metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.2D4, C6.4Q16, Dic6⋊3C4, C4.10D12, C6.5SD16, C4⋊C4.3S3, C4.2(C4×S3), C12.4(C2×C4), (C2×C4).36D6, (C2×C6).31D4, C2.6(D6⋊C4), C3⋊1(Q8⋊C4), C6.4(C22⋊C4), C2.2(D4.S3), (C2×Dic6).5C2, C2.2(C3⋊Q16), (C2×C12).11C22, C22.15(C3⋊D4), (C2×C3⋊C8).3C2, (C3×C4⋊C4).3C2, SmallGroup(96,17)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.SD16
G = < a,b,c | a6=b8=1, c2=a3b4, bab-1=a-1, ac=ca, cbc-1=a3b-1 >
Character table of C6.SD16
| 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 | 1 | -1 | -1 | -1 | 1 | -i | i | -i | i | -1 | i | i | 1 | -i | -i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | i | -i | -1 | 1 | -1 | -1 | 1 | i | -i | i | -i | -1 | i | i | 1 | -i | -i | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | -1 | 1 | -1 | -1 | 1 | -i | i | -i | i | -1 | -i | -i | 1 | i | i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | 1 | -1 | -1 | -1 | 1 | i | -i | i | -i | -1 | -i | -i | 1 | i | 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 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 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 |
| ρ12 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | -1 | √3 | -√3 | 1 | -√3 | √3 | orthogonal lifted from D12 |
| ρ14 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | -1 | -√3 | √3 | 1 | √3 | -√3 | orthogonal lifted from D12 |
| 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 | -2i | 2i | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 1 | i | i | -1 | -i | -i | complex lifted from C4×S3 |
| ρ18 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | √-3 | -√-3 | 1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ19 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -√-2 | -√-2 | √-2 | √-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ20 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | 2i | -2i | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 1 | -i | -i | -1 | i | 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 | 1 | -√-3 | √-3 | 1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ22 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | √-2 | √-2 | -√-2 | -√-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| 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 | symplectic lifted from C3⋊Q16, 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 D4.S3, Schur index 2 |
►Regular action on 96 points
(1 75 93 23 27 61)(2 62 28 24 94 76)(3 77 95 17 29 63)(4 64 30 18 96 78)(5 79 89 19 31 57)(6 58 32 20 90 80)(7 73 91 21 25 59)(8 60 26 22 92 74)(9 48 34 55 67 83)(10 84 68 56 35 41)(11 42 36 49 69 85)(12 86 70 50 37 43)(13 44 38 51 71 87)(14 88 72 52 39 45)(15 46 40 53 65 81)(16 82 66 54 33 47)
(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 55 19 13)(2 16 20 50)(3 53 21 11)(4 14 22 56)(5 51 23 9)(6 12 24 54)(7 49 17 15)(8 10 18 52)(25 42 77 65)(26 68 78 45)(27 48 79 71)(28 66 80 43)(29 46 73 69)(30 72 74 41)(31 44 75 67)(32 70 76 47)(33 58 86 94)(34 89 87 61)(35 64 88 92)(36 95 81 59)(37 62 82 90)(38 93 83 57)(39 60 84 96)(40 91 85 63)
G:=sub<Sym(96)| (1,75,93,23,27,61)(2,62,28,24,94,76)(3,77,95,17,29,63)(4,64,30,18,96,78)(5,79,89,19,31,57)(6,58,32,20,90,80)(7,73,91,21,25,59)(8,60,26,22,92,74)(9,48,34,55,67,83)(10,84,68,56,35,41)(11,42,36,49,69,85)(12,86,70,50,37,43)(13,44,38,51,71,87)(14,88,72,52,39,45)(15,46,40,53,65,81)(16,82,66,54,33,47), (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), (1,55,19,13)(2,16,20,50)(3,53,21,11)(4,14,22,56)(5,51,23,9)(6,12,24,54)(7,49,17,15)(8,10,18,52)(25,42,77,65)(26,68,78,45)(27,48,79,71)(28,66,80,43)(29,46,73,69)(30,72,74,41)(31,44,75,67)(32,70,76,47)(33,58,86,94)(34,89,87,61)(35,64,88,92)(36,95,81,59)(37,62,82,90)(38,93,83,57)(39,60,84,96)(40,91,85,63)>;
