metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.6D8, C6.3Q16, C12.1Q8, C4.1Dic6, C3⋊C8⋊1C4, C4⋊C4.1S3, C3⋊1(C2.D8), C6.2(C4⋊C4), C4.11(C4×S3), C12.1(C2×C4), (C2×C6).28D4, (C2×C4).33D6, C2.1(D4⋊S3), C4⋊Dic3.8C2, (C2×C12).8C22, C2.1(C3⋊Q16), C2.3(Dic3⋊C4), C22.12(C3⋊D4), (C2×C3⋊C8).1C2, (C3×C4⋊C4).1C2, SmallGroup(96,14)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.Q16
G = < a,b,c | a12=b4=1, c2=a9b2, bab-1=a7, cac-1=a5, cbc-1=a9b-1 >
Character table of C6.Q16
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | i | -i | -i | i | -1 | 1 | -1 | -1 | 1 | 1 | -1 | i | -i | -i | 1 | -1 | i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | i | -i | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | i | i | 1 | -1 | -i | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | i | -i | i | -i | -1 | 1 | -1 | 1 | -1 | -1 | 1 | i | -i | -i | 1 | -1 | i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | -i | i | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -i | i | i | 1 | -1 | -i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | √2 | -√2 | √2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ13 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -√2 | √2 | -√2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ14 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ15 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -√2 | -√2 | √2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ16 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | √2 | √2 | -√2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ17 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | -√3 | -√3 | √3 | 1 | -1 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ18 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | √3 | √3 | -√3 | 1 | -1 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ19 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | 2i | -2i | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | -i | i | i | -1 | 1 | -i | complex lifted from C4×S3 |
| ρ20 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | -2i | 2i | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | i | -i | -i | -1 | 1 | i | complex lifted from C4×S3 |
| ρ21 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | √-3 | -√-3 | √-3 | 1 | 1 | -√-3 | complex lifted from C3⋊D4 |
| ρ22 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -√-3 | √-3 | -√-3 | 1 | 1 | √-3 | complex lifted from C3⋊D4 |
| ρ23 | 4 | -4 | 4 | -4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
| ρ24 | 4 | -4 | -4 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C3⋊Q16, Schur index 2 |
►Regular action on 96 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 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 71 58 27)(2 66 59 34)(3 61 60 29)(4 68 49 36)(5 63 50 31)(6 70 51 26)(7 65 52 33)(8 72 53 28)(9 67 54 35)(10 62 55 30)(11 69 56 25)(12 64 57 32)(13 46 92 74)(14 41 93 81)(15 48 94 76)(16 43 95 83)(17 38 96 78)(18 45 85 73)(19 40 86 80)(20 47 87 75)(21 42 88 82)(22 37 89 77)(23 44 90 84)(24 39 91 79)
(1 13 55 89 7 19 49 95)(2 18 56 94 8 24 50 88)(3 23 57 87 9 17 51 93)(4 16 58 92 10 22 52 86)(5 21 59 85 11 15 53 91)(6 14 60 90 12 20 54 96)(25 45 66 82 31 39 72 76)(26 38 67 75 32 44 61 81)(27 43 68 80 33 37 62 74)(28 48 69 73 34 42 63 79)(29 41 70 78 35 47 64 84)(30 46 71 83 36 40 65 77)
