p-group, metabelian, nilpotent (class 4), monomial
Aliases: C4⋊1D16, C16⋊4D4, C8.15D8, C42.336D4, (C4×C16)⋊9C2, C4.7(C2×D8), (C2×D16)⋊6C2, C8⋊4D4⋊7C2, (C2×C4).83D8, C8.39(C2×D4), (C2×C8).252D4, C2.10(C2×D16), C4.1(C4⋊1D4), C2.12(C8⋊4D4), (C2×C16).83C22, (C2×C8).544C23, (C4×C8).401C22, (C2×D8).16C22, C22.130(C2×D8), (C2×C4).812(C2×D4), SmallGroup(128,978)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊D16
G = < a,b,c | a16=b4=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Subgroups: 392 in 108 conjugacy classes, 40 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, C23, C16, C42, C2×C8, D8, C2×D4, C4×C8, C2×C16, D16, C4⋊1D4, C2×D8, C2×D8, C4×C16, C8⋊4D4, C2×D16, C4⋊D16
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, D16, C4⋊1D4, C2×D8, C8⋊4D4, C2×D16, C4⋊D16
►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 62 33 19)(2 63 34 20)(3 64 35 21)(4 49 36 22)(5 50 37 23)(6 51 38 24)(7 52 39 25)(8 53 40 26)(9 54 41 27)(10 55 42 28)(11 56 43 29)(12 57 44 30)(13 58 45 31)(14 59 46 32)(15 60 47 17)(16 61 48 18)
(1 33)(2 48)(3 47)(4 46)(5 45)(6 44)(7 43)(8 42)(9 41)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 34)(17 21)(18 20)(22 32)(23 31)(24 30)(25 29)(26 28)(49 59)(50 58)(51 57)(52 56)(53 55)(60 64)(61 63)
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), (1,62,33,19)(2,63,34,20)(3,64,35,21)(4,49,36,22)(5,50,37,23)(6,51,38,24)(7,52,39,25)(8,53,40,26)(9,54,41,27)(10,55,42,28)(11,56,43,29)(12,57,44,30)(13,58,45,31)(14,59,46,32)(15,60,47,17)(16,61,48,18), (1,33)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,21)(18,20)(22,32)(23,31)(24,30)(25,29)(26,28)(49,59)(50,58)(51,57)(52,56)(53,55)(60,64)(61,63)>;
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), (1,62,33,19)(2,63,34,20)(3,64,35,21)(4,49,36,22)(5,50,37,23)(6,51,38,24)(7,52,39,25)(8,53,40,26)(9,54,41,27)(10,55,42,28)(11,56,43,29)(12,57,44,30)(13,58,45,31)(14,59,46,32)(15,60,47,17)(16,61,48,18), (1,33)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,21)(18,20)(22,32)(23,31)(24,30)(25,29)(26,28)(49,59)(50,58)(51,57)(52,56)(53,55)(60,64)(61,63) );
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)], [(1,62,33,19),(2,63,34,20),(3,64,35,21),(4,49,36,22),(5,50,37,23),(6,51,38,24),(7,52,39,25),(8,53,40,26),(9,54,41,27),(10,55,42,28),(11,56,43,29),(12,57,44,30),(13,58,45,31),(14,59,46,32),(15,60,47,17),(16,61,48,18)], [(1,33),(2,48),(3,47),(4,46),(5,45),(6,44),(7,43),(8,42),(9,41),(10,40),(11,39),(12,38),(13,37),(14,36),(15,35),(16,34),(17,21),(18,20),(22,32),(23,31),(24,30),(25,29),(26,28),(49,59),(50,58),(51,57),(52,56),(53,55),(60,64),(61,63)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 16 | 16 | 16 | 16 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | D4 | D4 | D4 | D8 | D8 | D16 |
| kernel | C4⋊D16 | C4×C16 | C8⋊4D4 | C2×D16 | C16 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 4 | 4 | 16 |
Matrix representation of C4⋊D16 ►in GL4(𝔽17) generated by
| 6 | 13 | 0 | 0 |
| 4 | 6 | 0 | 0 |
| 0 | 0 | 16 | 13 |
| 0 | 0 | 9 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 4 |
| 0 | 0 | 8 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 9 | 1 |
G:=sub<GL(4,GF(17))| [6,4,0,0,13,6,0,0,0,0,16,9,0,0,13,1],[1,0,0,0,0,1,0,0,0,0,1,8,0,0,4,16],[1,0,0,0,0,16,0,0,0,0,16,9,0,0,0,1] >;
C4⋊D16 in GAP, Magma, Sage, TeX
C_4\rtimes D_{16} % in TeX
G:=Group("C4:D16"); // GroupNames label
G:=SmallGroup(128,978);
// by ID
G=gap.SmallGroup(128,978);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,288,422,100,1123,360,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^16=b^4=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations