p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q16⋊3C4, C42.10C22, C8.5(C2×C4), C2.16(C4×D4), Q8.2(C2×C4), (C4×Q8).3C2, C8⋊C4.1C2, C4.Q8.2C2, C4.5(C4○D4), (C2×C4).102D4, (C2×Q16).6C2, C4⋊C4.54C22, (C2×C8).50C22, (C2×C4).77C23, C4.13(C22×C4), Q8⋊C4.6C2, C22.55(C2×D4), C2.5(C8.C22), (C2×Q8).47C22, SmallGroup(64,122)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q16⋊C4
G = < a,b,c | a8=c4=1, b2=a4, bab-1=a-1, cac-1=a5, bc=cb >
Character table of Q16⋊C4
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | trivial |
| ρ2 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 1 | -1 | 1 | -1 | -i | i | 1 | -1 | -i | i | i | -1 | -1 | -i | -i | 1 | 1 | i | -i | 1 | i | -1 | linear of order 4 |
| ρ10 | 1 | -1 | 1 | -1 | -i | i | 1 | -1 | -i | i | i | -1 | 1 | i | -i | 1 | -1 | -i | i | -1 | -i | 1 | linear of order 4 |
| ρ11 | 1 | -1 | 1 | -1 | i | -i | 1 | -1 | i | -i | i | 1 | 1 | -i | -i | -1 | -1 | i | i | 1 | -i | -1 | linear of order 4 |
| ρ12 | 1 | -1 | 1 | -1 | i | -i | 1 | -1 | i | -i | i | 1 | -1 | i | -i | -1 | 1 | -i | -i | -1 | i | 1 | linear of order 4 |
| ρ13 | 1 | -1 | 1 | -1 | i | -i | 1 | -1 | i | -i | -i | -1 | -1 | i | i | 1 | 1 | -i | i | 1 | -i | -1 | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | i | -i | 1 | -1 | i | -i | -i | -1 | 1 | -i | i | 1 | -1 | i | -i | -1 | i | 1 | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | -i | i | 1 | -1 | -i | i | -i | 1 | 1 | i | i | -1 | -1 | -i | -i | 1 | i | -1 | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | -i | i | 1 | -1 | -i | i | -i | 1 | -1 | -i | i | -1 | 1 | i | i | -1 | -i | 1 | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 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 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | 2 | -2 | -2i | -2i | -2 | 2 | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | -2 | 2 | -2 | 2i | 2i | -2 | 2 | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ22 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
►Regular action on 64 points
(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 13 5 9)(2 12 6 16)(3 11 7 15)(4 10 8 14)(17 55 21 51)(18 54 22 50)(19 53 23 49)(20 52 24 56)(25 45 29 41)(26 44 30 48)(27 43 31 47)(28 42 32 46)(33 61 37 57)(34 60 38 64)(35 59 39 63)(36 58 40 62)
(1 31 64 51)(2 28 57 56)(3 25 58 53)(4 30 59 50)(5 27 60 55)(6 32 61 52)(7 29 62 49)(8 26 63 54)(9 43 38 21)(10 48 39 18)(11 45 40 23)(12 42 33 20)(13 47 34 17)(14 44 35 22)(15 41 36 19)(16 46 37 24)
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), (1,13,5,9)(2,12,6,16)(3,11,7,15)(4,10,8,14)(17,55,21,51)(18,54,22,50)(19,53,23,49)(20,52,24,56)(25,45,29,41)(26,44,30,48)(27,43,31,47)(28,42,32,46)(33,61,37,57)(34,60,38,64)(35,59,39,63)(36,58,40,62), (1,31,64,51)(2,28,57,56)(3,25,58,53)(4,30,59,50)(5,27,60,55)(6,32,61,52)(7,29,62,49)(8,26,63,54)(9,43,38,21)(10,48,39,18)(11,45,40,23)(12,42,33,20)(13,47,34,17)(14,44,35,22)(15,41,36,19)(16,46,37,24)>;
