p-group, metabelian, nilpotent (class 3), monomial
Aliases: SD16:1C4, C42.9C22, C8:2(C2xC4), Q8:2(C2xC4), (C4xQ8):2C2, C8:C4:1C2, D4.2(C2xC4), C2.D8:11C2, (C4xD4).5C2, C2.15(C4xD4), C4.4(C4oD4), (C2xC4).101D4, Q8:C4:16C2, D4:C4.6C2, C4:C4.53C22, C2.4(C8:C22), (C2xC8).49C22, C4.12(C22xC4), (C2xC4).76C23, (C2xSD16).1C2, C22.54(C2xD4), (C2xD4).52C22, C2.4(C8.C22), (C2xQ8).46C22, SmallGroup(64,121)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for SD16:C4
G = < a,b,c | a8=b2=c4=1, bab=a3, cac-1=a5, bc=cb >
Subgroups: 101 in 60 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C42, C42, C22:C4, C4:C4, C4:C4, C2xC8, SD16, C22xC4, C2xD4, C2xQ8, C8:C4, D4:C4, Q8:C4, C2.D8, C4xD4, C4xQ8, C2xSD16, SD16:C4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22xC4, C2xD4, C4oD4, C4xD4, C8:C22, C8.C22, SD16:C4
Character table of SD16:C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 1 | -1 | 1 | -1 | -i | i | -i | i | -i | -i | i | -1 | 1 | i | i | -1 | -i | 1 | linear of order 4 |
| ρ10 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | i | -i | i | -i | -i | -i | i | 1 | -1 | i | -i | -1 | i | 1 | linear of order 4 |
| ρ11 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | i | -i | i | -i | i | i | -1 | 1 | -i | -i | 1 | i | -1 | linear of order 4 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | i | -i | i | -i | -i | i | i | 1 | -1 | -i | i | 1 | -i | -1 | linear of order 4 |
| ρ13 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | i | -i | i | -i | i | -i | -i | -1 | 1 | i | i | 1 | -i | -1 | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -i | i | -i | i | i | -i | -i | 1 | -1 | i | -i | 1 | i | -1 | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | i | -i | i | -i | i | i | -i | -1 | 1 | -i | -i | -1 | i | 1 | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | i | -i | i | i | i | -i | 1 | -1 | -i | i | -1 | -i | 1 | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ21 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8:C22 |
| 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 |
►On 32 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)
(1 32)(2 27)(3 30)(4 25)(5 28)(6 31)(7 26)(8 29)(9 21)(10 24)(11 19)(12 22)(13 17)(14 20)(15 23)(16 18)
(1 11 32 19)(2 16 25 24)(3 13 26 21)(4 10 27 18)(5 15 28 23)(6 12 29 20)(7 9 30 17)(8 14 31 22)
G:=sub<Sym(32)| (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), (1,32)(2,27)(3,30)(4,25)(5,28)(6,31)(7,26)(8,29)(9,21)(10,24)(11,19)(12,22)(13,17)(14,20)(15,23)(16,18), (1,11,32,19)(2,16,25,24)(3,13,26,21)(4,10,27,18)(5,15,28,23)(6,12,29,20)(7,9,30,17)(8,14,31,22)>;
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), (1,32)(2,27)(3,30)(4,25)(5,28)(6,31)(7,26)(8,29)(9,21)(10,24)(11,19)(12,22)(13,17)(14,20)(15,23)(16,18), (1,11,32,19)(2,16,25,24)(3,13,26,21)(4,10,27,18)(5,15,28,23)(6,12,29,20)(7,9,30,17)(8,14,31,22) );
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)], [(1,32),(2,27),(3,30),(4,25),(5,28),(6,31),(7,26),(8,29),(9,21),(10,24),(11,19),(12,22),(13,17),(14,20),(15,23),(16,18)], [(1,11,32,19),(2,16,25,24),(3,13,26,21),(4,10,27,18),(5,15,28,23),(6,12,29,20),(7,9,30,17),(8,14,31,22)]])
SD16:C4 is a maximal subgroup of
C42.352C23 C42.353C23 C42.354C23 C42.355C23 C42.357C23 C42.358C23 C42.360C23 C42.387C23 C42.390C23 SD16:6D4 SD16:8D4 C42.45C23 C42.46C23 C42.49C23 C42.50C23 C42.55C23 C42.57C23 C42.60C23 C42.62C23 C42.64C23 C42.508C23 C42.509C23 C42.511C23 C42.512C23 C42.513C23 C42.516C23 C42.517C23 C42.518C23 SD16:Q8 SD16:2Q8 SD16:3Q8 C42.75C23 C42.532C23 GL2(F3):C4
C42.D2p: C42.383D4 C4xC8:C22 C4xC8.C22 C42.228D4 C42.229D4 C42.230D4 C42.232D4 C42.233D4 ...
C2p.(C4xD4): C42.275C23 C42.276C23 C42.278C23 C42.280C23 D4.S3:C4 Q8:3(C4xS3) C24:C2:C4 SD16:Dic3 ...
C8:pD4:C2: C42.386C23 C42.391C23 SD16:D4 SD16:7D4 C42.43C23 C42.44C23 C42.56C23 C42.72C23 ...
SD16:C4 is a maximal quotient of
SD16:C8 D4.M4(2) Q8:2M4(2) C8:M4(2) C8:C42 C4.Q8:9C4 C4.Q8:10C4 C8:(C4:C4)
C42.D2p: D4:C42 Q8:C42 C42.16D6 C42.51D6 C42.56D6 C42.16D10 C42.51D10 C42.56D10 ...
C2p.(C4xD4): C2.(C4xD8) Q8:(C4:C4) (C2xSD16):14C4 (C2xSD16):15C4 D4:C4:C4 C2.(C4xQ16) C2.(C8:D4) C2.(C8:2D4) ...
Matrix representation of SD16:C4 ►in GL6(F17)
| 0 | 8 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 11 |
| 0 | 0 | 13 | 8 | 3 | 11 |
| 0 | 0 | 0 | 11 | 0 | 9 |
| 0 | 0 | 3 | 11 | 4 | 9 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
G:=sub<GL(6,GF(17))| [0,2,0,0,0,0,8,0,0,0,0,0,0,0,0,13,0,3,0,0,8,8,11,11,0,0,0,3,0,4,0,0,11,11,9,9],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;
SD16:C4 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\rtimes C_4 % in TeX
G:=Group("SD16:C4"); // GroupNames label
G:=SmallGroup(64,121);
// by ID
G=gap.SmallGroup(64,121);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,650,86,963,489,117]);
// Polycyclic
G:=Group<a,b,c|a^8=b^2=c^4=1,b*a*b=a^3,c*a*c^-1=a^5,b*c=c*b>;
// generators/relations
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