metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D40⋊17C4, C40.67D4, C8.14D20, Dic20⋊17C4, C42.267D10, (C4×C8)⋊10D5, C5⋊4(C8○D8), (C4×C40)⋊15C2, C8.22(C4×D5), C40⋊C2⋊11C4, C40.93(C2×C4), C10.39(C4×D4), C2.12(C4×D20), C4.76(C2×D20), D20.27(C2×C4), (C2×C8).324D10, C20.296(C2×D4), C40.6C4⋊17C2, D20⋊4C4⋊15C2, D40⋊7C2.10C2, D20.3C4⋊11C2, (C2×C20).791C23, C20.163(C22×C4), (C2×C40).407C22, (C4×C20).328C22, Dic10.28(C2×C4), C4○D20.35C22, C22.20(C4○D20), C4.Dic5.33C22, C4.62(C2×C4×D5), (C2×C10).62(C4○D4), (C2×C4).672(C22×D5), SmallGroup(320,327)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D40⋊17C4
G = < a,b,c | a40=b2=c4=1, bab=a-1, ac=ca, cbc-1=a10b >
Subgroups: 374 in 106 conjugacy classes, 47 normal (33 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, 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, C4×C8, C4≀C2, C8.C4, C8○D4, C4○D8, C5⋊2C8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C8○D8, C8×D5, C8⋊D5, C40⋊C2, D40, Dic20, C4.Dic5, C4×C20, C2×C40, C4○D20, D20⋊4C4, C40.6C4, C4×C40, D20.3C4, D40⋊7C2, D40⋊17C4
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○D8, C2×C4×D5, C2×D20, C4○D20, C4×D20, D40⋊17C4
►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 80)(2 79)(3 78)(4 77)(5 76)(6 75)(7 74)(8 73)(9 72)(10 71)(11 70)(12 69)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 61)(21 60)(22 59)(23 58)(24 57)(25 56)(26 55)(27 54)(28 53)(29 52)(30 51)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)
(1 31 21 11)(2 32 22 12)(3 33 23 13)(4 34 24 14)(5 35 25 15)(6 36 26 16)(7 37 27 17)(8 38 28 18)(9 39 29 19)(10 40 30 20)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(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), (1,80)(2,79)(3,78)(4,77)(5,76)(6,75)(7,74)(8,73)(9,72)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41), (1,31,21,11)(2,32,22,12)(3,33,23,13)(4,34,24,14)(5,35,25,15)(6,36,26,16)(7,37,27,17)(8,38,28,18)(9,39,29,19)(10,40,30,20)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(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), (1,80)(2,79)(3,78)(4,77)(5,76)(6,75)(7,74)(8,73)(9,72)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41), (1,31,21,11)(2,32,22,12)(3,33,23,13)(4,34,24,14)(5,35,25,15)(6,36,26,16)(7,37,27,17)(8,38,28,18)(9,39,29,19)(10,40,30,20)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(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)], [(1,80),(2,79),(3,78),(4,77),(5,76),(6,75),(7,74),(8,73),(9,72),(10,71),(11,70),(12,69),(13,68),(14,67),(15,66),(16,65),(17,64),(18,63),(19,62),(20,61),(21,60),(22,59),(23,58),(24,57),(25,56),(26,55),(27,54),(28,53),(29,52),(30,51),(31,50),(32,49),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41)], [(1,31,21,11),(2,32,22,12),(3,33,23,13),(4,34,24,14),(5,35,25,15),(6,36,26,16),(7,37,27,17),(8,38,28,18),(9,39,29,19),(10,40,30,20),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80)]])
92 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | 8L | 8M | 8N | 10A | ··· | 10F | 20A | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 20 | 20 | 1 | 1 | 2 | ··· | 2 | 20 | 20 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D5 | C4○D4 | D10 | D10 | C4×D5 | D20 | C8○D8 | C4○D20 | D40⋊17C4 |
| kernel | D40⋊17C4 | D20⋊4C4 | C40.6C4 | C4×C40 | D20.3C4 | D40⋊7C2 | C40⋊C2 | D40 | Dic20 | C40 | C4×C8 | C2×C10 | C42 | C2×C8 | C8 | C8 | C5 | C22 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 8 | 8 | 32 |
Matrix representation of D40⋊17C4 ►in GL2(𝔽41) generated by
| 28 | 0 |
| 0 | 22 |
| 0 | 22 |
| 28 | 0 |
| 32 | 0 |
| 0 | 40 |
G:=sub<GL(2,GF(41))| [28,0,0,22],[0,28,22,0],[32,0,0,40] >;
D40⋊17C4 in GAP, Magma, Sage, TeX
D_{40}\rtimes_{17}C_4 % in TeX
G:=Group("D40:17C4"); // GroupNames label
G:=SmallGroup(320,327);
// by ID
G=gap.SmallGroup(320,327);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,58,136,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^10*b>;
// generators/relations