direct product, p-group, metabelian, nilpotent (class 4), monomial
Aliases: C22×D16, C16⋊2C23, D8⋊1C23, C8.9C24, C23.63D8, C4.21(C2×D8), (C2×C4).94D8, C8.54(C2×D4), (C22×C16)⋊9C2, (C2×C8).262D4, (C2×C16)⋊18C22, (C2×D8)⋊45C22, (C22×D8)⋊14C2, C2.24(C22×D8), C4.15(C22×D4), C22.75(C2×D8), (C2×C8).571C23, (C22×C4).621D4, (C22×C8).541C22, (C2×C4).872(C2×D4), 2-Sylow(GO-(4,17)), SmallGroup(128,2140)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22×D16
G = < a,b,c,d | a2=b2=c16=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 660 in 220 conjugacy classes, 100 normal (9 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, D4, C23, C23, C16, C2×C8, D8, D8, C22×C4, C2×D4, C24, C2×C16, D16, C22×C8, C2×D8, C2×D8, C22×D4, C22×C16, C2×D16, C22×D8, C22×D16
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, D16, C2×D8, C22×D4, C2×D16, C22×D8, C22×D16
►On 64 points
Generators in S64
(1 44)(2 45)(3 46)(4 47)(5 48)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 61)(18 62)(19 63)(20 64)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)
(1 63)(2 64)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 33)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(31 40)(32 41)
(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 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(31 48)(32 47)
G:=sub<Sym(64)| (1,44)(2,45)(3,46)(4,47)(5,48)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,61)(18,62)(19,63)(20,64)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60), (1,63)(2,64)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,41), (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)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,48)(32,47)>;
G:=Group( (1,44)(2,45)(3,46)(4,47)(5,48)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,61)(18,62)(19,63)(20,64)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60), (1,63)(2,64)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,41), (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)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,48)(32,47) );
G=PermutationGroup([[(1,44),(2,45),(3,46),(4,47),(5,48),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,61),(18,62),(19,63),(20,64),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60)], [(1,63),(2,64),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,55),(10,56),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,33),(25,34),(26,35),(27,36),(28,37),(29,38),(30,39),(31,40),(32,41)], [(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),(17,46),(18,45),(19,44),(20,43),(21,42),(22,41),(23,40),(24,39),(25,38),(26,37),(27,36),(28,35),(29,34),(30,33),(31,48),(32,47)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | 4B | 4C | 4D | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | D4 | D4 | D8 | D8 | D16 |
| kernel | C22×D16 | C22×C16 | C2×D16 | C22×D8 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 12 | 2 | 3 | 1 | 6 | 2 | 16 |
Matrix representation of C22×D16 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 13 | 11 |
| 0 | 0 | 6 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,16,0,0,0,0,13,6,0,0,11,13],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1] >;
C22×D16 in GAP, Magma, Sage, TeX
C_2^2\times D_{16} % in TeX
G:=Group("C2^2xD16"); // GroupNames label
G:=SmallGroup(128,2140);
// by ID
G=gap.SmallGroup(128,2140);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,-2,-2,253,1684,851,242,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^16=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations