metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q16⋊3F5, Dic20⋊3C4, C8.8(C2×F5), C40.6(C2×C4), C5⋊(Q16⋊C4), (C5×Q16)⋊2C4, C5⋊Q16⋊4C4, (C2×F5).7D4, C2.24(D4×F5), (Q8×F5).2C2, Q8.4(C2×F5), C10.23(C4×D4), C40⋊C4.1C2, C8⋊F5.1C2, (D5×Q16).3C2, C4⋊F5.6C22, D10.68(C2×D4), D5⋊C8.5C22, Q8⋊F5.2C2, (C4×F5).5C22, C4.10(C22×F5), (Q8×D5).7C22, C20.10(C22×C4), Dic10.6(C2×C4), (C4×D5).32C23, (C8×D5).15C22, Dic5.6(C4○D4), D5.3(C8.C22), (C5×Q8).4(C2×C4), C5⋊2C8.12(C2×C4), SmallGroup(320,1077)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic20⋊C4
G = < a,b,c | a40=c4=1, b2=a20, bab-1=a-1, cac-1=a13, cbc-1=a20b >
Subgroups: 418 in 108 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, Q8, Q8, D5, C10, C42, C4⋊C4, C2×C8, Q16, Q16, C2×Q8, Dic5, Dic5, C20, C20, F5, D10, C8⋊C4, Q8⋊C4, C4.Q8, C4×Q8, C2×Q16, C5⋊2C8, C40, C5⋊C8, Dic10, Dic10, C4×D5, C4×D5, C5×Q8, C2×F5, C2×F5, Q16⋊C4, C8×D5, Dic20, C5⋊Q16, C5×Q16, D5⋊C8, C4×F5, C4×F5, C4⋊F5, C4⋊F5, Q8×D5, C8⋊F5, C40⋊C4, Q8⋊F5, D5×Q16, Q8×F5, Dic20⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, F5, C4×D4, C8.C22, C2×F5, Q16⋊C4, C22×F5, D4×F5, Dic20⋊C4
Character table of Dic20⋊C4
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5 | 8A | 8B | 8C | 8D | 10 | 20A | 20B | 20C | 40A | 40B | |
| size | 1 | 1 | 5 | 5 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 4 | 4 | 20 | 20 | 20 | 4 | 8 | 16 | 16 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | -1 | i | i | -i | -1 | 1 | -i | i | i | -i | 1 | -1 | 1 | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ10 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | i | -1 | -i | -i | i | 1 | 1 | i | -i | i | -i | 1 | 1 | -1 | -i | i | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ11 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | -1 | i | i | -i | 1 | -1 | i | -i | -i | i | 1 | -1 | 1 | -i | i | 1 | 1 | -1 | 1 | -1 | -1 | linear of order 4 |
| ρ12 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | i | -1 | -i | -i | i | -1 | -1 | -i | i | -i | i | 1 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ13 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | i | -1 | -i | -i | i | 1 | -1 | -i | i | i | -i | 1 | -1 | 1 | i | -i | 1 | 1 | -1 | 1 | -1 | -1 | linear of order 4 |
| ρ14 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -i | -1 | i | i | -i | -1 | -1 | i | -i | i | -i | 1 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ15 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | i | -1 | -i | -i | i | -1 | 1 | i | -i | -i | i | 1 | -1 | 1 | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ16 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -i | -1 | i | i | -i | 1 | 1 | -i | i | -i | i | 1 | 1 | -1 | i | -i | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | -2i | 2 | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 2i | 2 | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 4 | 4 | 0 | 0 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ22 | 4 | 4 | 0 | 0 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | -1 | orthogonal lifted from C2×F5 |
| ρ23 | 4 | 4 | 0 | 0 | 4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ24 | 4 | 4 | 0 | 0 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ27 | 8 | 8 | 0 | 0 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4×F5 |
| ρ28 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -√10 | √10 | symplectic faithful, Schur index 2 |
| ρ29 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | √10 | -√10 | symplectic faithful, Schur index 2 |
►On 80 points
Generators in S80
