p-group, metabelian, nilpotent (class 4), monomial
Aliases: C16⋊7D4, C22⋊1D16, C23.26D8, (C2×D16)⋊5C2, C8⋊7D4⋊2C2, C16⋊3C4⋊2C2, (C2×C4).59D8, C8.99(C2×D4), C2.6(C2×D16), C2.D16⋊1C2, (C22×C16)⋊8C2, (C2×C8).234D4, C8.44(C4○D4), C4.17(C4○D8), (C2×D8).5C22, C2.10(C4○D16), C2.18(C8⋊7D4), C4.90(C4⋊D4), (C2×C8).520C23, C2.D8.5C22, (C2×C16).81C22, C22.106(C2×D8), (C22×C4).590D4, (C22×C8).530C22, (C2×C4).788(C2×D4), SmallGroup(128,947)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C16⋊7D4
G = < a,b,c | a16=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 264 in 87 conjugacy classes, 34 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, D4, C23, C23, C16, C16, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, D4⋊C4, C2.D8, C2×C16, C2×C16, D16, C4⋊D4, C22×C8, C2×D8, C2.D16, C16⋊3C4, C8⋊7D4, C22×C16, C2×D16, C16⋊7D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, D16, C4⋊D4, C2×D8, C4○D8, C8⋊7D4, C2×D16, C4○D16, C16⋊7D4
►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 37 53 18)(2 36 54 17)(3 35 55 32)(4 34 56 31)(5 33 57 30)(6 48 58 29)(7 47 59 28)(8 46 60 27)(9 45 61 26)(10 44 62 25)(11 43 63 24)(12 42 64 23)(13 41 49 22)(14 40 50 21)(15 39 51 20)(16 38 52 19)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 38)(18 37)(19 36)(20 35)(21 34)(22 33)(23 48)(24 47)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(31 40)(32 39)(49 57)(50 56)(51 55)(52 54)(58 64)(59 63)(60 62)
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), (1,37,53,18)(2,36,54,17)(3,35,55,32)(4,34,56,31)(5,33,57,30)(6,48,58,29)(7,47,59,28)(8,46,60,27)(9,45,61,26)(10,44,62,25)(11,43,63,24)(12,42,64,23)(13,41,49,22)(14,40,50,21)(15,39,51,20)(16,38,52,19), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(49,57)(50,56)(51,55)(52,54)(58,64)(59,63)(60,62)>;
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), (1,37,53,18)(2,36,54,17)(3,35,55,32)(4,34,56,31)(5,33,57,30)(6,48,58,29)(7,47,59,28)(8,46,60,27)(9,45,61,26)(10,44,62,25)(11,43,63,24)(12,42,64,23)(13,41,49,22)(14,40,50,21)(15,39,51,20)(16,38,52,19), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(49,57)(50,56)(51,55)(52,54)(58,64)(59,63)(60,62) );
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)], [(1,37,53,18),(2,36,54,17),(3,35,55,32),(4,34,56,31),(5,33,57,30),(6,48,58,29),(7,47,59,28),(8,46,60,27),(9,45,61,26),(10,44,62,25),(11,43,63,24),(12,42,64,23),(13,41,49,22),(14,40,50,21),(15,39,51,20),(16,38,52,19)], [(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,38),(18,37),(19,36),(20,35),(21,34),(22,33),(23,48),(24,47),(25,46),(26,45),(27,44),(28,43),(29,42),(30,41),(31,40),(32,39),(49,57),(50,56),(51,55),(52,54),(58,64),(59,63),(60,62)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 16 | 16 | 2 | 2 | 2 | 2 | 16 | 16 | 2 | ··· | 2 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | D8 | D8 | C4○D8 | D16 | C4○D16 |
| kernel | C16⋊7D4 | C2.D16 | C16⋊3C4 | C8⋊7D4 | C22×C16 | C2×D16 | C16 | C2×C8 | C22×C4 | C8 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 8 |
Matrix representation of C16⋊7D4 ►in GL4(𝔽17) generated by
| 10 | 8 | 0 | 0 |
| 13 | 2 | 0 | 0 |
| 0 | 0 | 6 | 13 |
| 0 | 0 | 4 | 6 |
| 13 | 9 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [10,13,0,0,8,2,0,0,0,0,6,4,0,0,13,6],[13,0,0,0,9,4,0,0,0,0,16,0,0,0,0,1],[1,16,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;
C16⋊7D4 in GAP, Magma, Sage, TeX
C_{16}\rtimes_7D_4 % in TeX
G:=Group("C16:7D4"); // GroupNames label
G:=SmallGroup(128,947);
// by ID
G=gap.SmallGroup(128,947);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,288,422,1123,360,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^16=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations