p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4⋊C16, C8.34D8, C8.37SD16, C4.1M5(2), C4⋊C16⋊1C2, (C4×C16)⋊1C2, C4⋊C4.4C8, C4.40C4≀C2, C4⋊C8.10C4, C4.1(C2×C16), (C2×D4).4C8, (C8×D4).1C2, C2.2(D4⋊C8), (C4×D4).10C4, (C2×C8).294D4, C2.1(D4.C8), C2.5(C22⋊C16), (C4×C8).354C22, C42.252(C2×C4), (C2×C4).33M4(2), C4.45(D4⋊C4), C22.35(C22⋊C8), (C2×C4).45(C2×C8), (C2×C4).380(C22⋊C4), SmallGroup(128,61)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4⋊C16
G = < a,b,c | a4=b2=c16=1, bab=cac-1=a-1, cbc-1=ab >
Smallest permutation representation of D4⋊C16
►On 64 points
Generators in S64
(1 24 41 60)(2 61 42 25)(3 26 43 62)(4 63 44 27)(5 28 45 64)(6 49 46 29)(7 30 47 50)(8 51 48 31)(9 32 33 52)(10 53 34 17)(11 18 35 54)(12 55 36 19)(13 20 37 56)(14 57 38 21)(15 22 39 58)(16 59 40 23)
(1 52)(2 34)(3 54)(4 36)(5 56)(6 38)(7 58)(8 40)(9 60)(10 42)(11 62)(12 44)(13 64)(14 46)(15 50)(16 48)(17 25)(18 43)(19 27)(20 45)(21 29)(22 47)(23 31)(24 33)(26 35)(28 37)(30 39)(32 41)(49 57)(51 59)(53 61)(55 63)
(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)
G:=sub<Sym(64)| (1,24,41,60)(2,61,42,25)(3,26,43,62)(4,63,44,27)(5,28,45,64)(6,49,46,29)(7,30,47,50)(8,51,48,31)(9,32,33,52)(10,53,34,17)(11,18,35,54)(12,55,36,19)(13,20,37,56)(14,57,38,21)(15,22,39,58)(16,59,40,23), (1,52)(2,34)(3,54)(4,36)(5,56)(6,38)(7,58)(8,40)(9,60)(10,42)(11,62)(12,44)(13,64)(14,46)(15,50)(16,48)(17,25)(18,43)(19,27)(20,45)(21,29)(22,47)(23,31)(24,33)(26,35)(28,37)(30,39)(32,41)(49,57)(51,59)(53,61)(55,63), (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)>;
G:=Group( (1,24,41,60)(2,61,42,25)(3,26,43,62)(4,63,44,27)(5,28,45,64)(6,49,46,29)(7,30,47,50)(8,51,48,31)(9,32,33,52)(10,53,34,17)(11,18,35,54)(12,55,36,19)(13,20,37,56)(14,57,38,21)(15,22,39,58)(16,59,40,23), (1,52)(2,34)(3,54)(4,36)(5,56)(6,38)(7,58)(8,40)(9,60)(10,42)(11,62)(12,44)(13,64)(14,46)(15,50)(16,48)(17,25)(18,43)(19,27)(20,45)(21,29)(22,47)(23,31)(24,33)(26,35)(28,37)(30,39)(32,41)(49,57)(51,59)(53,61)(55,63), (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) );
G=PermutationGroup([[(1,24,41,60),(2,61,42,25),(3,26,43,62),(4,63,44,27),(5,28,45,64),(6,49,46,29),(7,30,47,50),(8,51,48,31),(9,32,33,52),(10,53,34,17),(11,18,35,54),(12,55,36,19),(13,20,37,56),(14,57,38,21),(15,22,39,58),(16,59,40,23)], [(1,52),(2,34),(3,54),(4,36),(5,56),(6,38),(7,58),(8,40),(9,60),(10,42),(11,62),(12,44),(13,64),(14,46),(15,50),(16,48),(17,25),(18,43),(19,27),(20,45),(21,29),(22,47),(23,31),(24,33),(26,35),(28,37),(30,39),(32,41),(49,57),(51,59),(53,61),(55,63)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 8M | 8N | 8O | 8P | 16A | ··· | 16P | 16Q | ··· | 16X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | D4 | D8 | SD16 | M4(2) | C4≀C2 | M5(2) | D4.C8 |
| kernel | D4⋊C16 | C4×C16 | C4⋊C16 | C8×D4 | C4⋊C8 | C4×D4 | C4⋊C4 | C2×D4 | D4 | C2×C8 | C8 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of D4⋊C16 ►in GL3(𝔽17) generated by
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 16 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 16 |
| 0 | 16 | 0 |
| 14 | 0 | 0 |
| 0 | 15 | 2 |
| 0 | 2 | 2 |
G:=sub<GL(3,GF(17))| [1,0,0,0,0,16,0,1,0],[1,0,0,0,0,16,0,16,0],[14,0,0,0,15,2,0,2,2] >;
D4⋊C16 in GAP, Magma, Sage, TeX
D_4\rtimes C_{16} % in TeX
G:=Group("D4:C16"); // GroupNames label
G:=SmallGroup(128,61);
// by ID
G=gap.SmallGroup(128,61);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,219,100,136,124]);
// Polycyclic
G:=Group<a,b,c|a^4=b^2=c^16=1,b*a*b=c*a*c^-1=a^-1,c*b*c^-1=a*b>;
// generators/relations
Export