p-group, metabelian, nilpotent (class 5), monomial
Aliases: C8.9D8, Q64⋊2C2, C16.4D4, C32.C22, SD64⋊2C2, C4.15D16, M6(2)⋊2C2, C22.6D16, C16.11C23, D16.3C22, Q32.3C22, C8.50(C2×D4), C4.18(C2×D8), (C2×C4).52D8, (C2×Q32)⋊11C2, C4○D16.4C2, C2.17(C2×D16), (C2×C8).142D4, (C2×C16).33C22, SmallGroup(128,996)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q64⋊C2
G = < a,b,c | a32=c2=1, b2=a16, bab-1=a-1, cac=a17, bc=cb >
Character table of Q64⋊C2
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 16A | 16B | 16C | 16D | 16E | 16F | 32A | 32B | 32C | 32D | 32E | 32F | 32G | 32H | |
| size | 1 | 1 | 2 | 16 | 2 | 2 | 16 | 16 | 16 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | 0 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | √2 | -√2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ12 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | -√2 | √2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ13 | 2 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | -√2 | -√2 | √2 | √2 | orthogonal lifted from D8 |
| ρ14 | 2 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | √2 | √2 | -√2 | -√2 | orthogonal lifted from D8 |
| ρ15 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | √2 | -√2 | ζ167-ζ16 | ζ167-ζ16 | -ζ165+ζ163 | -ζ165+ζ163 | ζ165-ζ163 | ζ165-ζ163 | -ζ167+ζ16 | -ζ167+ζ16 | orthogonal lifted from D16 |
| ρ16 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | -√2 | √2 | ζ165-ζ163 | ζ165-ζ163 | ζ167-ζ16 | ζ167-ζ16 | -ζ167+ζ16 | -ζ167+ζ16 | -ζ165+ζ163 | -ζ165+ζ163 | orthogonal lifted from D16 |
| ρ17 | 2 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | √2 | -√2 | ζ165-ζ163 | -ζ165+ζ163 | ζ167-ζ16 | -ζ167+ζ16 | -ζ167+ζ16 | ζ167-ζ16 | -ζ165+ζ163 | ζ165-ζ163 | orthogonal lifted from D16 |
| ρ18 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | -√2 | √2 | -ζ165+ζ163 | -ζ165+ζ163 | -ζ167+ζ16 | -ζ167+ζ16 | ζ167-ζ16 | ζ167-ζ16 | ζ165-ζ163 | ζ165-ζ163 | orthogonal lifted from D16 |
| ρ19 | 2 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | √2 | -√2 | -ζ165+ζ163 | ζ165-ζ163 | -ζ167+ζ16 | ζ167-ζ16 | ζ167-ζ16 | -ζ167+ζ16 | ζ165-ζ163 | -ζ165+ζ163 | orthogonal lifted from D16 |
| ρ20 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | √2 | -√2 | -ζ167+ζ16 | -ζ167+ζ16 | ζ165-ζ163 | ζ165-ζ163 | -ζ165+ζ163 | -ζ165+ζ163 | ζ167-ζ16 | ζ167-ζ16 | orthogonal lifted from D16 |
| ρ21 | 2 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | -√2 | √2 | -ζ167+ζ16 | ζ167-ζ16 | ζ165-ζ163 | -ζ165+ζ163 | -ζ165+ζ163 | ζ165-ζ163 | ζ167-ζ16 | -ζ167+ζ16 | orthogonal lifted from D16 |
| ρ22 | 2 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | -√2 | √2 | ζ167-ζ16 | -ζ167+ζ16 | -ζ165+ζ163 | ζ165-ζ163 | ζ165-ζ163 | -ζ165+ζ163 | -ζ167+ζ16 | ζ167-ζ16 | orthogonal lifted from D16 |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | -2ζ165+2ζ163 | -2ζ1615+2ζ169 | 2ζ165-2ζ163 | 2ζ1615-2ζ169 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ24 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 2ζ1615-2ζ169 | -2ζ165+2ζ163 | -2ζ1615+2ζ169 | 2ζ165-2ζ163 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 2ζ165-2ζ163 | 2ζ1615-2ζ169 | -2ζ165+2ζ163 | -2ζ1615+2ζ169 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | -2ζ1615+2ζ169 | 2ζ165-2ζ163 | 2ζ1615-2ζ169 | -2ζ165+2ζ163 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 64 points