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), (1,13,5,9)(2,12,6,16)(3,11,7,15)(4,10,8,14)(17,55,21,51)(18,54,22,50)(19,53,23,49)(20,52,24,56)(25,45,29,41)(26,44,30,48)(27,43,31,47)(28,42,32,46)(33,61,37,57)(34,60,38,64)(35,59,39,63)(36,58,40,62), (1,31,64,51)(2,28,57,56)(3,25,58,53)(4,30,59,50)(5,27,60,55)(6,32,61,52)(7,29,62,49)(8,26,63,54)(9,43,38,21)(10,48,39,18)(11,45,40,23)(12,42,33,20)(13,47,34,17)(14,44,35,22)(15,41,36,19)(16,46,37,24) );
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)], [(1,13,5,9),(2,12,6,16),(3,11,7,15),(4,10,8,14),(17,55,21,51),(18,54,22,50),(19,53,23,49),(20,52,24,56),(25,45,29,41),(26,44,30,48),(27,43,31,47),(28,42,32,46),(33,61,37,57),(34,60,38,64),(35,59,39,63),(36,58,40,62)], [(1,31,64,51),(2,28,57,56),(3,25,58,53),(4,30,59,50),(5,27,60,55),(6,32,61,52),(7,29,62,49),(8,26,63,54),(9,43,38,21),(10,48,39,18),(11,45,40,23),(12,42,33,20),(13,47,34,17),(14,44,35,22),(15,41,36,19),(16,46,37,24)]])
Q16⋊C4 is a maximal subgroup of
C42.354C23 C42.355C23 C42.359C23 C42.361C23 C42.385C23 C42.389C23 C42.390C23 Q16⋊9D4 Q16⋊10D4 C42.47C23 C42.48C23 C42.51C23 C42.52C23 C42.58C23 C42.63C23 C42.510C23 C42.512C23 C42.515C23 C42.518C23 Q16⋊4Q8 Q16⋊5Q8 C42.73C23 C42.531C23 CSU2(𝔽3)⋊C4
C42.D2p: Q32⋊C4 C42.383D4 C4×C8.C22 C42.229D4 C42.231D4 C42.234D4 C42.235D4 C42.258D4 ...
(Cp×Q16)⋊C4: C42.279C23 Q16⋊Dic3 Q16⋊Dic5 Dic20⋊C4 Q16⋊Dic7 ...
C2p.(C4×D4): C42.276C23 C42.281C23 C3⋊Q16⋊C4 Dic12⋊9C4 C5⋊Q16⋊5C4 Dic20⋊15C4 C7⋊Q16⋊C4 Dic28⋊9C4 ...
Q16⋊C4 is a maximal quotient of
Q16⋊5C8 Q8.M4(2) C8.M4(2) C8⋊C42 C2.D8⋊4C4 C4.(C4×Q8)
C42.D2p: Q8⋊C42 Dic12⋊C4 C42.59D6 Dic20⋊9C4 C42.59D10 Dic28⋊C4 C42.59D14 ...
(Cp×Q16)⋊C4: (C2×C4)⋊9Q16 (C2×Q16)⋊10C4 Q16⋊Dic3 Q16⋊Dic5 Dic20⋊C4 Q16⋊Dic7 ...
C2p.(C4×D4): Q8⋊C4⋊C4 C4.68(C4×D4) C2.(C8⋊D4) C3⋊Q16⋊C4 Dic12⋊9C4 C5⋊Q16⋊5C4 Dic20⋊15C4 C7⋊Q16⋊C4 ...
Matrix representation of Q16⋊C4 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 7 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 8 | 0 |
| 0 | 0 | 4 | 0 | 4 | 4 |
| 0 | 0 | 0 | 13 | 13 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 6 | 15 | 0 | 0 | 0 | 0 |
| 9 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 |
| 0 | 0 | 16 | 0 | 16 | 0 |
| 0 | 0 | 16 | 1 | 16 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 16 | 1 | 16 | 0 |
G:=sub<GL(6,GF(17))| [4,7,0,0,0,0,0,13,0,0,0,0,0,0,4,4,0,0,0,0,0,0,13,0,0,0,8,4,13,13,0,0,0,4,0,0],[6,9,0,0,0,0,15,11,0,0,0,0,0,0,1,0,16,16,0,0,0,0,0,1,0,0,2,1,16,16,0,0,0,16,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,1,0,16,0,0,15,16,1,1,0,0,0,0,0,16,0,0,0,0,1,0] >;
Q16⋊C4 in GAP, Magma, Sage, TeX
Q_{16}\rtimes C_4 % in TeX
G:=Group("Q16:C4"); // GroupNames label
G:=SmallGroup(64,122);
// by ID
G=gap.SmallGroup(64,122);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,199,650,86,963,489,117]);
// Polycyclic
G:=Group<a,b,c|a^8=c^4=1,b^2=a^4,b*a*b^-1=a^-1,c*a*c^-1=a^5,b*c=c*b>;
// generators/relations
Export
Subgroup lattice of Q16⋊C4 in TeX
Character table of Q16⋊C4 in TeX