direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4×Q16, C42.73C22, (C4×C8).6C2, C8.10(C2×C4), C4○3(C2.D8), C2.14(C4×D4), (C2×C4).53D4, Q8.1(C2×C4), (C4×Q8).2C2, C2.3(C2×Q16), C2.D8.9C2, C4.3(C4○D4), C2.5(C4○D8), (C2×Q16).7C2, C4○3(Q8⋊C4), C4⋊C4.52C22, (C2×C4).75C23, C4.11(C22×C4), (C2×C8).64C22, Q8⋊C4.8C2, C22.53(C2×D4), (C2×Q8).45C22, SmallGroup(64,120)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4×Q16
G = < a,b,c | a4=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >
Character table of C4×Q16
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 1 | -1 | -1 | 1 | -i | -i | i | i | -1 | 1 | i | -i | 1 | i | i | -1 | 1 | -1 | -i | -i | i | -1 | i | -1 | -i | 1 | -i | 1 | linear of order 4 |
| ρ10 | 1 | -1 | -1 | 1 | -i | -i | i | i | -1 | 1 | i | -i | 1 | i | -i | 1 | -1 | -1 | -i | i | -i | 1 | -i | 1 | i | -1 | i | -1 | linear of order 4 |
| ρ11 | 1 | -1 | -1 | 1 | i | i | -i | -i | -1 | 1 | -i | i | -1 | i | -i | -1 | 1 | 1 | -i | i | i | 1 | i | 1 | -i | -1 | -i | -1 | linear of order 4 |
| ρ12 | 1 | -1 | -1 | 1 | i | i | -i | -i | -1 | 1 | -i | i | -1 | i | i | 1 | -1 | 1 | -i | -i | -i | -1 | -i | -1 | i | 1 | i | 1 | linear of order 4 |
| ρ13 | 1 | -1 | -1 | 1 | -i | -i | i | i | -1 | 1 | i | -i | -1 | -i | -i | 1 | -1 | 1 | i | i | i | -1 | i | -1 | -i | 1 | -i | 1 | linear of order 4 |
| ρ14 | 1 | -1 | -1 | 1 | -i | -i | i | i | -1 | 1 | i | -i | -1 | -i | i | -1 | 1 | 1 | i | -i | -i | 1 | -i | 1 | i | -1 | i | -1 | linear of order 4 |
| ρ15 | 1 | -1 | -1 | 1 | i | i | -i | -i | -1 | 1 | -i | i | 1 | -i | i | 1 | -1 | -1 | i | -i | i | 1 | i | 1 | -i | -1 | -i | -1 | linear of order 4 |
| ρ16 | 1 | -1 | -1 | 1 | i | i | -i | -i | -1 | 1 | -i | i | 1 | -i | -i | -1 | 1 | -1 | i | i | -i | -1 | -i | -1 | i | 1 | i | 1 | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | √2 | -√2 | -√2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ20 | 2 | -2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | -√2 | √2 | √2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ21 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | -√2 | -√2 | √2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ22 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | √2 | √2 | -√2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ23 | 2 | 2 | -2 | -2 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√2 | -√-2 | √2 | √-2 | -√2 | -√-2 | √2 | complex lifted from C4○D8 |
| ρ24 | 2 | -2 | -2 | 2 | -2i | -2i | 2i | 2i | 2 | -2 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 2 | 2 | -2 | -2 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | -√2 | √-2 | √2 | -√-2 | -√2 | √-2 | √2 | complex lifted from C4○D8 |
| ρ26 | 2 | 2 | -2 | -2 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | √2 | -√-2 | -√2 | √-2 | √2 | -√-2 | -√2 | complex lifted from C4○D8 |
| ρ27 | 2 | 2 | -2 | -2 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √2 | √-2 | -√2 | -√-2 | √2 | √-2 | -√2 | complex lifted from C4○D8 |
| ρ28 | 2 | -2 | -2 | 2 | 2i | 2i | -2i | -2i | 2 | -2 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►Regular action on 64 points
(1 32 60 51)(2 25 61 52)(3 26 62 53)(4 27 63 54)(5 28 64 55)(6 29 57 56)(7 30 58 49)(8 31 59 50)(9 48 39 18)(10 41 40 19)(11 42 33 20)(12 43 34 21)(13 44 35 22)(14 45 36 23)(15 46 37 24)(16 47 38 17)