Generators in S64
(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)
(1 40 17 56)(2 39 18 55)(3 38 19 54)(4 37 20 53)(5 36 21 52)(6 35 22 51)(7 34 23 50)(8 33 24 49)(9 64 25 48)(10 63 26 47)(11 62 27 46)(12 61 28 45)(13 60 29 44)(14 59 30 43)(15 58 31 42)(16 57 32 41)
(1 55)(2 40)(3 57)(4 42)(5 59)(6 44)(7 61)(8 46)(9 63)(10 48)(11 33)(12 50)(13 35)(14 52)(15 37)(16 54)(17 39)(18 56)(19 41)(20 58)(21 43)(22 60)(23 45)(24 62)(25 47)(26 64)(27 49)(28 34)(29 51)(30 36)(31 53)(32 38)
G:=sub<Sym(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), (1,40,17,56)(2,39,18,55)(3,38,19,54)(4,37,20,53)(5,36,21,52)(6,35,22,51)(7,34,23,50)(8,33,24,49)(9,64,25,48)(10,63,26,47)(11,62,27,46)(12,61,28,45)(13,60,29,44)(14,59,30,43)(15,58,31,42)(16,57,32,41), (1,55)(2,40)(3,57)(4,42)(5,59)(6,44)(7,61)(8,46)(9,63)(10,48)(11,33)(12,50)(13,35)(14,52)(15,37)(16,54)(17,39)(18,56)(19,41)(20,58)(21,43)(22,60)(23,45)(24,62)(25,47)(26,64)(27,49)(28,34)(29,51)(30,36)(31,53)(32,38)>;
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), (1,40,17,56)(2,39,18,55)(3,38,19,54)(4,37,20,53)(5,36,21,52)(6,35,22,51)(7,34,23,50)(8,33,24,49)(9,64,25,48)(10,63,26,47)(11,62,27,46)(12,61,28,45)(13,60,29,44)(14,59,30,43)(15,58,31,42)(16,57,32,41), (1,55)(2,40)(3,57)(4,42)(5,59)(6,44)(7,61)(8,46)(9,63)(10,48)(11,33)(12,50)(13,35)(14,52)(15,37)(16,54)(17,39)(18,56)(19,41)(20,58)(21,43)(22,60)(23,45)(24,62)(25,47)(26,64)(27,49)(28,34)(29,51)(30,36)(31,53)(32,38) );
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)], [(1,40,17,56),(2,39,18,55),(3,38,19,54),(4,37,20,53),(5,36,21,52),(6,35,22,51),(7,34,23,50),(8,33,24,49),(9,64,25,48),(10,63,26,47),(11,62,27,46),(12,61,28,45),(13,60,29,44),(14,59,30,43),(15,58,31,42),(16,57,32,41)], [(1,55),(2,40),(3,57),(4,42),(5,59),(6,44),(7,61),(8,46),(9,63),(10,48),(11,33),(12,50),(13,35),(14,52),(15,37),(16,54),(17,39),(18,56),(19,41),(20,58),(21,43),(22,60),(23,45),(24,62),(25,47),(26,64),(27,49),(28,34),(29,51),(30,36),(31,53),(32,38)]])
Matrix representation of Q64⋊C2 ►in GL4(𝔽97) generated by
| 41 | 44 | 26 | 1 |
| 76 | 30 | 96 | 26 |
| 49 | 29 | 91 | 39 |
| 8 | 73 | 7 | 32 |
| 82 | 79 | 1 | 26 |
| 20 | 66 | 26 | 96 |
| 1 | 92 | 65 | 90 |
| 76 | 84 | 31 | 78 |
| 20 | 48 | 95 | 0 |
| 76 | 48 | 0 | 95 |
| 35 | 80 | 77 | 49 |
| 62 | 17 | 21 | 49 |
G:=sub<GL(4,GF(97))| [41,76,49,8,44,30,29,73,26,96,91,7,1,26,39,32],[82,20,1,76,79,66,92,84,1,26,65,31,26,96,90,78],[20,76,35,62,48,48,80,17,95,0,77,21,0,95,49,49] >;
Q64⋊C2 in GAP, Magma, Sage, TeX
Q_{64}\rtimes C_2 % in TeX
G:=Group("Q64:C2"); // GroupNames label
G:=SmallGroup(128,996);
// by ID
G=gap.SmallGroup(128,996);
# by ID
G:=PCGroup([7,-2,2,2,-2,-2,-2,-2,141,456,1430,675,346,192,1684,851,242,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c|a^32=c^2=1,b^2=a^16,b*a*b^-1=a^-1,c*a*c=a^17,b*c=c*b>;
// generators/relations
Export
Subgroup lattice of Q64⋊C2 in TeX
Character table of Q64⋊C2 in TeX