p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4.3Q16, C4.4SD16, C42.77C22, (C4×C8).4C2, C4⋊Q8.7C2, (C2×C4).75D4, C2.9(C2×Q16), C4.14(C4○D4), C4⋊C4.17C22, (C2×C8).67C22, Q8⋊C4.1C2, C2.15(C2×SD16), (C2×C4).112C23, C22.108(C2×D4), (C2×Q8).20C22, C2.10(C4.4D4), SmallGroup(64,168)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.SD16
G = < a,b,c | a4=b8=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b3 >
Character table of C4.SD16
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | 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 |
| ρ10 | 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 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | √2 | -√2 | √2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | √2 | -√2 | -√2 | -√2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ13 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | -√2 | √2 | -√2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | -√2 | √2 | √2 | √2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 2i | 0 | -2i | -2i | complex lifted from C4○D4 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 0 | 2i | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | 0 | -2i | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | -2i | 0 | 2i | 2i | complex lifted from C4○D4 |
| ρ19 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | -√-2 | √-2 | √-2 | -√-2 | complex lifted from SD16 |
| ρ20 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | √-2 | -√-2 | -√-2 | √-2 | complex lifted from SD16 |
| ρ21 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | -√-2 | -√-2 | √-2 | √-2 | -√-2 | √-2 | complex lifted from SD16 |
| ρ22 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | √-2 | √-2 | -√-2 | -√-2 | √-2 | -√-2 | complex lifted from SD16 |
►Regular action on 64 points
(1 38 29 15)(2 39 30 16)(3 40 31 9)(4 33 32 10)(5 34 25 11)(6 35 26 12)(7 36 27 13)(8 37 28 14)(17 53 41 57)(18 54 42 58)(19 55 43 59)(20 56 44 60)(21 49 45 61)(22 50 46 62)(23 51 47 63)(24 52 48 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 54 29 58)(2 61 30 49)(3 52 31 64)(4 59 32 55)(5 50 25 62)(6 57 26 53)(7 56 27 60)(8 63 28 51)(9 48 40 24)(10 19 33 43)(11 46 34 22)(12 17 35 41)(13 44 36 20)(14 23 37 47)(15 42 38 18)(16 21 39 45)
G:=sub<Sym(64)| (1,38,29,15)(2,39,30,16)(3,40,31,9)(4,33,32,10)(5,34,25,11)(6,35,26,12)(7,36,27,13)(8,37,28,14)(17,53,41,57)(18,54,42,58)(19,55,43,59)(20,56,44,60)(21,49,45,61)(22,50,46,62)(23,51,47,63)(24,52,48,64), (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), (1,54,29,58)(2,61,30,49)(3,52,31,64)(4,59,32,55)(5,50,25,62)(6,57,26,53)(7,56,27,60)(8,63,28,51)(9,48,40,24)(10,19,33,43)(11,46,34,22)(12,17,35,41)(13,44,36,20)(14,23,37,47)(15,42,38,18)(16,21,39,45)>;
G:=Group( (1,38,29,15)(2,39,30,16)(3,40,31,9)(4,33,32,10)(5,34,25,11)(6,35,26,12)(7,36,27,13)(8,37,28,14)(17,53,41,57)(18,54,42,58)(19,55,43,59)(20,56,44,60)(21,49,45,61)(22,50,46,62)(23,51,47,63)(24,52,48,64), (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), (1,54,29,58)(2,61,30,49)(3,52,31,64)(4,59,32,55)(5,50,25,62)(6,57,26,53)(7,56,27,60)(8,63,28,51)(9,48,40,24)(10,19,33,43)(11,46,34,22)(12,17,35,41)(13,44,36,20)(14,23,37,47)(15,42,38,18)(16,21,39,45) );
G=PermutationGroup([[(1,38,29,15),(2,39,30,16),(3,40,31,9),(4,33,32,10),(5,34,25,11),(6,35,26,12),(7,36,27,13),(8,37,28,14),(17,53,41,57),(18,54,42,58),(19,55,43,59),(20,56,44,60),(21,49,45,61),(22,50,46,62),(23,51,47,63),(24,52,48,64)], [(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)], [(1,54,29,58),(2,61,30,49),(3,52,31,64),(4,59,32,55),(5,50,25,62),(6,57,26,53),(7,56,27,60),(8,63,28,51),(9,48,40,24),(10,19,33,43),(11,46,34,22),(12,17,35,41),(13,44,36,20),(14,23,37,47),(15,42,38,18),(16,21,39,45)]])
C4.SD16 is a maximal subgroup of
C8⋊5Q16 C82⋊12C2 C82⋊3C2 C8⋊5SD16 C8.9SD16 C42.664C23 C42.665C23 C42.666C23 C42.667C23 C8⋊3Q16 C42.355D4 C42.241D4 C42.243D4 C42.365D4 C42.367D4 C42.259D4 C42.262D4 C42.264D4 C42.265D4 C42.267D4 C42.268D4 C42.281D4 C42.282D4 C42.283D4
C4p.Q16: C8.7Q16 C12.14Q16 C12.9Q16 C12.Q16 C20.14Q16 C20.Q16 C20.11Q16 C28.14Q16 ...
C4⋊C4.D2p: D4.SD16 Q8.Q16 D4.3Q16 C42.199C23 D4.5SD16 D4⋊3Q16 Q8⋊3Q16 C42.207C23 ...
C4.SD16 is a maximal quotient of
C42.55Q8 C2.(C8⋊8D4) (C2×C4).19Q16 (C2×C4).23Q16
C42.D2p: C42.431D4 C42.436D4 C12.14Q16 C12.9Q16 C12.Q16 C20.14Q16 C20.Q16 C20.11Q16 ...
C4⋊C4.D2p: (C2×Q8)⋊Q8 C4⋊C4.95D4 Dic3.1Q16 Dic5.3Q16 Dic7.1Q16 ...
Matrix representation of C4.SD16 ►in GL4(𝔽17) generated by
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 14 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 5 | 5 |
| 0 | 0 | 12 | 5 |
| 7 | 1 | 0 | 0 |
| 1 | 10 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 16 | 0 |
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,1,0,0,0,0,1],[3,3,0,0,14,3,0,0,0,0,5,12,0,0,5,5],[7,1,0,0,1,10,0,0,0,0,0,16,0,0,16,0] >;
C4.SD16 in GAP, Magma, Sage, TeX
C_4.{\rm SD}_{16} % in TeX
G:=Group("C4.SD16"); // GroupNames label
G:=SmallGroup(64,168);
// by ID
G=gap.SmallGroup(64,168);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,103,362,50,963,117,1444,88]);
// Polycyclic
G:=Group<a,b,c|a^4=b^8=1,c^2=a^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^2*b^3>;
// generators/relations
Export
Subgroup lattice of C4.SD16 in TeX
Character table of C4.SD16 in TeX