p-group, metabelian, nilpotent (class 5), monomial
Aliases: D16⋊2C4, C2.1D32, C16.15D4, C4.1SD32, C2.1SD64, C8.15SD16, C22.8D16, (C2×C32)⋊3C2, C16⋊3C4⋊1C2, (C2×C4).65D8, C16.10(C2×C4), (C2×D16).1C2, (C2×C8).237D4, C2.6(C2.D16), C8.13(C22⋊C4), (C2×C16).75C22, C4.13(D4⋊C4), SmallGroup(128,147)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D16⋊2C4
G = < a,b,c | a16=b2=c4=1, bab=cac-1=a-1, cbc-1=a13b >
Smallest permutation representation of D16⋊2C4
►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 63)(2 62)(3 61)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 54)(11 53)(12 52)(13 51)(14 50)(15 49)(16 64)(17 42)(18 41)(19 40)(20 39)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)
(1 30 64 47)(2 29 49 46)(3 28 50 45)(4 27 51 44)(5 26 52 43)(6 25 53 42)(7 24 54 41)(8 23 55 40)(9 22 56 39)(10 21 57 38)(11 20 58 37)(12 19 59 36)(13 18 60 35)(14 17 61 34)(15 32 62 33)(16 31 63 48)
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), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,64)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43), (1,30,64,47)(2,29,49,46)(3,28,50,45)(4,27,51,44)(5,26,52,43)(6,25,53,42)(7,24,54,41)(8,23,55,40)(9,22,56,39)(10,21,57,38)(11,20,58,37)(12,19,59,36)(13,18,60,35)(14,17,61,34)(15,32,62,33)(16,31,63,48)>;
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), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,64)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43), (1,30,64,47)(2,29,49,46)(3,28,50,45)(4,27,51,44)(5,26,52,43)(6,25,53,42)(7,24,54,41)(8,23,55,40)(9,22,56,39)(10,21,57,38)(11,20,58,37)(12,19,59,36)(13,18,60,35)(14,17,61,34)(15,32,62,33)(16,31,63,48) );
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)], [(1,63),(2,62),(3,61),(4,60),(5,59),(6,58),(7,57),(8,56),(9,55),(10,54),(11,53),(12,52),(13,51),(14,50),(15,49),(16,64),(17,42),(18,41),(19,40),(20,39),(21,38),(22,37),(23,36),(24,35),(25,34),(26,33),(27,48),(28,47),(29,46),(30,45),(31,44),(32,43)], [(1,30,64,47),(2,29,49,46),(3,28,50,45),(4,27,51,44),(5,26,52,43),(6,25,53,42),(7,24,54,41),(8,23,55,40),(9,22,56,39),(10,21,57,38),(11,20,58,37),(12,19,59,36),(13,18,60,35),(14,17,61,34),(15,32,62,33),(16,31,63,48)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 16A | ··· | 16H | 32A | ··· | 32P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 32 | ··· | 32 |
| size | 1 | 1 | 1 | 1 | 16 | 16 | 2 | 2 | 16 | 16 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | SD16 | D8 | SD32 | D16 | D32 | SD64 |
| kernel | D16⋊2C4 | C16⋊3C4 | C2×C32 | C2×D16 | D16 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 |
Matrix representation of D16⋊2C4 ►in GL4(𝔽97) generated by
| 1 | 95 | 0 | 0 |
| 1 | 96 | 0 | 0 |
| 0 | 0 | 69 | 52 |
| 0 | 0 | 71 | 24 |
| 96 | 2 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 69 | 52 |
| 0 | 0 | 95 | 28 |
| 0 | 17 | 0 | 0 |
| 57 | 0 | 0 | 0 |
| 0 | 0 | 67 | 43 |
| 0 | 0 | 40 | 30 |
G:=sub<GL(4,GF(97))| [1,1,0,0,95,96,0,0,0,0,69,71,0,0,52,24],[96,0,0,0,2,1,0,0,0,0,69,95,0,0,52,28],[0,57,0,0,17,0,0,0,0,0,67,40,0,0,43,30] >;
D16⋊2C4 in GAP, Magma, Sage, TeX
D_{16}\rtimes_2C_4 % in TeX
G:=Group("D16:2C4"); // GroupNames label
G:=SmallGroup(128,147);
// by ID
G=gap.SmallGroup(128,147);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,422,219,268,1684,851,242,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c|a^16=b^2=c^4=1,b*a*b=c*a*c^-1=a^-1,c*b*c^-1=a^13*b>;
// generators/relations
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