metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C80⋊8C4, C16⋊7F5, D5.M5(2), C20.14C42, C5⋊C8.C8, C5⋊C16⋊5C4, (C2×F5).C8, C2.3(C8×F5), C10.2(C4×C8), D5⋊C8.5C4, C5⋊2C16⋊16C4, C5⋊1(C16⋊5C4), (C4×F5).4C4, (C8×F5).2C2, C4.14(C4×F5), C8.38(C2×F5), C40.33(C2×C4), D5⋊C16.2C2, D10.5(C2×C8), (D5×C16).10C2, Dic5.8(C2×C8), (C8×D5).60C22, C5⋊2C8.33(C2×C4), (C4×D5).68(C2×C4), SmallGroup(320,182)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C16⋊7F5
G = < a,b,c | a16=b5=c4=1, ab=ba, cac-1=a9, cbc-1=b3 >
Smallest permutation representation of C16⋊7F5
►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 38 75 24 63)(2 39 76 25 64)(3 40 77 26 49)(4 41 78 27 50)(5 42 79 28 51)(6 43 80 29 52)(7 44 65 30 53)(8 45 66 31 54)(9 46 67 32 55)(10 47 68 17 56)(11 48 69 18 57)(12 33 70 19 58)(13 34 71 20 59)(14 35 72 21 60)(15 36 73 22 61)(16 37 74 23 62)
(2 10)(4 12)(6 14)(8 16)(17 39 68 64)(18 48 69 57)(19 41 70 50)(20 34 71 59)(21 43 72 52)(22 36 73 61)(23 45 74 54)(24 38 75 63)(25 47 76 56)(26 40 77 49)(27 33 78 58)(28 42 79 51)(29 35 80 60)(30 44 65 53)(31 37 66 62)(32 46 67 55)
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), (1,38,75,24,63)(2,39,76,25,64)(3,40,77,26,49)(4,41,78,27,50)(5,42,79,28,51)(6,43,80,29,52)(7,44,65,30,53)(8,45,66,31,54)(9,46,67,32,55)(10,47,68,17,56)(11,48,69,18,57)(12,33,70,19,58)(13,34,71,20,59)(14,35,72,21,60)(15,36,73,22,61)(16,37,74,23,62), (2,10)(4,12)(6,14)(8,16)(17,39,68,64)(18,48,69,57)(19,41,70,50)(20,34,71,59)(21,43,72,52)(22,36,73,61)(23,45,74,54)(24,38,75,63)(25,47,76,56)(26,40,77,49)(27,33,78,58)(28,42,79,51)(29,35,80,60)(30,44,65,53)(31,37,66,62)(32,46,67,55)>;
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), (1,38,75,24,63)(2,39,76,25,64)(3,40,77,26,49)(4,41,78,27,50)(5,42,79,28,51)(6,43,80,29,52)(7,44,65,30,53)(8,45,66,31,54)(9,46,67,32,55)(10,47,68,17,56)(11,48,69,18,57)(12,33,70,19,58)(13,34,71,20,59)(14,35,72,21,60)(15,36,73,22,61)(16,37,74,23,62), (2,10)(4,12)(6,14)(8,16)(17,39,68,64)(18,48,69,57)(19,41,70,50)(20,34,71,59)(21,43,72,52)(22,36,73,61)(23,45,74,54)(24,38,75,63)(25,47,76,56)(26,40,77,49)(27,33,78,58)(28,42,79,51)(29,35,80,60)(30,44,65,53)(31,37,66,62)(32,46,67,55) );
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)], [(1,38,75,24,63),(2,39,76,25,64),(3,40,77,26,49),(4,41,78,27,50),(5,42,79,28,51),(6,43,80,29,52),(7,44,65,30,53),(8,45,66,31,54),(9,46,67,32,55),(10,47,68,17,56),(11,48,69,18,57),(12,33,70,19,58),(13,34,71,20,59),(14,35,72,21,60),(15,36,73,22,61),(16,37,74,23,62)], [(2,10),(4,12),(6,14),(8,16),(17,39,68,64),(18,48,69,57),(19,41,70,50),(20,34,71,59),(21,43,72,52),(22,36,73,61),(23,45,74,54),(24,38,75,63),(25,47,76,56),(26,40,77,49),(27,33,78,58),(28,42,79,51),(29,35,80,60),(30,44,65,53),(31,37,66,62),(32,46,67,55)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 10 | 16A | 16B | 16C | 16D | 16E | ··· | 16P | 20A | 20B | 40A | 40B | 40C | 40D | 80A | ··· | 80H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 16 | 16 | 16 | 16 | 16 | ··· | 16 | 20 | 20 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 5 | 5 | 10 | 10 | 10 | 10 | 4 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 4 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C4 | C8 | C8 | M5(2) | F5 | C2×F5 | C4×F5 | C8×F5 | C16⋊7F5 |
| kernel | C16⋊7F5 | D5×C16 | D5⋊C16 | C8×F5 | C5⋊2C16 | C80 | C5⋊C16 | D5⋊C8 | C4×F5 | C5⋊C8 | C2×F5 | D5 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 8 | 8 | 8 | 1 | 1 | 2 | 4 | 8 |
Matrix representation of C16⋊7F5 ►in GL4(𝔽241) generated by
| 154 | 174 | 0 | 67 |
| 0 | 87 | 174 | 67 |
| 67 | 174 | 87 | 0 |
| 67 | 0 | 174 | 154 |
| 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 240 |
| 0 | 1 | 0 | 240 |
| 0 | 0 | 1 | 240 |
| 1 | 0 | 240 | 0 |
| 0 | 0 | 240 | 1 |
| 0 | 1 | 240 | 0 |
| 0 | 0 | 240 | 0 |
G:=sub<GL(4,GF(241))| [154,0,67,67,174,87,174,0,0,174,87,174,67,67,0,154],[0,1,0,0,0,0,1,0,0,0,0,1,240,240,240,240],[1,0,0,0,0,0,1,0,240,240,240,240,0,1,0,0] >;
C16⋊7F5 in GAP, Magma, Sage, TeX
C_{16}\rtimes_7F_5 % in TeX
G:=Group("C16:7F5"); // GroupNames label
G:=SmallGroup(320,182);
// by ID
G=gap.SmallGroup(320,182);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,477,64,80,102,6278,3156]);
// Polycyclic
G:=Group<a,b,c|a^16=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^9,c*b*c^-1=b^3>;
// generators/relations
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