metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.5D4, C10.1Q8, Dic5⋊1C4, C2.1Dic10, C22.4D10, C5⋊2(C4⋊C4), C2.4(C4×D5), (C2×C4).1D5, (C2×C20).1C2, C10.11(C2×C4), C2.1(C5⋊D4), (C2×C10).4C22, (C2×Dic5).1C2, SmallGroup(80,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.D4
G = < a,b,c | a10=b4=1, c2=a5, bab-1=cac-1=a-1, cbc-1=b-1 >
Character table of C10.D4
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | 1 | -i | i | i | -1 | -i | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | i | i | i | i | -i | -i | -i | -i | linear of order 4 |
| ρ6 | 1 | -1 | -1 | 1 | i | -i | -i | -1 | i | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -i | -i | -i | -i | i | i | i | i | linear of order 4 |
| ρ7 | 1 | -1 | -1 | 1 | i | -i | i | 1 | -i | -1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -i | -i | -i | -i | i | i | i | i | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | -i | i | -i | 1 | i | -1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | i | i | i | i | -i | -i | -i | -i | linear of order 4 |
| ρ9 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | orthogonal lifted from D10 |
| ρ12 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | orthogonal lifted from D10 |
| ρ13 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | -ζ4ζ53+ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | ζ4ζ53-ζ4ζ52 | ζ43ζ54-ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | symplectic lifted from Dic10, Schur index 2 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | ζ4ζ53-ζ4ζ52 | ζ43ζ54-ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | ζ4ζ53-ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | symplectic lifted from Dic10, Schur index 2 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | ζ43ζ54-ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | ζ4ζ53-ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | ζ43ζ54-ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | symplectic lifted from Dic10, Schur index 2 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | -ζ43ζ54+ζ43ζ5 | ζ4ζ53-ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | ζ43ζ54-ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | symplectic lifted from Dic10, Schur index 2 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | ζ53-ζ52 | -ζ54+ζ5 | ζ54-ζ5 | -ζ53+ζ52 | -ζ54+ζ5 | ζ54-ζ5 | -ζ53+ζ52 | ζ53-ζ52 | complex lifted from C5⋊D4 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | -ζ54+ζ5 | -ζ53+ζ52 | ζ53-ζ52 | ζ54-ζ5 | -ζ53+ζ52 | ζ53-ζ52 | ζ54-ζ5 | -ζ54+ζ5 | complex lifted from C5⋊D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | -1-√5/2 | -1+√5/2 | 1+√5/2 | -ζ53+ζ52 | ζ54-ζ5 | -ζ54+ζ5 | ζ53-ζ52 | ζ54-ζ5 | -ζ54+ζ5 | ζ53-ζ52 | -ζ53+ζ52 | complex lifted from C5⋊D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | -1+√5/2 | -1-√5/2 | 1-√5/2 | ζ54-ζ5 | ζ53-ζ52 | -ζ53+ζ52 | -ζ54+ζ5 | ζ53-ζ52 | -ζ53+ζ52 | -ζ54+ζ5 | ζ54-ζ5 | complex lifted from C5⋊D4 |
| ρ23 | 2 | -2 | -2 | 2 | 2i | -2i | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | -1+√5/2 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | ζ4ζ53+ζ4ζ52 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | ζ4ζ54+ζ4ζ5 | complex lifted from C4×D5 |
| ρ24 | 2 | -2 | -2 | 2 | -2i | 2i | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | -1-√5/2 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | ζ43ζ54+ζ43ζ5 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | ζ43ζ53+ζ43ζ52 | complex lifted from C4×D5 |
| ρ25 | 2 | -2 | -2 | 2 | -2i | 2i | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | -1+√5/2 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | ζ43ζ53+ζ43ζ52 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | ζ43ζ54+ζ43ζ5 | complex lifted from C4×D5 |
