6(C8.26D4), C40.70(C2×C4), (C2×C8).69D10, C4.212(D4×D5), C10.85(C4×D4), C8.C4⋊4D5, C5⋊2C8.51D4, D20⋊7C4⋊7C2, D20.24(C2×C4), C20.371(C2×D4), D40⋊7C2.4C2, (C2×C40).41C22, D20.2C4⋊10C2, (C2×C20).310C23, C20.113(C22×C4), Dic10.25(C2×C4), C4○D20.17C22, C2.15(D20⋊8C4), C22.1(Q8⋊2D5), (C4×Dic5).46C22, (C5×M4(2)).20C22, C4.46(C2×C4×D5), (C5×C8.C4)⋊4C2, (C2×C10).1(C4○D4), (C2×C5⋊2C8).78C22, (C2×C4).413(C22×D5), 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 [×3], C4 [×2], C4 [×3], C22, C22 [×2], C5, C8 [×2], C8 [×4], C2×C4, C2×C4 [×3], D4 [×4], Q8 [×2], D5 [×2], C10, C10, C42, C2×C8, C2×C8 [×3], M4(2) [×2], M4(2) [×2], D8, SD16 [×2], Q16, C4○D4 [×2], Dic5 [×3], C20 [×2], D10 [×2], C2×C10, C8⋊C4, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, C5⋊2C8 [×2], C40 [×2], C40 [×2], Dic10 [×2], C4×D5 [×2], D20 [×2], C2×Dic5, C5⋊D4 [×2], C2×C20, C8.26D4, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], D40, Dic20, C2×C5⋊2C8, C4×Dic5, C2×C40, C5×M4(2) [×2], C4○D20 [×2], C40⋊8C4, D20⋊7C4 [×2], C5×C8.C4, D40⋊7C2, D20.2C4 [×2], D40⋊16C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C4×D5 [×2], C22×D5, C8.26D4, C2×C4×D5, D4×D5, 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 76)(2 75)(3 74)(4 73)(5 72)(6 71)(7 70)(8 69)(9 68)(10 67)(11 66)(12 65)(13 64)(14 63)(15 62)(16 61)(17 60)(18 59)(19 58)(20 57)(21 56)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 80)(38 79)(39 78)(40 77)
(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 53)(43 71)(44 60)(45 49)(46 78)(47 67)(48 56)(50 74)(51 63)(54 70)(55 59)(57 77)(58 66)(61 73)(64 80)(65 69)(68 76)(75 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), (1,76)(2,75)(3,74)(4,73)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,80)(38,79)(39,78)(40,77), (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)(41,53)(43,71)(44,60)(45,49)(46,78)(47,67)(48,56)(50,74)(51,63)(54,70)(55,59)(57,77)(58,66)(61,73)(64,80)(65,69)(68,76)(75,79)>;
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), (1,76)(2,75)(3,74)(4,73)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,80)(38,79)(39,78)(40,77), (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)(41,53)(43,71)(44,60)(45,49)(46,78)(47,67)(48,56)(50,74)(51,63)(54,70)(55,59)(57,77)(58,66)(61,73)(64,80)(65,69)(68,76)(75,79) );
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)], [(1,76),(2,75),(3,74),(4,73),(5,72),(6,71),(7,70),(8,69),(9,68),(10,67),(11,66),(12,65),(13,64),(14,63),(15,62),(16,61),(17,60),(18,59),(19,58),(20,57),(21,56),(22,55),(23,54),(24,53),(25,52),(26,51),(27,50),(28,49),(29,48),(30,47),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41),(37,80),(38,79),(39,78),(40,77)], [(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),(41,53),(43,71),(44,60),(45,49),(46,78),(47,67),(48,56),(50,74),(51,63),(54,70),(55,59),(57,77),(58,66),(61,73),(64,80),(65,69),(68,76),(75,79)])
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 | C4○D4 | D10 | D10 | C4×D5 | C8.26D4 | D4×D5 | Q8⋊2D5 | D40⋊16C4 |
| kernel | D40⋊16C4 | C40⋊8C4 | D20⋊7C4 | C5×C8.C4 | D40⋊7C2 | D20.2C4 | C40⋊C2 | D40 | Dic20 | C5⋊2C8 | C8.C4 | C2×C10 | C2×C8 | 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