(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 72 21 52)(2 71 22 51)(3 70 23 50)(4 69 24 49)(5 68 25 48)(6 67 26 47)(7 66 27 46)(8 65 28 45)(9 64 29 44)(10 63 30 43)(11 62 31 42)(12 61 32 41)(13 60 33 80)(14 59 34 79)(15 58 35 78)(16 57 36 77)(17 56 37 76)(18 55 38 75)(19 54 39 74)(20 53 40 73)
(1 57 21 77)(2 54 30 50)(3 51 39 63)(4 48 8 76)(5 45 17 49)(6 42 26 62)(7 79 35 75)(9 73 13 61)(10 70 22 74)(11 67 31 47)(12 64 40 60)(14 58 18 46)(15 55 27 59)(16 52 36 72)(19 43 23 71)(20 80 32 44)(24 68 28 56)(25 65 37 69)(29 53 33 41)(34 78 38 66)
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), (1,72,21,52)(2,71,22,51)(3,70,23,50)(4,69,24,49)(5,68,25,48)(6,67,26,47)(7,66,27,46)(8,65,28,45)(9,64,29,44)(10,63,30,43)(11,62,31,42)(12,61,32,41)(13,60,33,80)(14,59,34,79)(15,58,35,78)(16,57,36,77)(17,56,37,76)(18,55,38,75)(19,54,39,74)(20,53,40,73), (1,57,21,77)(2,54,30,50)(3,51,39,63)(4,48,8,76)(5,45,17,49)(6,42,26,62)(7,79,35,75)(9,73,13,61)(10,70,22,74)(11,67,31,47)(12,64,40,60)(14,58,18,46)(15,55,27,59)(16,52,36,72)(19,43,23,71)(20,80,32,44)(24,68,28,56)(25,65,37,69)(29,53,33,41)(34,78,38,66)>;
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), (1,72,21,52)(2,71,22,51)(3,70,23,50)(4,69,24,49)(5,68,25,48)(6,67,26,47)(7,66,27,46)(8,65,28,45)(9,64,29,44)(10,63,30,43)(11,62,31,42)(12,61,32,41)(13,60,33,80)(14,59,34,79)(15,58,35,78)(16,57,36,77)(17,56,37,76)(18,55,38,75)(19,54,39,74)(20,53,40,73), (1,57,21,77)(2,54,30,50)(3,51,39,63)(4,48,8,76)(5,45,17,49)(6,42,26,62)(7,79,35,75)(9,73,13,61)(10,70,22,74)(11,67,31,47)(12,64,40,60)(14,58,18,46)(15,55,27,59)(16,52,36,72)(19,43,23,71)(20,80,32,44)(24,68,28,56)(25,65,37,69)(29,53,33,41)(34,78,38,66) );
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)], [(1,72,21,52),(2,71,22,51),(3,70,23,50),(4,69,24,49),(5,68,25,48),(6,67,26,47),(7,66,27,46),(8,65,28,45),(9,64,29,44),(10,63,30,43),(11,62,31,42),(12,61,32,41),(13,60,33,80),(14,59,34,79),(15,58,35,78),(16,57,36,77),(17,56,37,76),(18,55,38,75),(19,54,39,74),(20,53,40,73)], [(1,57,21,77),(2,54,30,50),(3,51,39,63),(4,48,8,76),(5,45,17,49),(6,42,26,62),(7,79,35,75),(9,73,13,61),(10,70,22,74),(11,67,31,47),(12,64,40,60),(14,58,18,46),(15,55,27,59),(16,52,36,72),(19,43,23,71),(20,80,32,44),(24,68,28,56),(25,65,37,69),(29,53,33,41),(34,78,38,66)]])
Matrix representation of Dic20⋊C4 ►in GL8(𝔽41)
| 40 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 39 | 27 | 0 |
| 0 | 0 | 0 | 0 | 29 | 29 | 27 | 13 |
| 0 | 0 | 0 | 0 | 2 | 16 | 0 | 4 |
| 0 | 0 | 0 | 0 | 37 | 39 | 12 | 10 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 | 0 | 40 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 39 | 37 | 37 |
| 0 | 0 | 0 | 0 | 39 | 25 | 0 | 37 |
| 0 | 0 | 0 | 0 | 39 | 2 | 14 | 0 |
| 0 | 0 | 0 | 0 | 16 | 14 | 2 | 18 |
G:=sub<GL(8,GF(41))| [40,40,40,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,29,2,37,0,0,0,0,39,29,16,39,0,0,0,0,27,27,0,12,0,0,0,0,0,13,4,10],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,40,0,0,0,0,0,1,0,40,0,0,0,0,40,1,0,0,0,0,0,0,0,2,0,40],[0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,25,39,39,16,0,0,0,0,39,25,2,14,0,0,0,0,37,0,14,2,0,0,0,0,37,37,0,18] >;
Dic20⋊C4 in GAP, Magma, Sage, TeX
{\rm Dic}_{20}\rtimes C_4 % in TeX
G:=Group("Dic20:C4"); // GroupNames label
G:=SmallGroup(320,1077);
// by ID
G=gap.SmallGroup(320,1077);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,758,219,184,851,438,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^40=c^4=1,b^2=a^20,b*a*b^-1=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^20*b>;
// generators/relations
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