p-group, metabelian, nilpotent (class 4), monomial
Aliases: C16⋊4D4, C4⋊1D16, C8.15D8, C42.336D4, (C4×C16)⋊9C2, C4.7(C2×D8), (C2×D16)⋊6C2, C8⋊4D4⋊7C2, (C2×C4).83D8, C8.39(C2×D4), C2.10(C2×D16), (C2×C8).252D4, C4.1(C4⋊1D4), C2.12(C8⋊4D4), (C4×C8).401C22, (C2×C8).544C23, (C2×C16).83C22, (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 C16⋊4D4
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 [×2], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×12], C8 [×4], C2×C4, C2×C4 [×2], D4 [×16], C23 [×4], C16 [×4], C42, C2×C8 [×2], D8 [×12], C2×D4 [×8], C4×C8, C2×C16 [×2], D16 [×8], C4⋊1D4 [×2], C2×D8 [×4], C2×D8 [×2], C4×C16, C8⋊4D4 [×2], C2×D16 [×4], C16⋊4D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×4], C2×D4 [×3], D16 [×4], C4⋊1D4, C2×D8 [×2], C8⋊4D4, C2×D16 [×2], C16⋊4D4
►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 23 64 48)(2 24 49 33)(3 25 50 34)(4 26 51 35)(5 27 52 36)(6 28 53 37)(7 29 54 38)(8 30 55 39)(9 31 56 40)(10 32 57 41)(11 17 58 42)(12 18 59 43)(13 19 60 44)(14 20 61 45)(15 21 62 46)(16 22 63 47)
(1 64)(2 63)(3 62)(4 61)(5 60)(6 59)(7 58)(8 57)(9 56)(10 55)(11 54)(12 53)(13 52)(14 51)(15 50)(16 49)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(30 32)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)
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), (1,23,64,48)(2,24,49,33)(3,25,50,34)(4,26,51,35)(5,27,52,36)(6,28,53,37)(7,29,54,38)(8,30,55,39)(9,31,56,40)(10,32,57,41)(11,17,58,42)(12,18,59,43)(13,19,60,44)(14,20,61,45)(15,21,62,46)(16,22,63,47), (1,64)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(30,32)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)>;
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), (1,23,64,48)(2,24,49,33)(3,25,50,34)(4,26,51,35)(5,27,52,36)(6,28,53,37)(7,29,54,38)(8,30,55,39)(9,31,56,40)(10,32,57,41)(11,17,58,42)(12,18,59,43)(13,19,60,44)(14,20,61,45)(15,21,62,46)(16,22,63,47), (1,64)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(30,32)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41) );
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)], [(1,23,64,48),(2,24,49,33),(3,25,50,34),(4,26,51,35),(5,27,52,36),(6,28,53,37),(7,29,54,38),(8,30,55,39),(9,31,56,40),(10,32,57,41),(11,17,58,42),(12,18,59,43),(13,19,60,44),(14,20,61,45),(15,21,62,46),(16,22,63,47)], [(1,64),(2,63),(3,62),(4,61),(5,60),(6,59),(7,58),(8,57),(9,56),(10,55),(11,54),(12,53),(13,52),(14,51),(15,50),(16,49),(17,29),(18,28),(19,27),(20,26),(21,25),(22,24),(30,32),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41)])
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 | C16⋊4D4 | 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 C16⋊4D4 ►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] >;
C16⋊4D4 in GAP, Magma, Sage, TeX
C_{16}\rtimes_4D_4 % in TeX
G:=Group("C16:4D4"); // 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