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 [×3], C2 [×3], C4 [×5], C22 [×3], C22 [×5], C5, C8, C2×C4 [×2], C2×C4 [×6], D4 [×3], C23, C23, D5, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4 [×2], Dic5 [×3], C20 [×2], D10 [×3], C2×C10 [×3], C2×C10 [×2], C2.C42, C22⋊C8, C4⋊D4, C40, D20, C2×Dic5 [×5], C5⋊D4 [×2], C2×C20 [×2], 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 [×3], C4 [×2], C22, C2×C4, D4 [×2], 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 65)(3 67)(5 69)(7 71)(9 73)(11 75)(13 77)(15 79)(17 41)(19 43)(21 45)(23 47)(25 49)(27 51)(29 53)(31 55)(33 57)(35 59)(37 61)(39 63)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)
(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 65 79)(2 14)(3 77 67 13)(4 76)(5 11 69 75)(6 10)(7 73 71 9)(8 72)(12 68)(16 64)(17 39 41 63)(18 38)(19 61 43 37)(20 60)(21 35 45 59)(22 34)(23 57 47 33)(24 56)(25 31 49 55)(26 30)(27 53 51 29)(28 52)(32 48)(36 44)(40 80)(42 62)(46 58)(50 54)(66 78)(70 74)
G:=sub<Sym(80)| (1,65)(3,67)(5,69)(7,71)(9,73)(11,75)(13,77)(15,79)(17,41)(19,43)(21,45)(23,47)(25,49)(27,51)(29,53)(31,55)(33,57)(35,59)(37,61)(39,63), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64), (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), (1,15,65,79)(2,14)(3,77,67,13)(4,76)(5,11,69,75)(6,10)(7,73,71,9)(8,72)(12,68)(16,64)(17,39,41,63)(18,38)(19,61,43,37)(20,60)(21,35,45,59)(22,34)(23,57,47,33)(24,56)(25,31,49,55)(26,30)(27,53,51,29)(28,52)(32,48)(36,44)(40,80)(42,62)(46,58)(50,54)(66,78)(70,74)>;
G:=Group( (1,65)(3,67)(5,69)(7,71)(9,73)(11,75)(13,77)(15,79)(17,41)(19,43)(21,45)(23,47)(25,49)(27,51)(29,53)(31,55)(33,57)(35,59)(37,61)(39,63), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64), (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), (1,15,65,79)(2,14)(3,77,67,13)(4,76)(5,11,69,75)(6,10)(7,73,71,9)(8,72)(12,68)(16,64)(17,39,41,63)(18,38)(19,61,43,37)(20,60)(21,35,45,59)(22,34)(23,57,47,33)(24,56)(25,31,49,55)(26,30)(27,53,51,29)(28,52)(32,48)(36,44)(40,80)(42,62)(46,58)(50,54)(66,78)(70,74) );
G=PermutationGroup([(1,65),(3,67),(5,69),(7,71),(9,73),(11,75),(13,77),(15,79),(17,41),(19,43),(21,45),(23,47),(25,49),(27,51),(29,53),(31,55),(33,57),(35,59),(37,61),(39,63)], [(1,65),(2,66),(3,67),(4,68),(5,69),(6,70),(7,71),(8,72),(9,73),(10,74),(11,75),(12,76),(13,77),(14,78),(15,79),(16,80),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64)], [(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)], [(1,15,65,79),(2,14),(3,77,67,13),(4,76),(5,11,69,75),(6,10),(7,73,71,9),(8,72),(12,68),(16,64),(17,39,41,63),(18,38),(19,61,43,37),(20,60),(21,35,45,59),(22,34),(23,57,47,33),(24,56),(25,31,49,55),(26,30),(27,53,51,29),(28,52),(32,48),(36,44),(40,80),(42,62),(46,58),(50,54),(66,78),(70,74)])
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