metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C80⋊2C2, C16⋊2D5, C5⋊1SD32, C10.2D8, C2.4D40, C4.2D20, D40.1C2, C20.25D4, C8.14D10, Dic20⋊1C2, C40.15C22, SmallGroup(160,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C16⋊D5
G = < a,b,c | a16=b5=c2=1, ab=ba, cac=a7, cbc=b-1 >
Smallest permutation representation of C16⋊D5
►On 80 points
Generators in S80
(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)(65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
(1 55 66 21 47)(2 56 67 22 48)(3 57 68 23 33)(4 58 69 24 34)(5 59 70 25 35)(6 60 71 26 36)(7 61 72 27 37)(8 62 73 28 38)(9 63 74 29 39)(10 64 75 30 40)(11 49 76 31 41)(12 50 77 32 42)(13 51 78 17 43)(14 52 79 18 44)(15 53 80 19 45)(16 54 65 20 46)
(1 47)(2 38)(3 45)(4 36)(5 43)(6 34)(7 41)(8 48)(9 39)(10 46)(11 37)(12 44)(13 35)(14 42)(15 33)(16 40)(17 59)(18 50)(19 57)(20 64)(21 55)(22 62)(23 53)(24 60)(25 51)(26 58)(27 49)(28 56)(29 63)(30 54)(31 61)(32 52)(65 75)(67 73)(68 80)(69 71)(70 78)(72 76)(77 79)
G:=sub<Sym(80)| (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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,55,66,21,47)(2,56,67,22,48)(3,57,68,23,33)(4,58,69,24,34)(5,59,70,25,35)(6,60,71,26,36)(7,61,72,27,37)(8,62,73,28,38)(9,63,74,29,39)(10,64,75,30,40)(11,49,76,31,41)(12,50,77,32,42)(13,51,78,17,43)(14,52,79,18,44)(15,53,80,19,45)(16,54,65,20,46), (1,47)(2,38)(3,45)(4,36)(5,43)(6,34)(7,41)(8,48)(9,39)(10,46)(11,37)(12,44)(13,35)(14,42)(15,33)(16,40)(17,59)(18,50)(19,57)(20,64)(21,55)(22,62)(23,53)(24,60)(25,51)(26,58)(27,49)(28,56)(29,63)(30,54)(31,61)(32,52)(65,75)(67,73)(68,80)(69,71)(70,78)(72,76)(77,79)>;
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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,55,66,21,47)(2,56,67,22,48)(3,57,68,23,33)(4,58,69,24,34)(5,59,70,25,35)(6,60,71,26,36)(7,61,72,27,37)(8,62,73,28,38)(9,63,74,29,39)(10,64,75,30,40)(11,49,76,31,41)(12,50,77,32,42)(13,51,78,17,43)(14,52,79,18,44)(15,53,80,19,45)(16,54,65,20,46), (1,47)(2,38)(3,45)(4,36)(5,43)(6,34)(7,41)(8,48)(9,39)(10,46)(11,37)(12,44)(13,35)(14,42)(15,33)(16,40)(17,59)(18,50)(19,57)(20,64)(21,55)(22,62)(23,53)(24,60)(25,51)(26,58)(27,49)(28,56)(29,63)(30,54)(31,61)(32,52)(65,75)(67,73)(68,80)(69,71)(70,78)(72,76)(77,79) );
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),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,55,66,21,47),(2,56,67,22,48),(3,57,68,23,33),(4,58,69,24,34),(5,59,70,25,35),(6,60,71,26,36),(7,61,72,27,37),(8,62,73,28,38),(9,63,74,29,39),(10,64,75,30,40),(11,49,76,31,41),(12,50,77,32,42),(13,51,78,17,43),(14,52,79,18,44),(15,53,80,19,45),(16,54,65,20,46)], [(1,47),(2,38),(3,45),(4,36),(5,43),(6,34),(7,41),(8,48),(9,39),(10,46),(11,37),(12,44),(13,35),(14,42),(15,33),(16,40),(17,59),(18,50),(19,57),(20,64),(21,55),(22,62),(23,53),(24,60),(25,51),(26,58),(27,49),(28,56),(29,63),(30,54),(31,61),(32,52),(65,75),(67,73),(68,80),(69,71),(70,78),(72,76),(77,79)]])
C16⋊D5 is a maximal subgroup of
D80⋊7C2 D80⋊C2 C16.D10 D16⋊D5 D5×SD32 SD32⋊3D5 Q32⋊D5 D40.S3 C24.D10 C48⋊D5
C16⋊D5 is a maximal quotient of
C40.78D4 C80⋊14C4 D40⋊7C4 D40.S3 C24.D10 C48⋊D5
43 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 5A | 5B | 8A | 8B | 10A | 10B | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 40A | ··· | 40H | 80A | ··· | 80P |
| order | 1 | 2 | 2 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 40 | 2 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
43 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | D5 | D8 | D10 | SD32 | D20 | D40 | C16⋊D5 |
| kernel | C16⋊D5 | C80 | D40 | Dic20 | C20 | C16 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 16 |
Matrix representation of C16⋊D5 ►in GL2(𝔽241) generated by
| 6 | 147 |
| 162 | 153 |
| 240 | 191 |
| 240 | 190 |
| 240 | 0 |
| 240 | 1 |
G:=sub<GL(2,GF(241))| [6,162,147,153],[240,240,191,190],[240,240,0,1] >;
C16⋊D5 in GAP, Magma, Sage, TeX
C_{16}\rtimes D_5 % in TeX
G:=Group("C16:D5"); // GroupNames label
G:=SmallGroup(160,7);
// by ID
G=gap.SmallGroup(160,7);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,73,79,506,50,579,69,4613]);
// Polycyclic
G:=Group<a,b,c|a^16=b^5=c^2=1,a*b=b*a,c*a*c=a^7,c*b*c=b^-1>;
// generators/relations
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