G:=sub<Sym(96)| (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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,71,58,27)(2,66,59,34)(3,61,60,29)(4,68,49,36)(5,63,50,31)(6,70,51,26)(7,65,52,33)(8,72,53,28)(9,67,54,35)(10,62,55,30)(11,69,56,25)(12,64,57,32)(13,46,92,74)(14,41,93,81)(15,48,94,76)(16,43,95,83)(17,38,96,78)(18,45,85,73)(19,40,86,80)(20,47,87,75)(21,42,88,82)(22,37,89,77)(23,44,90,84)(24,39,91,79), (1,13,55,89,7,19,49,95)(2,18,56,94,8,24,50,88)(3,23,57,87,9,17,51,93)(4,16,58,92,10,22,52,86)(5,21,59,85,11,15,53,91)(6,14,60,90,12,20,54,96)(25,45,66,82,31,39,72,76)(26,38,67,75,32,44,61,81)(27,43,68,80,33,37,62,74)(28,48,69,73,34,42,63,79)(29,41,70,78,35,47,64,84)(30,46,71,83,36,40,65,77)>;
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)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,71,58,27)(2,66,59,34)(3,61,60,29)(4,68,49,36)(5,63,50,31)(6,70,51,26)(7,65,52,33)(8,72,53,28)(9,67,54,35)(10,62,55,30)(11,69,56,25)(12,64,57,32)(13,46,92,74)(14,41,93,81)(15,48,94,76)(16,43,95,83)(17,38,96,78)(18,45,85,73)(19,40,86,80)(20,47,87,75)(21,42,88,82)(22,37,89,77)(23,44,90,84)(24,39,91,79), (1,13,55,89,7,19,49,95)(2,18,56,94,8,24,50,88)(3,23,57,87,9,17,51,93)(4,16,58,92,10,22,52,86)(5,21,59,85,11,15,53,91)(6,14,60,90,12,20,54,96)(25,45,66,82,31,39,72,76)(26,38,67,75,32,44,61,81)(27,43,68,80,33,37,62,74)(28,48,69,73,34,42,63,79)(29,41,70,78,35,47,64,84)(30,46,71,83,36,40,65,77) );
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),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,71,58,27),(2,66,59,34),(3,61,60,29),(4,68,49,36),(5,63,50,31),(6,70,51,26),(7,65,52,33),(8,72,53,28),(9,67,54,35),(10,62,55,30),(11,69,56,25),(12,64,57,32),(13,46,92,74),(14,41,93,81),(15,48,94,76),(16,43,95,83),(17,38,96,78),(18,45,85,73),(19,40,86,80),(20,47,87,75),(21,42,88,82),(22,37,89,77),(23,44,90,84),(24,39,91,79)], [(1,13,55,89,7,19,49,95),(2,18,56,94,8,24,50,88),(3,23,57,87,9,17,51,93),(4,16,58,92,10,22,52,86),(5,21,59,85,11,15,53,91),(6,14,60,90,12,20,54,96),(25,45,66,82,31,39,72,76),(26,38,67,75,32,44,61,81),(27,43,68,80,33,37,62,74),(28,48,69,73,34,42,63,79),(29,41,70,78,35,47,64,84),(30,46,71,83,36,40,65,77)]])
C6.Q16 is a maximal subgroup of
Dic3⋊4D8 D4.S3⋊C4 Dic3.D8 D4.Dic6 D6.D8 D6⋊D8 D6⋊C8⋊11C2 C3⋊C8⋊1D4 Dic3⋊4Q16 Q8⋊3Dic6 Q8.4Dic6 D6.Q16 C3⋊(C8⋊D4) D6⋊1Q16 C8⋊Dic3⋊C2 Q8⋊3(C4×S3) C24⋊3Q8 Dic6.Q8 C8.8Dic6 (S3×C8)⋊C4 C8⋊(C4×S3) C4.Q8⋊S3 C6.(C4○D8) D12.Q8 C24⋊2Q8 Dic3.Q16 C24⋊4Q8 S3×C2.D8 C8⋊S3⋊C4 D6.5D8 D6.2Q16 D12⋊2Q8 C4⋊C4.225D6 (C2×C6).40D8 C4⋊C4.230D6 C4⋊C4.232D6 C4⋊C4.233D6 C4⋊C4.234D6 C4⋊C4.236D6 C12.50D8 D4.3Dic6 C4×D4⋊S3 C42.51D6 Q8⋊5Dic6 Q8.5Dic6 C42.56D6 C4×C3⋊Q16 (C2×C6).D8 C6.Q16⋊C2 C3⋊C8⋊22D4 C3⋊C8⋊5D4 (C2×C6).Q16 (C2×Q8).51D6 C3⋊C8⋊6D4 C3⋊C8.29D4 Dic6.4Q8 C42.68D6 C42.215D6 D12.4Q8 C12.17D8 C42.76D6 D12⋊6Q8 Dic6⋊5Q8 C36.Q8 C6.18D24 C12.8Dic6 C12.9Dic6 C30.20D8 C60.5Q8 C60.1Q8 Dic5.4Dic6
C6.Q16 is a maximal quotient of
C12.53D8 C6.6D16 C6.SD32 C24.7Q8 C12.C42 C36.Q8 C6.18D24 C12.8Dic6 C12.9Dic6 C30.20D8 C60.5Q8 C60.1Q8 Dic5.4Dic6
Matrix representation of C6.Q16 ►in GL4(𝔽73) generated by
| 0 | 1 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 72 | 2 |
| 0 | 0 | 72 | 1 |
| 7 | 14 | 0 | 0 |
| 59 | 66 | 0 | 0 |
| 0 | 0 | 36 | 41 |
| 0 | 0 | 20 | 37 |
| 60 | 30 | 0 | 0 |
| 43 | 13 | 0 | 0 |
| 0 | 0 | 32 | 41 |
| 0 | 0 | 16 | 0 |
G:=sub<GL(4,GF(73))| [0,72,0,0,1,72,0,0,0,0,72,72,0,0,2,1],[7,59,0,0,14,66,0,0,0,0,36,20,0,0,41,37],[60,43,0,0,30,13,0,0,0,0,32,16,0,0,41,0] >;
C6.Q16 in GAP, Magma, Sage, TeX
C_6.Q_{16} % in TeX
G:=Group("C6.Q16"); // GroupNames label
G:=SmallGroup(96,14);
// by ID
G=gap.SmallGroup(96,14);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,121,31,297,69,2309]);
// Polycyclic
G:=Group<a,b,c|a^12=b^4=1,c^2=a^9*b^2,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=a^9*b^-1>;
// generators/relations
Export
Subgroup lattice of C6.Q16 in TeX
Character table of C6.Q16 in TeX