G:=Group( (1,75,93,23,27,61)(2,62,28,24,94,76)(3,77,95,17,29,63)(4,64,30,18,96,78)(5,79,89,19,31,57)(6,58,32,20,90,80)(7,73,91,21,25,59)(8,60,26,22,92,74)(9,48,34,55,67,83)(10,84,68,56,35,41)(11,42,36,49,69,85)(12,86,70,50,37,43)(13,44,38,51,71,87)(14,88,72,52,39,45)(15,46,40,53,65,81)(16,82,66,54,33,47), (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), (1,55,19,13)(2,16,20,50)(3,53,21,11)(4,14,22,56)(5,51,23,9)(6,12,24,54)(7,49,17,15)(8,10,18,52)(25,42,77,65)(26,68,78,45)(27,48,79,71)(28,66,80,43)(29,46,73,69)(30,72,74,41)(31,44,75,67)(32,70,76,47)(33,58,86,94)(34,89,87,61)(35,64,88,92)(36,95,81,59)(37,62,82,90)(38,93,83,57)(39,60,84,96)(40,91,85,63) );
G=PermutationGroup([[(1,75,93,23,27,61),(2,62,28,24,94,76),(3,77,95,17,29,63),(4,64,30,18,96,78),(5,79,89,19,31,57),(6,58,32,20,90,80),(7,73,91,21,25,59),(8,60,26,22,92,74),(9,48,34,55,67,83),(10,84,68,56,35,41),(11,42,36,49,69,85),(12,86,70,50,37,43),(13,44,38,51,71,87),(14,88,72,52,39,45),(15,46,40,53,65,81),(16,82,66,54,33,47)], [(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)], [(1,55,19,13),(2,16,20,50),(3,53,21,11),(4,14,22,56),(5,51,23,9),(6,12,24,54),(7,49,17,15),(8,10,18,52),(25,42,77,65),(26,68,78,45),(27,48,79,71),(28,66,80,43),(29,46,73,69),(30,72,74,41),(31,44,75,67),(32,70,76,47),(33,58,86,94),(34,89,87,61),(35,64,88,92),(36,95,81,59),(37,62,82,90),(38,93,83,57),(39,60,84,96),(40,91,85,63)]])
C6.SD16 is a maximal subgroup of
C12⋊Q8⋊C2 Dic6.D4 (C2×C8).200D6 D4⋊(C4×S3) D4⋊2S3⋊C4 D4⋊3D12 D4.D12 D12.D4 Dic3.1Q16 Dic3⋊Q16 (C2×Q8).36D6 S3×Q8⋊C4 (S3×Q8)⋊C4 Q8⋊3D12 D6⋊Q16 Dic3⋊SD16 Dic3⋊8SD16 Dic12⋊9C4 Dic6⋊Q8 Dic6.Q8 D6.2SD16 C8⋊8D12 C8.2D12 C6.(C4○D8) Dic3⋊5Q16 Dic3.Q16 Dic6.2Q8 D6.2Q16 C8⋊3D12 D6⋊2Q16 C2.D8⋊7S3 C24⋊C2⋊C4 C4○D12⋊C4 C4⋊C4.230D6 C4⋊C4.231D6 C4⋊C4.233D6 C4.(C2×D12) C4⋊C4.237D6 D4.1D12 C4×D4.S3 C42.51D6 D4.2D12 Q8.6D12 C4×C3⋊Q16 C42.59D6 C12⋊7Q16 D12⋊17D4 Dic6⋊17D4 C3⋊C8⋊23D4 C3⋊C8⋊5D4 D12.37D4 Dic6.37D4 C3⋊C8.29D4 C3⋊C8.6D4 Dic6.4Q8 C42.216D6 C42.71D6 C42.82D6 Dic6⋊5Q8 C12.Q16 Dic6⋊6Q8 C18.Q16 Dic6⋊Dic3 C12.73D12 C62.114D4 Dic6⋊Dic5 Dic30⋊12C4 Dic30⋊9C4 Dic6⋊F5
C6.SD16 is a maximal quotient of
C4⋊Dic3⋊C4 C4.Dic12 Dic6⋊2C8 C12.2D8 C12.C42 C18.Q16 Dic6⋊Dic3 C12.73D12 C62.114D4 Dic6⋊Dic5 Dic30⋊12C4 Dic30⋊9C4 Dic6⋊F5
Matrix representation of C6.SD16 ►in GL5(𝔽73)
| 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 72 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 27 | 0 | 0 | 0 | 0 |
| 0 | 54 | 68 | 0 | 0 |
| 0 | 14 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 55 | 12 |
| 27 | 0 | 0 | 0 | 0 |
| 0 | 30 | 60 | 0 | 0 |
| 0 | 13 | 43 | 0 | 0 |
| 0 | 0 | 0 | 11 | 66 |
| 0 | 0 | 0 | 38 | 62 |
G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,72,1,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,72],[27,0,0,0,0,0,54,14,0,0,0,68,19,0,0,0,0,0,0,55,0,0,0,4,12],[27,0,0,0,0,0,30,13,0,0,0,60,43,0,0,0,0,0,11,38,0,0,0,66,62] >;
C6.SD16 in GAP, Magma, Sage, TeX
C_6.{\rm SD}_{16} % in TeX
G:=Group("C6.SD16"); // GroupNames label
G:=SmallGroup(96,17);
// by ID
G=gap.SmallGroup(96,17);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,121,31,579,297,69,2309]);
// Polycyclic
G:=Group<a,b,c|a^6=b^8=1,c^2=a^3*b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b^-1>;
// generators/relations
Export
Subgroup lattice of C6.SD16 in TeX
Character table of C6.SD16 in TeX