metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D40⋊3C4, Q16⋊4F5, C5⋊C8.3D4, Q8⋊D5⋊4C4, C8.9(C2×F5), C40.7(C2×C4), C8⋊F5⋊3C2, (C5×Q16)⋊3C4, Q8.F5⋊4C2, C2.26(D4×F5), Q8.6(C2×F5), C5⋊3(C8.26D4), Q8⋊2F5⋊4C2, D20.6(C2×C4), C10.25(C4×D4), C40.C4⋊3C2, D5⋊C8.6C22, (C4×F5).6C22, C4.12(C22×F5), D10.6(C4○D4), Q8.D10.3C2, C4.F5.6C22, C20.12(C22×C4), Dic5.77(C2×D4), (C8×D5).16C22, (C4×D5).34C23, Q8⋊2D5.7C22, (C5×Q8).6(C2×C4), C5⋊2C8.14(C2×C4), SmallGroup(320,1079)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16⋊F5
G = < a,b,c,d | a8=c5=d4=1, b2=a4, bab-1=a-1, ac=ca, dad-1=a5, bc=cb, dbd-1=a6b, dcd-1=c3 >
Subgroups: 418 in 104 conjugacy classes, 40 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, Q8, D5, C10, C42, C2×C8, M4(2), D8, SD16, Q16, C4○D4, Dic5, C20, C20, F5, D10, D10, C8⋊C4, C4≀C2, C8.C4, C8○D4, C4○D8, C5⋊2C8, C40, C5⋊C8, C5⋊C8, C4×D5, C4×D5, D20, D20, C5×Q8, C2×F5, C8.26D4, C8×D5, D40, Q8⋊D5, C5×Q16, D5⋊C8, D5⋊C8, C4.F5, C4.F5, C4×F5, Q8⋊2D5, C8⋊F5, C40.C4, Q8⋊2F5, Q8.D10, Q8.F5, Q16⋊F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×F5, C8.26D4, C22×F5, D4×F5, Q16⋊F5
Character table of Q16⋊F5
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10 | 20A | 20B | 20C | 40A | 40B | |
| size | 1 | 1 | 10 | 20 | 20 | 2 | 4 | 4 | 5 | 5 | 20 | 20 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 4 | 8 | 16 | 16 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -i | i | 1 | 1 | -i | i | -i | i | -1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ10 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | -i | i | 1 | -1 | i | -i | i | -i | 1 | i | -i | -i | i | 1 | 1 | -1 | 1 | -1 | -1 | linear of order 4 |
| ρ11 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -i | i | 1 | -1 | i | -i | i | -i | 1 | -i | i | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ12 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -i | i | 1 | 1 | -i | i | -i | i | -1 | -i | i | -i | i | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ13 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | i | -i | 1 | -1 | -i | i | -i | i | 1 | -i | i | i | -i | 1 | 1 | -1 | 1 | -1 | -1 | linear of order 4 |
| ρ14 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | i | -i | 1 | 1 | i | -i | i | -i | -1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ15 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | i | -i | 1 | 1 | i | -i | i | -i | -1 | i | -i | i | -i | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ16 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | i | -i | 1 | -1 | -i | i | -i | i | 1 | i | -i | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 4 |
| ρ17 | 2 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 2 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 2 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 4 | 4 | 0 | 0 | 0 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ22 | 4 | 4 | 0 | 0 | 0 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | -1 | orthogonal lifted from C2×F5 |
