metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C16⋊4F5, C80⋊10C4, C20.16C42, D10.2M4(2), Dic5.2M4(2), C5⋊C16⋊2C4, C5⋊2C16⋊8C4, D5⋊C8.2C4, C5⋊2(C16⋊C4), (C4×F5).2C4, C4.25(C4×F5), C8.31(C2×F5), C40.35(C2×C4), C8⋊F5.3C2, C80⋊C2.4C2, C2.5(C8⋊F5), C10.2(C8⋊C4), C8.F5.3C2, (C8×D5).34C22, C5⋊2C8.17(C2×C4), (C4×D5).41(C2×C4), SmallGroup(320,184)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C16⋊4F5
G = < a,b,c | a16=b5=c4=1, ab=ba, cac-1=a5, cbc-1=b3 >
Smallest permutation representation of C16⋊4F5
►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 61 38 23 74)(2 62 39 24 75)(3 63 40 25 76)(4 64 41 26 77)(5 49 42 27 78)(6 50 43 28 79)(7 51 44 29 80)(8 52 45 30 65)(9 53 46 31 66)(10 54 47 32 67)(11 55 48 17 68)(12 56 33 18 69)(13 57 34 19 70)(14 58 35 20 71)(15 59 36 21 72)(16 60 37 22 73)
(2 14 10 6)(3 11)(4 8 12 16)(7 15)(17 63 48 76)(18 60 41 65)(19 57 34 70)(20 54 43 75)(21 51 36 80)(22 64 45 69)(23 61 38 74)(24 58 47 79)(25 55 40 68)(26 52 33 73)(27 49 42 78)(28 62 35 67)(29 59 44 72)(30 56 37 77)(31 53 46 66)(32 50 39 71)
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), (1,61,38,23,74)(2,62,39,24,75)(3,63,40,25,76)(4,64,41,26,77)(5,49,42,27,78)(6,50,43,28,79)(7,51,44,29,80)(8,52,45,30,65)(9,53,46,31,66)(10,54,47,32,67)(11,55,48,17,68)(12,56,33,18,69)(13,57,34,19,70)(14,58,35,20,71)(15,59,36,21,72)(16,60,37,22,73), (2,14,10,6)(3,11)(4,8,12,16)(7,15)(17,63,48,76)(18,60,41,65)(19,57,34,70)(20,54,43,75)(21,51,36,80)(22,64,45,69)(23,61,38,74)(24,58,47,79)(25,55,40,68)(26,52,33,73)(27,49,42,78)(28,62,35,67)(29,59,44,72)(30,56,37,77)(31,53,46,66)(32,50,39,71)>;
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), (1,61,38,23,74)(2,62,39,24,75)(3,63,40,25,76)(4,64,41,26,77)(5,49,42,27,78)(6,50,43,28,79)(7,51,44,29,80)(8,52,45,30,65)(9,53,46,31,66)(10,54,47,32,67)(11,55,48,17,68)(12,56,33,18,69)(13,57,34,19,70)(14,58,35,20,71)(15,59,36,21,72)(16,60,37,22,73), (2,14,10,6)(3,11)(4,8,12,16)(7,15)(17,63,48,76)(18,60,41,65)(19,57,34,70)(20,54,43,75)(21,51,36,80)(22,64,45,69)(23,61,38,74)(24,58,47,79)(25,55,40,68)(26,52,33,73)(27,49,42,78)(28,62,35,67)(29,59,44,72)(30,56,37,77)(31,53,46,66)(32,50,39,71) );
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)], [(1,61,38,23,74),(2,62,39,24,75),(3,63,40,25,76),(4,64,41,26,77),(5,49,42,27,78),(6,50,43,28,79),(7,51,44,29,80),(8,52,45,30,65),(9,53,46,31,66),(10,54,47,32,67),(11,55,48,17,68),(12,56,33,18,69),(13,57,34,19,70),(14,58,35,20,71),(15,59,36,21,72),(16,60,37,22,73)], [(2,14,10,6),(3,11),(4,8,12,16),(7,15),(17,63,48,76),(18,60,41,65),(19,57,34,70),(20,54,43,75),(21,51,36,80),(22,64,45,69),(23,61,38,74),(24,58,47,79),(25,55,40,68),(26,52,33,73),(27,49,42,78),(28,62,35,67),(29,59,44,72),(30,56,37,77),(31,53,46,66),(32,50,39,71)]])
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⋊4F5 |
| kernel | C16⋊4F5 | 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⋊4F5 ►in GL4(𝔽241) generated by
| 120 | 240 | 88 | 152 |
| 89 | 209 | 88 | 177 |
| 64 | 153 | 32 | 152 |
| 89 | 153 | 1 | 121 |
| 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))| [120,89,64,89,240,209,153,153,88,88,32,1,152,177,152,121],[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⋊4F5 in GAP, Magma, Sage, TeX
C_{16}\rtimes_4F_5 % in TeX
G:=Group("C16:4F5"); // GroupNames label
G:=SmallGroup(320,184);
// by ID
G=gap.SmallGroup(320,184);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,1123,192,102,6278,3156]);
// Polycyclic
G:=Group<a,b,c|a^16=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^3>;
// generators/relations
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