metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.44D4, D20.26D4, Dic10.26D4, C4.66(D4×D5), (C2×SD16)⋊3D5, (C2×C8).92D10, (C10×SD16)⋊1C2, (C2×D4).78D10, C20.D4⋊8C2, C20.180(C2×D4), C5⋊5(D4.3D4), C8.32(C5⋊D4), (C2×Q8).59D10, C40.6C4⋊10C2, D20.3C4⋊4C2, C20.C23⋊4C2, C20.10D4⋊7C2, (C2×C40).48C22, C2.21(C20⋊2D4), (C2×C20).454C23, D4.D10.2C2, C4○D20.47C22, (Q8×C10).83C22, C10.118(C4⋊D4), (D4×C10).103C22, C4.Dic5.20C22, C22.21(D4⋊2D5), C4.84(C2×C5⋊D4), (C2×C4).127(C22×D5), (C2×C10).159(C4○D4), SmallGroup(320,804)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.44D4
G = < a,b,c | a40=c2=1, b4=a20, bab-1=a19, cac=a9, cbc=a20b3 >
Subgroups: 366 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C5⋊2C8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, D4.3D4, C8×D5, C8⋊D5, C4.Dic5, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C2×C40, C5×SD16, C4○D20, D4×C10, Q8×C10, C40.6C4, C20.D4, C20.10D4, D20.3C4, D4.D10, C20.C23, C10×SD16, C40.44D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C5⋊D4, C22×D5, D4.3D4, D4×D5, D4⋊2D5, C2×C5⋊D4, C20⋊2D4, C40.44D4
►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 43 11 73 21 63 31 53)(2 62 12 52 22 42 32 72)(3 41 13 71 23 61 33 51)(4 60 14 50 24 80 34 70)(5 79 15 69 25 59 35 49)(6 58 16 48 26 78 36 68)(7 77 17 67 27 57 37 47)(8 56 18 46 28 76 38 66)(9 75 19 65 29 55 39 45)(10 54 20 44 30 74 40 64)
(1 53)(2 62)(3 71)(4 80)(5 49)(6 58)(7 67)(8 76)(9 45)(10 54)(11 63)(12 72)(13 41)(14 50)(15 59)(16 68)(17 77)(18 46)(19 55)(20 64)(21 73)(22 42)(23 51)(24 60)(25 69)(26 78)(27 47)(28 56)(29 65)(30 74)(31 43)(32 52)(33 61)(34 70)(35 79)(36 48)(37 57)(38 66)(39 75)(40 44)
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), (1,43,11,73,21,63,31,53)(2,62,12,52,22,42,32,72)(3,41,13,71,23,61,33,51)(4,60,14,50,24,80,34,70)(5,79,15,69,25,59,35,49)(6,58,16,48,26,78,36,68)(7,77,17,67,27,57,37,47)(8,56,18,46,28,76,38,66)(9,75,19,65,29,55,39,45)(10,54,20,44,30,74,40,64), (1,53)(2,62)(3,71)(4,80)(5,49)(6,58)(7,67)(8,76)(9,45)(10,54)(11,63)(12,72)(13,41)(14,50)(15,59)(16,68)(17,77)(18,46)(19,55)(20,64)(21,73)(22,42)(23,51)(24,60)(25,69)(26,78)(27,47)(28,56)(29,65)(30,74)(31,43)(32,52)(33,61)(34,70)(35,79)(36,48)(37,57)(38,66)(39,75)(40,44)>;
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), (1,43,11,73,21,63,31,53)(2,62,12,52,22,42,32,72)(3,41,13,71,23,61,33,51)(4,60,14,50,24,80,34,70)(5,79,15,69,25,59,35,49)(6,58,16,48,26,78,36,68)(7,77,17,67,27,57,37,47)(8,56,18,46,28,76,38,66)(9,75,19,65,29,55,39,45)(10,54,20,44,30,74,40,64), (1,53)(2,62)(3,71)(4,80)(5,49)(6,58)(7,67)(8,76)(9,45)(10,54)(11,63)(12,72)(13,41)(14,50)(15,59)(16,68)(17,77)(18,46)(19,55)(20,64)(21,73)(22,42)(23,51)(24,60)(25,69)(26,78)(27,47)(28,56)(29,65)(30,74)(31,43)(32,52)(33,61)(34,70)(35,79)(36,48)(37,57)(38,66)(39,75)(40,44) );
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)], [(1,43,11,73,21,63,31,53),(2,62,12,52,22,42,32,72),(3,41,13,71,23,61,33,51),(4,60,14,50,24,80,34,70),(5,79,15,69,25,59,35,49),(6,58,16,48,26,78,36,68),(7,77,17,67,27,57,37,47),(8,56,18,46,28,76,38,66),(9,75,19,65,29,55,39,45),(10,54,20,44,30,74,40,64)], [(1,53),(2,62),(3,71),(4,80),(5,49),(6,58),(7,67),(8,76),(9,45),(10,54),(11,63),(12,72),(13,41),(14,50),(15,59),(16,68),(17,77),(18,46),(19,55),(20,64),(21,73),(22,42),(23,51),(24,60),(25,69),(26,78),(27,47),(28,56),(29,65),(30,74),(31,43),(32,52),(33,61),(34,70),(35,79),(36,48),(37,57),(38,66),(39,75),(40,44)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 8 | 20 | 2 | 2 | 8 | 20 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 40 | 40 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | D4.3D4 | D4×D5 | D4⋊2D5 | C40.44D4 |
| kernel | C40.44D4 | C40.6C4 | C20.D4 | C20.10D4 | D20.3C4 | D4.D10 | C20.C23 | C10×SD16 | C40 | Dic10 | D20 | C2×SD16 | C2×C10 | C2×C8 | C2×D4 | C2×Q8 | C8 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 8 |
Matrix representation of C40.44D4 ►in GL4(𝔽41) generated by
| 0 | 25 | 30 | 30 |
| 40 | 38 | 30 | 21 |
| 0 | 0 | 0 | 23 |
| 0 | 0 | 40 | 28 |
| 34 | 17 | 16 | 27 |
| 38 | 9 | 33 | 2 |
| 13 | 39 | 16 | 16 |
| 0 | 38 | 34 | 23 |
| 7 | 38 | 3 | 16 |
| 3 | 18 | 31 | 9 |
| 28 | 0 | 34 | 13 |
| 0 | 38 | 34 | 23 |
G:=sub<GL(4,GF(41))| [0,40,0,0,25,38,0,0,30,30,0,40,30,21,23,28],[34,38,13,0,17,9,39,38,16,33,16,34,27,2,16,23],[7,3,28,0,38,18,0,38,3,31,34,34,16,9,13,23] >;
C40.44D4 in GAP, Magma, Sage, TeX
C_{40}._{44}D_4 % in TeX
G:=Group("C40.44D4"); // GroupNames label
G:=SmallGroup(320,804);
// by ID
G=gap.SmallGroup(320,804);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,555,1123,297,136,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=c^2=1,b^4=a^20,b*a*b^-1=a^19,c*a*c=a^9,c*b*c=a^20*b^3>;
// generators/relations