| ρ26 | 2 | -2 | -2 | 2 | 2i | -2i | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | -1-√5/2 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | ζ4ζ54+ζ4ζ5 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | ζ4ζ53+ζ4ζ52 | complex lifted from C4×D5 |
►Regular action 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 65 30 60)(2 64 21 59)(3 63 22 58)(4 62 23 57)(5 61 24 56)(6 70 25 55)(7 69 26 54)(8 68 27 53)(9 67 28 52)(10 66 29 51)(11 50 71 31)(12 49 72 40)(13 48 73 39)(14 47 74 38)(15 46 75 37)(16 45 76 36)(17 44 77 35)(18 43 78 34)(19 42 79 33)(20 41 80 32)
(1 75 6 80)(2 74 7 79)(3 73 8 78)(4 72 9 77)(5 71 10 76)(11 29 16 24)(12 28 17 23)(13 27 18 22)(14 26 19 21)(15 25 20 30)(31 51 36 56)(32 60 37 55)(33 59 38 54)(34 58 39 53)(35 57 40 52)(41 65 46 70)(42 64 47 69)(43 63 48 68)(44 62 49 67)(45 61 50 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), (1,65,30,60)(2,64,21,59)(3,63,22,58)(4,62,23,57)(5,61,24,56)(6,70,25,55)(7,69,26,54)(8,68,27,53)(9,67,28,52)(10,66,29,51)(11,50,71,31)(12,49,72,40)(13,48,73,39)(14,47,74,38)(15,46,75,37)(16,45,76,36)(17,44,77,35)(18,43,78,34)(19,42,79,33)(20,41,80,32), (1,75,6,80)(2,74,7,79)(3,73,8,78)(4,72,9,77)(5,71,10,76)(11,29,16,24)(12,28,17,23)(13,27,18,22)(14,26,19,21)(15,25,20,30)(31,51,36,56)(32,60,37,55)(33,59,38,54)(34,58,39,53)(35,57,40,52)(41,65,46,70)(42,64,47,69)(43,63,48,68)(44,62,49,67)(45,61,50,66)>;
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), (1,65,30,60)(2,64,21,59)(3,63,22,58)(4,62,23,57)(5,61,24,56)(6,70,25,55)(7,69,26,54)(8,68,27,53)(9,67,28,52)(10,66,29,51)(11,50,71,31)(12,49,72,40)(13,48,73,39)(14,47,74,38)(15,46,75,37)(16,45,76,36)(17,44,77,35)(18,43,78,34)(19,42,79,33)(20,41,80,32), (1,75,6,80)(2,74,7,79)(3,73,8,78)(4,72,9,77)(5,71,10,76)(11,29,16,24)(12,28,17,23)(13,27,18,22)(14,26,19,21)(15,25,20,30)(31,51,36,56)(32,60,37,55)(33,59,38,54)(34,58,39,53)(35,57,40,52)(41,65,46,70)(42,64,47,69)(43,63,48,68)(44,62,49,67)(45,61,50,66) );
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)], [(1,65,30,60),(2,64,21,59),(3,63,22,58),(4,62,23,57),(5,61,24,56),(6,70,25,55),(7,69,26,54),(8,68,27,53),(9,67,28,52),(10,66,29,51),(11,50,71,31),(12,49,72,40),(13,48,73,39),(14,47,74,38),(15,46,75,37),(16,45,76,36),(17,44,77,35),(18,43,78,34),(19,42,79,33),(20,41,80,32)], [(1,75,6,80),(2,74,7,79),(3,73,8,78),(4,72,9,77),(5,71,10,76),(11,29,16,24),(12,28,17,23),(13,27,18,22),(14,26,19,21),(15,25,20,30),(31,51,36,56),(32,60,37,55),(33,59,38,54),(34,58,39,53),(35,57,40,52),(41,65,46,70),(42,64,47,69),(43,63,48,68),(44,62,49,67),(45,61,50,66)]])
C10.D4 is a maximal subgroup of
C4×Dic10 C20.6Q8 C42⋊D5 C42⋊2D5 C23.11D10 Dic5.14D4 C23.D10 Dic5⋊4D4 D10.12D4 D10⋊D4 Dic5⋊3Q8 C20⋊Q8 Dic5.Q8 C4.Dic10 D5×C4⋊C4 D10.13D4 D10⋊Q8 C4⋊C4⋊D5 C20.48D4 C4×C5⋊D4 C23.23D10 C23.18D10 Dic5⋊D4 Dic5⋊Q8 D10⋊3Q8 C30.Q8 Dic15⋊5C4 C30.4Q8 C50.D4 Dic5⋊Dic5 C10.Dic10 C102.22C22 D5.Dic10
C10.D4 is a maximal quotient of
C10.D8 C20.Q8 C20.8Q8 C20.53D4 C10.10C42 C30.Q8 Dic15⋊5C4 C30.4Q8 C50.D4 Dic5⋊Dic5 C10.Dic10 C102.22C22 D5.Dic10
Matrix representation of C10.D4 ►in GL4(𝔽41) generated by
| 1 | 34 | 0 | 0 |
| 7 | 34 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 7 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 32 | 0 | 0 | 0 |
| 19 | 9 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [1,7,0,0,34,34,0,0,0,0,1,0,0,0,0,1],[1,7,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[32,19,0,0,0,9,0,0,0,0,40,0,0,0,0,1] >;
C10.D4 in GAP, Magma, Sage, TeX
C_{10}.D_4 % in TeX
G:=Group("C10.D4"); // GroupNames label
G:=SmallGroup(80,12);
// by ID
G=gap.SmallGroup(80,12);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,40,101,26,1604]);
// Polycyclic
G:=Group<a,b,c|a^10=b^4=1,c^2=a^5,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C10.D4 in TeX
Character table of C10.D4 in TeX