metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22.2D40, C23.31D20, (C2×D20)⋊2C4, C4⋊Dic5⋊4C4, C22⋊C8⋊2D5, (C2×C10).1D8, C10.21C4≀C2, (C2×C20).440D4, C20⋊7D4.1C2, (C2×C10).2SD16, (C22×C10).40D4, (C22×C4).55D10, C5⋊4(C22.SD16), C2.7(D20⋊7C4), C2.3(D20⋊5C4), C10.27(C23⋊C4), C22.2(C40⋊C2), C10.26(D4⋊C4), (C22×C20).41C22, C10.10C42⋊26C2, C2.7(C23.1D10), C22.59(D10⋊C4), (C5×C22⋊C8)⋊2C2, (C2×C4).13(C4×D5), (C2×C20).198(C2×C4), (C2×C4).211(C5⋊D4), (C2×C10).106(C22⋊C4), SmallGroup(320,28)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.2D40
G = < a,b,c,d | a2=b2=c40=1, d2=a, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=ac-1 >
Subgroups: 494 in 90 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C2.C42, C22⋊C8, C4⋊D4, C40, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C22.SD16, C4⋊Dic5, D10⋊C4, C2×C40, C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, C10.10C42, C5×C22⋊C8, C20⋊7D4, C22.2D40
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D8, SD16, D10, C23⋊C4, D4⋊C4, C4≀C2, C4×D5, D20, C5⋊D4, C22.SD16, C40⋊C2, D40, D10⋊C4, C23.1D10, D20⋊5C4, D20⋊7C4, C22.2D40
►On 80 points
(1 66)(3 68)(5 70)(7 72)(9 74)(11 76)(13 78)(15 80)(17 42)(19 44)(21 46)(23 48)(25 50)(27 52)(29 54)(31 56)(33 58)(35 60)(37 62)(39 64)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 73)(9 74)(10 75)(11 76)(12 77)(13 78)(14 79)(15 80)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 57)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 65)
(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 15 66 80)(2 14)(3 78 68 13)(4 77)(5 11 70 76)(6 10)(7 74 72 9)(8 73)(12 69)(16 65)(17 39 42 64)(18 38)(19 62 44 37)(20 61)(21 35 46 60)(22 34)(23 58 48 33)(24 57)(25 31 50 56)(26 30)(27 54 52 29)(28 53)(32 49)(36 45)(40 41)(43 63)(47 59)(51 55)(67 79)(71 75)
G:=sub<Sym(80)| (1,66)(3,68)(5,70)(7,72)(9,74)(11,76)(13,78)(15,80)(17,42)(19,44)(21,46)(23,48)(25,50)(27,52)(29,54)(31,56)(33,58)(35,60)(37,62)(39,64), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,65), (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), (1,15,66,80)(2,14)(3,78,68,13)(4,77)(5,11,70,76)(6,10)(7,74,72,9)(8,73)(12,69)(16,65)(17,39,42,64)(18,38)(19,62,44,37)(20,61)(21,35,46,60)(22,34)(23,58,48,33)(24,57)(25,31,50,56)(26,30)(27,54,52,29)(28,53)(32,49)(36,45)(40,41)(43,63)(47,59)(51,55)(67,79)(71,75)>;
G:=Group( (1,66)(3,68)(5,70)(7,72)(9,74)(11,76)(13,78)(15,80)(17,42)(19,44)(21,46)(23,48)(25,50)(27,52)(29,54)(31,56)(33,58)(35,60)(37,62)(39,64), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,65), (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), (1,15,66,80)(2,14)(3,78,68,13)(4,77)(5,11,70,76)(6,10)(7,74,72,9)(8,73)(12,69)(16,65)(17,39,42,64)(18,38)(19,62,44,37)(20,61)(21,35,46,60)(22,34)(23,58,48,33)(24,57)(25,31,50,56)(26,30)(27,54,52,29)(28,53)(32,49)(36,45)(40,41)(43,63)(47,59)(51,55)(67,79)(71,75) );
G=PermutationGroup([[(1,66),(3,68),(5,70),(7,72),(9,74),(11,76),(13,78),(15,80),(17,42),(19,44),(21,46),(23,48),(25,50),(27,52),(29,54),(31,56),(33,58),(35,60),(37,62),(39,64)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,73),(9,74),(10,75),(11,76),(12,77),(13,78),(14,79),(15,80),(16,41),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,49),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(31,56),(32,57),(33,58),(34,59),(35,60),(36,61),(37,62),(38,63),(39,64),(40,65)], [(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)], [(1,15,66,80),(2,14),(3,78,68,13),(4,77),(5,11,70,76),(6,10),(7,74,72,9),(8,73),(12,69),(16,65),(17,39,42,64),(18,38),(19,62,44,37),(20,61),(21,35,46,60),(22,34),(23,58,48,33),(24,57),(25,31,50,56),(26,30),(27,54,52,29),(28,53),(32,49),(36,45),(40,41),(43,63),(47,59),(51,55),(67,79),(71,75)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 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 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 40 | 2 | 2 | 4 | 20 | 20 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D5 | D8 | SD16 | D10 | C4≀C2 | C4×D5 | C5⋊D4 | D20 | C40⋊C2 | D40 | C23⋊C4 | C23.1D10 | D20⋊7C4 |
| kernel | C22.2D40 | C10.10C42 | C5×C22⋊C8 | C20⋊7D4 | C4⋊Dic5 | C2×D20 | C2×C20 | C22×C10 | C22⋊C8 | C2×C10 | C2×C10 | C22×C4 | C10 | C2×C4 | C2×C4 | C23 | C22 | C22 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 1 | 4 | 4 |
Matrix representation of C22.2D40 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 11 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 6 | 39 | 0 | 0 |
| 2 | 20 | 0 | 0 |
| 0 | 0 | 19 | 31 |
| 0 | 0 | 37 | 22 |
| 14 | 27 | 0 | 0 |
| 11 | 27 | 0 | 0 |
| 0 | 0 | 9 | 27 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,11,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[6,2,0,0,39,20,0,0,0,0,19,37,0,0,31,22],[14,11,0,0,27,27,0,0,0,0,9,0,0,0,27,40] >;
C22.2D40 in GAP, Magma, Sage, TeX
C_2^2._2D_{40} % in TeX
G:=Group("C2^2.2D40"); // GroupNames label
G:=SmallGroup(320,28);
// by ID
G=gap.SmallGroup(320,28);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,85,92,422,387,100,1123,570,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^40=1,d^2=a,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations