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 51 36 77 22)(2 52 37 78 23)(3 53 38 79 24)(4 54 39 80 25)(5 55 40 65 26)(6 56 41 66 27)(7 57 42 67 28)(8 58 43 68 29)(9 59 44 69 30)(10 60 45 70 31)(11 61 46 71 32)(12 62 47 72 17)(13 63 48 73 18)(14 64 33 74 19)(15 49 34 75 20)(16 50 35 76 21)
(2 14 10 6)(3 11)(4 8 12 16)(7 15)(17 76 54 43)(18 73 63 48)(19 70 56 37)(20 67 49 42)(21 80 58 47)(22 77 51 36)(23 74 60 41)(24 71 53 46)(25 68 62 35)(26 65 55 40)(27 78 64 45)(28 75 57 34)(29 72 50 39)(30 69 59 44)(31 66 52 33)(32 79 61 38)
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), (1,51,36,77,22)(2,52,37,78,23)(3,53,38,79,24)(4,54,39,80,25)(5,55,40,65,26)(6,56,41,66,27)(7,57,42,67,28)(8,58,43,68,29)(9,59,44,69,30)(10,60,45,70,31)(11,61,46,71,32)(12,62,47,72,17)(13,63,48,73,18)(14,64,33,74,19)(15,49,34,75,20)(16,50,35,76,21), (2,14,10,6)(3,11)(4,8,12,16)(7,15)(17,76,54,43)(18,73,63,48)(19,70,56,37)(20,67,49,42)(21,80,58,47)(22,77,51,36)(23,74,60,41)(24,71,53,46)(25,68,62,35)(26,65,55,40)(27,78,64,45)(28,75,57,34)(29,72,50,39)(30,69,59,44)(31,66,52,33)(32,79,61,38)>;
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), (1,51,36,77,22)(2,52,37,78,23)(3,53,38,79,24)(4,54,39,80,25)(5,55,40,65,26)(6,56,41,66,27)(7,57,42,67,28)(8,58,43,68,29)(9,59,44,69,30)(10,60,45,70,31)(11,61,46,71,32)(12,62,47,72,17)(13,63,48,73,18)(14,64,33,74,19)(15,49,34,75,20)(16,50,35,76,21), (2,14,10,6)(3,11)(4,8,12,16)(7,15)(17,76,54,43)(18,73,63,48)(19,70,56,37)(20,67,49,42)(21,80,58,47)(22,77,51,36)(23,74,60,41)(24,71,53,46)(25,68,62,35)(26,65,55,40)(27,78,64,45)(28,75,57,34)(29,72,50,39)(30,69,59,44)(31,66,52,33)(32,79,61,38) );
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)], [(1,51,36,77,22),(2,52,37,78,23),(3,53,38,79,24),(4,54,39,80,25),(5,55,40,65,26),(6,56,41,66,27),(7,57,42,67,28),(8,58,43,68,29),(9,59,44,69,30),(10,60,45,70,31),(11,61,46,71,32),(12,62,47,72,17),(13,63,48,73,18),(14,64,33,74,19),(15,49,34,75,20),(16,50,35,76,21)], [(2,14,10,6),(3,11),(4,8,12,16),(7,15),(17,76,54,43),(18,73,63,48),(19,70,56,37),(20,67,49,42),(21,80,58,47),(22,77,51,36),(23,74,60,41),(24,71,53,46),(25,68,62,35),(26,65,55,40),(27,78,64,45),(28,75,57,34),(29,72,50,39),(30,69,59,44),(31,66,52,33),(32,79,61,38)])
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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