(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 12 5 16)(2 11 6 15)(3 10 7 14)(4 9 8 13)(17 51 21 55)(18 50 22 54)(19 49 23 53)(20 56 24 52)(25 42 29 46)(26 41 30 45)(27 48 31 44)(28 47 32 43)(33 57 37 61)(34 64 38 60)(35 63 39 59)(36 62 40 58)
G:=sub<Sym(64)| (1,32,60,51)(2,25,61,52)(3,26,62,53)(4,27,63,54)(5,28,64,55)(6,29,57,56)(7,30,58,49)(8,31,59,50)(9,48,39,18)(10,41,40,19)(11,42,33,20)(12,43,34,21)(13,44,35,22)(14,45,36,23)(15,46,37,24)(16,47,38,17), (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), (1,12,5,16)(2,11,6,15)(3,10,7,14)(4,9,8,13)(17,51,21,55)(18,50,22,54)(19,49,23,53)(20,56,24,52)(25,42,29,46)(26,41,30,45)(27,48,31,44)(28,47,32,43)(33,57,37,61)(34,64,38,60)(35,63,39,59)(36,62,40,58)>;
G:=Group( (1,32,60,51)(2,25,61,52)(3,26,62,53)(4,27,63,54)(5,28,64,55)(6,29,57,56)(7,30,58,49)(8,31,59,50)(9,48,39,18)(10,41,40,19)(11,42,33,20)(12,43,34,21)(13,44,35,22)(14,45,36,23)(15,46,37,24)(16,47,38,17), (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), (1,12,5,16)(2,11,6,15)(3,10,7,14)(4,9,8,13)(17,51,21,55)(18,50,22,54)(19,49,23,53)(20,56,24,52)(25,42,29,46)(26,41,30,45)(27,48,31,44)(28,47,32,43)(33,57,37,61)(34,64,38,60)(35,63,39,59)(36,62,40,58) );
G=PermutationGroup([[(1,32,60,51),(2,25,61,52),(3,26,62,53),(4,27,63,54),(5,28,64,55),(6,29,57,56),(7,30,58,49),(8,31,59,50),(9,48,39,18),(10,41,40,19),(11,42,33,20),(12,43,34,21),(13,44,35,22),(14,45,36,23),(15,46,37,24),(16,47,38,17)], [(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)], [(1,12,5,16),(2,11,6,15),(3,10,7,14),(4,9,8,13),(17,51,21,55),(18,50,22,54),(19,49,23,53),(20,56,24,52),(25,42,29,46),(26,41,30,45),(27,48,31,44),(28,47,32,43),(33,57,37,61),(34,64,38,60),(35,63,39,59),(36,62,40,58)]])
C4×Q16 is a maximal subgroup of
Q16⋊1C8 Q16⋊C8 Q16⋊5C8 C8⋊9Q16 Q8.M4(2) C8⋊6Q16 C8.M4(2) Q16⋊2D4 Q16.4D4 Q16.5D4 Q16⋊Q8 C4.Q32 Q16.Q8 C42.384D4 C42.224D4 C42.451D4 C42.226D4 C42.354C23 C42.355C23 C42.358C23 C42.361C23 C42.308D4 C42.367D4 C42.256D4 C42.387C23 C42.389C23 C42.390C23 Q16⋊4D4 Q16⋊5D4 Q16⋊12D4 Q16⋊13D4 D4⋊5Q16 C42.465C23 C42.469C23 C42.476C23 C42.477C23 C42.482C23 C42.485C23 D4⋊6Q16 C42.491C23 C42.493C23 C42.497C23 Q8⋊5Q16 C42.505C23 C42.506C23 C42.515C23 C42.516C23 C42.518C23 Q16⋊6Q8 Q16⋊4Q8 Q16⋊5Q8 C42.527C23 Q8⋊6Q16 C42.530C23 C42.75C23 C42.532C23
C2p.(C4×D4): SD32⋊3C4 Q32⋊4C4 C42.276C23 C42.279C23 C42.280C23 Dic3⋊4Q16 Dic3⋊5Q16 Dic5⋊4Q16 ...
C4×Q16 is a maximal quotient of
C8⋊9Q16 C8⋊6Q16 C2.D8⋊5C4 C8⋊5(C4⋊C4)
C2p.(C4×D4): Q8⋊(C4⋊C4) (C2×C4)⋊9Q16 C2.(C4×Q16) C2.(C8⋊8D4) (C2×C4)⋊6Q16 Dic3⋊4Q16 Dic3⋊5Q16 Dic5⋊4Q16 ...
Matrix representation of C4×Q16 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 11 | 11 |
| 0 | 0 | 3 | 0 |
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 13 | 9 |
| 0 | 0 | 0 | 4 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,16,0,0,0,0,16],[0,16,0,0,1,0,0,0,0,0,11,3,0,0,11,0],[0,16,0,0,16,0,0,0,0,0,13,0,0,0,9,4] >;
C4×Q16 in GAP, Magma, Sage, TeX
C_4\times Q_{16} % in TeX
G:=Group("C4xQ16"); // GroupNames label
G:=SmallGroup(64,120);
// by ID
G=gap.SmallGroup(64,120);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,199,230,963,489,117]);
// Polycyclic
G:=Group<a,b,c|a^4=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C4×Q16 in TeX
Character table of C4×Q16 in TeX