p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8:2Q8, C4.7Q16, C42.25C22, C4:C8.7C2, C4:Q8.6C2, (C4xQ8).7C2, C2.7(C2xQ16), C4.14(C2xQ8), C2.D8.4C2, (C2xC4).133D4, (C2xC8).6C22, C4.26(C4oD4), C4:C4.14C22, Q8:C4.3C2, C22.98(C2xD4), C2.15(C8:C22), (C2xC4).102C23, C2.15(C22:Q8), (C2xQ8).52C22, SmallGroup(64,158)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.Q16
G = < a,b,c | a4=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=a2b-1 >
Quotients: C1, C2, C22, D4, Q8, C23, Q16, C2xD4, C2xQ8, C4oD4, C22:Q8, C2xQ16, C8:C22, C4.Q16
Character table of C4.Q16
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ12 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ13 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ14 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ15 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ16 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | -2 | 2 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | -2 | 2 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ19 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8:C22 |
►Regular action on 64 points
(1 17 28 46)(2 47 29 18)(3 19 30 48)(4 41 31 20)(5 21 32 42)(6 43 25 22)(7 23 26 44)(8 45 27 24)(9 38 50 62)(10 63 51 39)(11 40 52 64)(12 57 53 33)(13 34 54 58)(14 59 55 35)(15 36 56 60)(16 61 49 37)
(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 55 5 51)(2 13 6 9)(3 53 7 49)(4 11 8 15)(10 28 14 32)(12 26 16 30)(17 35 21 39)(18 58 22 62)(19 33 23 37)(20 64 24 60)(25 50 29 54)(27 56 31 52)(34 43 38 47)(36 41 40 45)(42 63 46 59)(44 61 48 57)
G:=sub<Sym(64)| (1,17,28,46)(2,47,29,18)(3,19,30,48)(4,41,31,20)(5,21,32,42)(6,43,25,22)(7,23,26,44)(8,45,27,24)(9,38,50,62)(10,63,51,39)(11,40,52,64)(12,57,53,33)(13,34,54,58)(14,59,55,35)(15,36,56,60)(16,61,49,37), (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), (1,55,5,51)(2,13,6,9)(3,53,7,49)(4,11,8,15)(10,28,14,32)(12,26,16,30)(17,35,21,39)(18,58,22,62)(19,33,23,37)(20,64,24,60)(25,50,29,54)(27,56,31,52)(34,43,38,47)(36,41,40,45)(42,63,46,59)(44,61,48,57)>;
G:=Group( (1,17,28,46)(2,47,29,18)(3,19,30,48)(4,41,31,20)(5,21,32,42)(6,43,25,22)(7,23,26,44)(8,45,27,24)(9,38,50,62)(10,63,51,39)(11,40,52,64)(12,57,53,33)(13,34,54,58)(14,59,55,35)(15,36,56,60)(16,61,49,37), (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), (1,55,5,51)(2,13,6,9)(3,53,7,49)(4,11,8,15)(10,28,14,32)(12,26,16,30)(17,35,21,39)(18,58,22,62)(19,33,23,37)(20,64,24,60)(25,50,29,54)(27,56,31,52)(34,43,38,47)(36,41,40,45)(42,63,46,59)(44,61,48,57) );
G=PermutationGroup([[(1,17,28,46),(2,47,29,18),(3,19,30,48),(4,41,31,20),(5,21,32,42),(6,43,25,22),(7,23,26,44),(8,45,27,24),(9,38,50,62),(10,63,51,39),(11,40,52,64),(12,57,53,33),(13,34,54,58),(14,59,55,35),(15,36,56,60),(16,61,49,37)], [(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)], [(1,55,5,51),(2,13,6,9),(3,53,7,49),(4,11,8,15),(10,28,14,32),(12,26,16,30),(17,35,21,39),(18,58,22,62),(19,33,23,37),(20,64,24,60),(25,50,29,54),(27,56,31,52),(34,43,38,47),(36,41,40,45),(42,63,46,59),(44,61,48,57)]])
C4.Q16 is a maximal subgroup of
C42.191C23 Q8:2SD16 D4:Q16 Q8:Q16 Q8.Q16 D4.3Q16 C42.199C23 C8:8Q16 Q8.2SD16 Q8.2Q16 C8:Q16 C42.249C23 C42.255C23 C42.447D4 C42.448D4 C42.22C23 C42.224D4 C42.450D4 C42.230D4 C42.234D4 C42.355C23 C42.358C23 C42.282D4 C42.285D4 C42.289D4 C42.290D4 C42.424C23 C42.425C23 C42.295D4 C42.297D4 C42.300D4 C42.302D4 C42.25C23 C42.29C23 D4:5Q16 C42.470C23 C42.43C23 C42.48C23 C42.478C23 C42.482C23 D4:6Q16 C42.489C23 C42.57C23 C42.63C23 C42.497C23 C42.498C23 C42.502C23 Q8:5Q16 C42.508C23 C42.513C23 C42.516C23 C42.518C23 SD16:4Q8 Q8xQ16 SD16:Q8 SD16:2Q8 SL2(F3):Q8
C4p.Q16: Q8.1Q16 C8.3Q16 Dic6:3Q8 Q8:5Dic6 Dic6:5Q8 C20.7Q16 C20.23Q16 Dic10:5Q8 ...
(CpxQ8):Q8: C42.220D4 C42.21C23 Q8:3Dic6 Dic5.9Q16 Dic7.Q16 ...
C4p:Q8.C2: Q16:4Q8 Q16:5Q8 Dic3.Q16 Dic10:2Q8 Dic14:2Q8 ...
C4.Q16 is a maximal quotient of
Q8:(C4:C4) C2.D8:5C4 C42.29Q8 C4:C4:Q8 (C2xC4).23Q16
(CpxQ8):Q8: (C2xQ8):Q8 Q8:3Dic6 Q8:5Dic6 Dic5.9Q16 C20.23Q16 Dic7.Q16 C28.23Q16 ...
C42.D2p: C42.99D4 C42.121D4 Dic6:3Q8 Dic6:5Q8 C20.7Q16 Dic10:5Q8 C28.7Q16 Dic14:5Q8 ...
(C2xC8).D2p: (C2xC8).52D4 (C2xC4).21Q16 (C2xC8).60D4 Dic3.Q16 Dic10:2Q8 Dic14:2Q8 ...
Matrix representation of C4.Q16 ►in GL4(F17) generated by
| 4 | 2 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 5 | 3 | 0 | 0 |
| 14 | 12 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
G:=sub<GL(4,GF(17))| [4,0,0,0,2,13,0,0,0,0,16,0,0,0,0,16],[5,14,0,0,3,12,0,0,0,0,15,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0] >;
C4.Q16 in GAP, Magma, Sage, TeX
C_4.Q_{16} % in TeX
G:=Group("C4.Q16"); // GroupNames label
G:=SmallGroup(64,158);
// by ID
G=gap.SmallGroup(64,158);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,240,121,55,362,158,1444,376,88]);
// Polycyclic
G:=Group<a,b,c|a^4=b^8=1,c^2=b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b^-1>;
// generators/relations
Export
Subgroup lattice of C4.Q16 in TeX
Character table of C4.Q16 in TeX