p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8.8SD16, C42.246C23, C4⋊C4.68D4, C8⋊2C8⋊18C2, (C2×C8).98D4, C8⋊2Q8⋊24C2, (C2×D4).62D4, C8⋊6D4.4C2, C4.62(C2×SD16), C4⋊Q8.67C22, C4⋊C8.188C22, (C4×C8).149C22, D4⋊2Q8.11C2, C4.6Q16⋊18C2, C2.8(C8.D4), (C4×D4).48C22, C4.44(C8.C22), C2.11(D4.D4), C2.14(D4.4D4), C22.207(C4⋊D4), (C2×C4).31(C4○D4), (C2×C4).1281(C2×D4), SmallGroup(128,427)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.8SD16
G = < a,b,c | a8=b8=c2=1, bab-1=a3, cac=a5, cbc=a4b3 >
Subgroups: 176 in 77 conjugacy classes, 34 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4×C8, C22⋊C8, D4⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C2.D8, C4×D4, C4⋊Q8, C2×M4(2), C4.6Q16, C8⋊2C8, C8⋊6D4, D4⋊2Q8, C8⋊2Q8, C8.8SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C4⋊D4, C2×SD16, C8.C22, D4.D4, C8.D4, D4.4D4, C8.8SD16
Character table of C8.8SD16
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | |
| size | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 4 | 8 | 16 | 16 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 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 | 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 | 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 | 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 | 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 | -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 | -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 | 1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 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 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 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 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | complex lifted from C4○D4 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | complex lifted from C4○D4 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | -√-2 | -√-2 | √-2 | 0 | 0 | √-2 | complex lifted from SD16 |
| ρ16 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | √-2 | √-2 | -√-2 | 0 | 0 | -√-2 | complex lifted from SD16 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | -√-2 | √-2 | √-2 | 0 | 0 | -√-2 | complex lifted from SD16 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | √-2 | -√-2 | -√-2 | 0 | 0 | √-2 | complex lifted from SD16 |
| ρ19 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 0 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4.4D4 |
| ρ20 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | 0 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4.4D4 |
| ρ21 | 4 | -4 | 4 | -4 | 0 | 0 | -4 | 0 | 4 | 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 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
►On 64 points
Generators in S64
(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 27 54 64 42 21 38 12)(2 30 55 59 43 24 39 15)(3 25 56 62 44 19 40 10)(4 28 49 57 45 22 33 13)(5 31 50 60 46 17 34 16)(6 26 51 63 47 20 35 11)(7 29 52 58 48 23 36 14)(8 32 53 61 41 18 37 9)
(2 6)(4 8)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)(25 58)(26 63)(27 60)(28 57)(29 62)(30 59)(31 64)(32 61)(33 53)(34 50)(35 55)(36 52)(37 49)(38 54)(39 51)(40 56)(41 45)(43 47)
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), (1,27,54,64,42,21,38,12)(2,30,55,59,43,24,39,15)(3,25,56,62,44,19,40,10)(4,28,49,57,45,22,33,13)(5,31,50,60,46,17,34,16)(6,26,51,63,47,20,35,11)(7,29,52,58,48,23,36,14)(8,32,53,61,41,18,37,9), (2,6)(4,8)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21)(25,58)(26,63)(27,60)(28,57)(29,62)(30,59)(31,64)(32,61)(33,53)(34,50)(35,55)(36,52)(37,49)(38,54)(39,51)(40,56)(41,45)(43,47)>;
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), (1,27,54,64,42,21,38,12)(2,30,55,59,43,24,39,15)(3,25,56,62,44,19,40,10)(4,28,49,57,45,22,33,13)(5,31,50,60,46,17,34,16)(6,26,51,63,47,20,35,11)(7,29,52,58,48,23,36,14)(8,32,53,61,41,18,37,9), (2,6)(4,8)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21)(25,58)(26,63)(27,60)(28,57)(29,62)(30,59)(31,64)(32,61)(33,53)(34,50)(35,55)(36,52)(37,49)(38,54)(39,51)(40,56)(41,45)(43,47) );
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)], [(1,27,54,64,42,21,38,12),(2,30,55,59,43,24,39,15),(3,25,56,62,44,19,40,10),(4,28,49,57,45,22,33,13),(5,31,50,60,46,17,34,16),(6,26,51,63,47,20,35,11),(7,29,52,58,48,23,36,14),(8,32,53,61,41,18,37,9)], [(2,6),(4,8),(9,18),(10,23),(11,20),(12,17),(13,22),(14,19),(15,24),(16,21),(25,58),(26,63),(27,60),(28,57),(29,62),(30,59),(31,64),(32,61),(33,53),(34,50),(35,55),(36,52),(37,49),(38,54),(39,51),(40,56),(41,45),(43,47)]])
Matrix representation of C8.8SD16 ►in GL8(𝔽17)
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 | 2 | 0 |
| 0 | 0 | 0 | 0 | 14 | 3 | 0 | 2 |
| 0 | 0 | 0 | 0 | 0 | 16 | 14 | 14 |
| 0 | 0 | 0 | 0 | 1 | 0 | 3 | 14 |
| 12 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 1 | 14 | 15 |
| 0 | 0 | 0 | 0 | 1 | 10 | 15 | 3 |
| 0 | 0 | 0 | 0 | 10 | 1 | 16 | 7 |
| 0 | 0 | 0 | 0 | 1 | 7 | 7 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 14 | 16 | 0 |
| 0 | 0 | 0 | 0 | 3 | 14 | 0 | 16 |
G:=sub<GL(8,GF(17))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,3,14,0,1,0,0,0,0,3,3,16,0,0,0,0,0,2,0,14,3,0,0,0,0,0,2,14,14],[12,12,0,0,0,0,0,0,5,12,0,0,0,0,0,0,0,0,5,5,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,0,7,1,10,1,0,0,0,0,1,10,1,7,0,0,0,0,14,15,16,7,0,0,0,0,15,3,7,1],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,14,3,0,0,0,0,0,1,14,14,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16] >;
C8.8SD16 in GAP, Magma, Sage, TeX
C_8._8{\rm SD}_{16} % in TeX
G:=Group("C8.8SD16"); // GroupNames label
G:=SmallGroup(128,427);
// by ID
G=gap.SmallGroup(128,427);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,672,141,288,422,387,1123,136,2804,718,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^3,c*a*c=a^5,c*b*c=a^4*b^3>;
// generators/relations
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