direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic10, C10⋊Q8, C4.11D10, C10.1C23, C22.8D10, C20.11C22, Dic5.1C22, C5⋊1(C2×Q8), (C2×C4).4D5, (C2×C20).4C2, C2.3(C22×D5), (C2×C10).8C22, (C2×Dic5).3C2, SmallGroup(80,35)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic10
G = < a,b,c | a2=b20=1, c2=b10, ab=ba, ac=ca, cbc-1=b-1 >
Character table of C2×Dic10
| 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 | -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 | 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 | 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 | linear of order 2 |
| ρ9 | 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 |
| 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 D10 |
| 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 D5 |
| 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 D10 |
| ρ14 | 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 |
| ρ15 | 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 |
| ρ16 | 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 |
| ρ17 | 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 |
| ρ18 | 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 |
| 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 | -ζ4ζ53+ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | ζ4ζ53-ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | ζ43ζ54-ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | symplectic lifted from Dic10, Schur index 2 |
| 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 | -ζ4ζ53+ζ4ζ52 | ζ43ζ54-ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | ζ4ζ53-ζ4ζ52 | ζ43ζ54-ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | symplectic lifted from Dic10, Schur index 2 |
| 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 | ζ4ζ53-ζ4ζ52 | ζ43ζ54-ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | symplectic lifted from Dic10, Schur index 2 |
| 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 | -ζ43ζ54+ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | ζ43ζ54-ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | symplectic lifted from Dic10, Schur index 2 |
| ρ23 | 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 | ζ43ζ54-ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | ζ4ζ53-ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | symplectic lifted from Dic10, Schur index 2 |
| ρ24 | 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 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | symplectic lifted from Dic10, Schur index 2 |
| ρ25 | 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 | -ζ43ζ54+ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | ζ4ζ53-ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | symplectic lifted from Dic10, Schur index 2 |
| ρ26 | 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 | -ζ4ζ53+ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | symplectic lifted from Dic10, Schur index 2 |
►Regular action on 80 points
Generators in S80
(1 37)(2 38)(3 39)(4 40)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 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 46 11 56)(2 45 12 55)(3 44 13 54)(4 43 14 53)(5 42 15 52)(6 41 16 51)(7 60 17 50)(8 59 18 49)(9 58 19 48)(10 57 20 47)(21 62 31 72)(22 61 32 71)(23 80 33 70)(24 79 34 69)(25 78 35 68)(26 77 36 67)(27 76 37 66)(28 75 38 65)(29 74 39 64)(30 73 40 63)
G:=sub<Sym(80)| (1,37)(2,38)(3,39)(4,40)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (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), (1,46,11,56)(2,45,12,55)(3,44,13,54)(4,43,14,53)(5,42,15,52)(6,41,16,51)(7,60,17,50)(8,59,18,49)(9,58,19,48)(10,57,20,47)(21,62,31,72)(22,61,32,71)(23,80,33,70)(24,79,34,69)(25,78,35,68)(26,77,36,67)(27,76,37,66)(28,75,38,65)(29,74,39,64)(30,73,40,63)>;
G:=Group( (1,37)(2,38)(3,39)(4,40)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (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), (1,46,11,56)(2,45,12,55)(3,44,13,54)(4,43,14,53)(5,42,15,52)(6,41,16,51)(7,60,17,50)(8,59,18,49)(9,58,19,48)(10,57,20,47)(21,62,31,72)(22,61,32,71)(23,80,33,70)(24,79,34,69)(25,78,35,68)(26,77,36,67)(27,76,37,66)(28,75,38,65)(29,74,39,64)(30,73,40,63) );
G=PermutationGroup([[(1,37),(2,38),(3,39),(4,40),(5,21),(6,22),(7,23),(8,24),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80)], [(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)], [(1,46,11,56),(2,45,12,55),(3,44,13,54),(4,43,14,53),(5,42,15,52),(6,41,16,51),(7,60,17,50),(8,59,18,49),(9,58,19,48),(10,57,20,47),(21,62,31,72),(22,61,32,71),(23,80,33,70),(24,79,34,69),(25,78,35,68),(26,77,36,67),(27,76,37,66),(28,75,38,65),(29,74,39,64),(30,73,40,63)]])
C2×Dic10 is a maximal subgroup of
C10.Q16 C20.44D4 C4.12D20 Dic5.D4 C20⋊2Q8 C4.D20 Dic5.14D4 Dic5.5D4 Dic5⋊3Q8 C20⋊Q8 D10⋊Q8 D10⋊2Q8 C8.D10 C20.48D4 C20.17D4 Dic5⋊Q8 D4.9D10 C2×Q8×D5 D4.10D10
C2×Dic10 is a maximal quotient of
C20⋊2Q8 C20.6Q8 Dic5.14D4 C20⋊Q8 C4.Dic10 C20.48D4
Matrix representation of C2×Dic10 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 40 |
| 0 | 0 | 8 | 34 |
| 40 | 11 | 0 | 0 |
| 11 | 1 | 0 | 0 |
| 0 | 0 | 38 | 8 |
| 0 | 0 | 40 | 3 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[0,40,0,0,1,0,0,0,0,0,1,8,0,0,40,34],[40,11,0,0,11,1,0,0,0,0,38,40,0,0,8,3] >;
C2×Dic10 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{10} % in TeX
G:=Group("C2xDic10"); // GroupNames label
G:=SmallGroup(80,35);
// by ID
G=gap.SmallGroup(80,35);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,40,182,42,1604]);
// Polycyclic
G:=Group<a,b,c|a^2=b^20=1,c^2=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C2×Dic10 in TeX
Character table of C2×Dic10 in TeX