metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C80⋊9C4, C16⋊3F5, C20.15C42, D10.1M4(2), Dic5.1M4(2), C5⋊C16⋊1C4, C5⋊2C16⋊7C4, D5⋊C8.1C4, C5⋊1(C16⋊C4), (C4×F5).1C4, C4.24(C4×F5), C8.30(C2×F5), C40.34(C2×C4), C8⋊F5.2C2, C80⋊C2.3C2, C2.4(C8⋊F5), C10.1(C8⋊C4), C8.F5.2C2, (C8×D5).33C22, C5⋊2C8.16(C2×C4), (C4×D5).40(C2×C4), SmallGroup(320,183)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C16⋊F5
G = < a,b,c | a16=b5=c4=1, ab=ba, cac-1=a13, cbc-1=b3 >
Smallest permutation representation of C16⋊F5
►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 27 43 55 70)(2 28 44 56 71)(3 29 45 57 72)(4 30 46 58 73)(5 31 47 59 74)(6 32 48 60 75)(7 17 33 61 76)(8 18 34 62 77)(9 19 35 63 78)(10 20 36 64 79)(11 21 37 49 80)(12 22 38 50 65)(13 23 39 51 66)(14 24 40 52 67)(15 25 41 53 68)(16 26 42 54 69)
(2 6 10 14)(3 11)(4 16 12 8)(7 15)(17 41 76 53)(18 46 69 50)(19 35 78 63)(20 40 71 60)(21 45 80 57)(22 34 73 54)(23 39 66 51)(24 44 75 64)(25 33 68 61)(26 38 77 58)(27 43 70 55)(28 48 79 52)(29 37 72 49)(30 42 65 62)(31 47 74 59)(32 36 67 56)
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), (1,27,43,55,70)(2,28,44,56,71)(3,29,45,57,72)(4,30,46,58,73)(5,31,47,59,74)(6,32,48,60,75)(7,17,33,61,76)(8,18,34,62,77)(9,19,35,63,78)(10,20,36,64,79)(11,21,37,49,80)(12,22,38,50,65)(13,23,39,51,66)(14,24,40,52,67)(15,25,41,53,68)(16,26,42,54,69), (2,6,10,14)(3,11)(4,16,12,8)(7,15)(17,41,76,53)(18,46,69,50)(19,35,78,63)(20,40,71,60)(21,45,80,57)(22,34,73,54)(23,39,66,51)(24,44,75,64)(25,33,68,61)(26,38,77,58)(27,43,70,55)(28,48,79,52)(29,37,72,49)(30,42,65,62)(31,47,74,59)(32,36,67,56)>;
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), (1,27,43,55,70)(2,28,44,56,71)(3,29,45,57,72)(4,30,46,58,73)(5,31,47,59,74)(6,32,48,60,75)(7,17,33,61,76)(8,18,34,62,77)(9,19,35,63,78)(10,20,36,64,79)(11,21,37,49,80)(12,22,38,50,65)(13,23,39,51,66)(14,24,40,52,67)(15,25,41,53,68)(16,26,42,54,69), (2,6,10,14)(3,11)(4,16,12,8)(7,15)(17,41,76,53)(18,46,69,50)(19,35,78,63)(20,40,71,60)(21,45,80,57)(22,34,73,54)(23,39,66,51)(24,44,75,64)(25,33,68,61)(26,38,77,58)(27,43,70,55)(28,48,79,52)(29,37,72,49)(30,42,65,62)(31,47,74,59)(32,36,67,56) );
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)], [(1,27,43,55,70),(2,28,44,56,71),(3,29,45,57,72),(4,30,46,58,73),(5,31,47,59,74),(6,32,48,60,75),(7,17,33,61,76),(8,18,34,62,77),(9,19,35,63,78),(10,20,36,64,79),(11,21,37,49,80),(12,22,38,50,65),(13,23,39,51,66),(14,24,40,52,67),(15,25,41,53,68),(16,26,42,54,69)], [(2,6,10,14),(3,11),(4,16,12,8),(7,15),(17,41,76,53),(18,46,69,50),(19,35,78,63),(20,40,71,60),(21,45,80,57),(22,34,73,54),(23,39,66,51),(24,44,75,64),(25,33,68,61),(26,38,77,58),(27,43,70,55),(28,48,79,52),(29,37,72,49),(30,42,65,62),(31,47,74,59),(32,36,67,56)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 4E | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 10 | 16A | 16B | 16C | ··· | 16H | 20A | 20B | 40A | 40B | 40C | 40D | 80A | ··· | 80H |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 16 | 16 | 16 | ··· | 16 | 20 | 20 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 10 | 1 | 1 | 10 | 20 | 20 | 4 | 2 | 2 | 10 | 10 | 20 | 20 | 4 | 4 | 4 | 20 | ··· | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C4 | M4(2) | M4(2) | F5 | C2×F5 | C16⋊C4 | C4×F5 | C8⋊F5 | C16⋊F5 |
| kernel | C16⋊F5 | C80⋊C2 | C8.F5 | C8⋊F5 | C5⋊2C16 | C80 | C5⋊C16 | D5⋊C8 | C4×F5 | Dic5 | D10 | C16 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 8 |
Matrix representation of C16⋊F5 ►in GL4(𝔽241) generated by
| 237 | 233 | 222 | 11 |
| 230 | 226 | 222 | 211 |
| 30 | 19 | 15 | 11 |
| 230 | 19 | 8 | 4 |
| 240 | 240 | 240 | 240 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 240 | 240 | 240 | 240 |
G:=sub<GL(4,GF(241))| [237,230,30,230,233,226,19,19,222,222,15,8,11,211,11,4],[240,1,0,0,240,0,1,0,240,0,0,1,240,0,0,0],[1,0,0,240,0,0,1,240,0,0,0,240,0,1,0,240] >;
C16⋊F5 in GAP, Magma, Sage, TeX
C_{16}\rtimes F_5 % in TeX
G:=Group("C16:F5"); // GroupNames label
G:=SmallGroup(320,183);
// by ID
G=gap.SmallGroup(320,183);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,1123,80,102,6278,3156]);
// Polycyclic
G:=Group<a,b,c|a^16=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^13,c*b*c^-1=b^3>;
// generators/relations
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