p-group, metabelian, nilpotent (class 4), monomial
Aliases: C16⋊5D4, C4⋊1SD32, C8.17D8, C42.338D4, C4.9(C2×D8), (C4×C16)⋊13C2, (C2×C4).85D8, C8.41(C2×D4), C4⋊Q16⋊7C2, (C2×C8).254D4, (C2×SD32)⋊16C2, C8⋊4D4.10C2, C4.3(C4⋊1D4), C2.16(C2×SD32), C2.14(C8⋊4D4), (C4×C8).403C22, (C2×C16).99C22, (C2×C8).546C23, (C2×D8).17C22, C22.132(C2×D8), (C2×Q16).18C22, (C2×C4).814(C2×D4), SmallGroup(128,980)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C16⋊5D4
G = < a,b,c | a16=b4=c2=1, ab=ba, cac=a7, cbc=b-1 >
Subgroups: 296 in 98 conjugacy classes, 40 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C16, C42, C4⋊C4, C2×C8, D8, Q16, C2×D4, C2×Q8, C4×C8, C2×C16, SD32, C4⋊1D4, C4⋊Q8, C2×D8, C2×D8, C2×Q16, C2×Q16, C4×C16, C8⋊4D4, C4⋊Q16, C2×SD32, C16⋊5D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, SD32, C4⋊1D4, C2×D8, C8⋊4D4, C2×SD32, C16⋊5D4
►On 64 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)
(1 48 21 50)(2 33 22 51)(3 34 23 52)(4 35 24 53)(5 36 25 54)(6 37 26 55)(7 38 27 56)(8 39 28 57)(9 40 29 58)(10 41 30 59)(11 42 31 60)(12 43 32 61)(13 44 17 62)(14 45 18 63)(15 46 19 64)(16 47 20 49)
(1 50)(2 57)(3 64)(4 55)(5 62)(6 53)(7 60)(8 51)(9 58)(10 49)(11 56)(12 63)(13 54)(14 61)(15 52)(16 59)(17 36)(18 43)(19 34)(20 41)(21 48)(22 39)(23 46)(24 37)(25 44)(26 35)(27 42)(28 33)(29 40)(30 47)(31 38)(32 45)
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), (1,48,21,50)(2,33,22,51)(3,34,23,52)(4,35,24,53)(5,36,25,54)(6,37,26,55)(7,38,27,56)(8,39,28,57)(9,40,29,58)(10,41,30,59)(11,42,31,60)(12,43,32,61)(13,44,17,62)(14,45,18,63)(15,46,19,64)(16,47,20,49), (1,50)(2,57)(3,64)(4,55)(5,62)(6,53)(7,60)(8,51)(9,58)(10,49)(11,56)(12,63)(13,54)(14,61)(15,52)(16,59)(17,36)(18,43)(19,34)(20,41)(21,48)(22,39)(23,46)(24,37)(25,44)(26,35)(27,42)(28,33)(29,40)(30,47)(31,38)(32,45)>;
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), (1,48,21,50)(2,33,22,51)(3,34,23,52)(4,35,24,53)(5,36,25,54)(6,37,26,55)(7,38,27,56)(8,39,28,57)(9,40,29,58)(10,41,30,59)(11,42,31,60)(12,43,32,61)(13,44,17,62)(14,45,18,63)(15,46,19,64)(16,47,20,49), (1,50)(2,57)(3,64)(4,55)(5,62)(6,53)(7,60)(8,51)(9,58)(10,49)(11,56)(12,63)(13,54)(14,61)(15,52)(16,59)(17,36)(18,43)(19,34)(20,41)(21,48)(22,39)(23,46)(24,37)(25,44)(26,35)(27,42)(28,33)(29,40)(30,47)(31,38)(32,45) );
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)], [(1,48,21,50),(2,33,22,51),(3,34,23,52),(4,35,24,53),(5,36,25,54),(6,37,26,55),(7,38,27,56),(8,39,28,57),(9,40,29,58),(10,41,30,59),(11,42,31,60),(12,43,32,61),(13,44,17,62),(14,45,18,63),(15,46,19,64),(16,47,20,49)], [(1,50),(2,57),(3,64),(4,55),(5,62),(6,53),(7,60),(8,51),(9,58),(10,49),(11,56),(12,63),(13,54),(14,61),(15,52),(16,59),(17,36),(18,43),(19,34),(20,41),(21,48),(22,39),(23,46),(24,37),(25,44),(26,35),(27,42),(28,33),(29,40),(30,47),(31,38),(32,45)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 16 | 16 | 2 | ··· | 2 | 16 | 16 | 2 | ··· | 2 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D8 | D8 | SD32 |
| kernel | C16⋊5D4 | C4×C16 | C8⋊4D4 | C4⋊Q16 | C2×SD32 | C16 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 4 | 4 | 16 |
Matrix representation of C16⋊5D4 ►in GL4(𝔽17) generated by
| 14 | 14 | 0 | 0 |
| 3 | 14 | 0 | 0 |
| 0 | 0 | 8 | 14 |
| 0 | 0 | 10 | 11 |
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 16 | 16 |
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [14,3,0,0,14,14,0,0,0,0,8,10,0,0,14,11],[0,1,0,0,16,0,0,0,0,0,1,16,0,0,2,16],[0,16,0,0,16,0,0,0,0,0,1,0,0,0,2,16] >;
C16⋊5D4 in GAP, Magma, Sage, TeX
C_{16}\rtimes_5D_4 % in TeX
G:=Group("C16:5D4"); // GroupNames label
G:=SmallGroup(128,980);
// by ID
G=gap.SmallGroup(128,980);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,736,422,100,1123,360,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^16=b^4=c^2=1,a*b=b*a,c*a*c=a^7,c*b*c=b^-1>;
// generators/relations