| ρ23 | 4 | 4 | 0 | 0 | 0 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ24 | 4 | 4 | 0 | 0 | 0 | 4 | -4 | 4 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8.26D4 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8.26D4 |
| ρ27 | 8 | 8 | 0 | 0 | 0 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4×F5 |
| ρ28 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -√10 | √10 | orthogonal faithful |
| ρ29 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | √10 | -√10 | orthogonal faithful |
►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 60 5 64)(2 59 6 63)(3 58 7 62)(4 57 8 61)(9 30 13 26)(10 29 14 25)(11 28 15 32)(12 27 16 31)(17 33 21 37)(18 40 22 36)(19 39 23 35)(20 38 24 34)(41 74 45 78)(42 73 46 77)(43 80 47 76)(44 79 48 75)(49 71 53 67)(50 70 54 66)(51 69 55 65)(52 68 56 72)
(1 10 37 77 66)(2 11 38 78 67)(3 12 39 79 68)(4 13 40 80 69)(5 14 33 73 70)(6 15 34 74 71)(7 16 35 75 72)(8 9 36 76 65)(17 42 50 60 29)(18 43 51 61 30)(19 44 52 62 31)(20 45 53 63 32)(21 46 54 64 25)(22 47 55 57 26)(23 48 56 58 27)(24 41 49 59 28)
(2 6)(4 8)(9 40 65 80)(10 37 66 77)(11 34 67 74)(12 39 68 79)(13 36 69 76)(14 33 70 73)(15 38 71 78)(16 35 72 75)(17 56 46 31)(18 53 47 28)(19 50 48 25)(20 55 41 30)(21 52 42 27)(22 49 43 32)(23 54 44 29)(24 51 45 26)(57 59 61 63)(58 64 62 60)
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), (1,60,5,64)(2,59,6,63)(3,58,7,62)(4,57,8,61)(9,30,13,26)(10,29,14,25)(11,28,15,32)(12,27,16,31)(17,33,21,37)(18,40,22,36)(19,39,23,35)(20,38,24,34)(41,74,45,78)(42,73,46,77)(43,80,47,76)(44,79,48,75)(49,71,53,67)(50,70,54,66)(51,69,55,65)(52,68,56,72), (1,10,37,77,66)(2,11,38,78,67)(3,12,39,79,68)(4,13,40,80,69)(5,14,33,73,70)(6,15,34,74,71)(7,16,35,75,72)(8,9,36,76,65)(17,42,50,60,29)(18,43,51,61,30)(19,44,52,62,31)(20,45,53,63,32)(21,46,54,64,25)(22,47,55,57,26)(23,48,56,58,27)(24,41,49,59,28), (2,6)(4,8)(9,40,65,80)(10,37,66,77)(11,34,67,74)(12,39,68,79)(13,36,69,76)(14,33,70,73)(15,38,71,78)(16,35,72,75)(17,56,46,31)(18,53,47,28)(19,50,48,25)(20,55,41,30)(21,52,42,27)(22,49,43,32)(23,54,44,29)(24,51,45,26)(57,59,61,63)(58,64,62,60)>;
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), (1,60,5,64)(2,59,6,63)(3,58,7,62)(4,57,8,61)(9,30,13,26)(10,29,14,25)(11,28,15,32)(12,27,16,31)(17,33,21,37)(18,40,22,36)(19,39,23,35)(20,38,24,34)(41,74,45,78)(42,73,46,77)(43,80,47,76)(44,79,48,75)(49,71,53,67)(50,70,54,66)(51,69,55,65)(52,68,56,72), (1,10,37,77,66)(2,11,38,78,67)(3,12,39,79,68)(4,13,40,80,69)(5,14,33,73,70)(6,15,34,74,71)(7,16,35,75,72)(8,9,36,76,65)(17,42,50,60,29)(18,43,51,61,30)(19,44,52,62,31)(20,45,53,63,32)(21,46,54,64,25)(22,47,55,57,26)(23,48,56,58,27)(24,41,49,59,28), (2,6)(4,8)(9,40,65,80)(10,37,66,77)(11,34,67,74)(12,39,68,79)(13,36,69,76)(14,33,70,73)(15,38,71,78)(16,35,72,75)(17,56,46,31)(18,53,47,28)(19,50,48,25)(20,55,41,30)(21,52,42,27)(22,49,43,32)(23,54,44,29)(24,51,45,26)(57,59,61,63)(58,64,62,60) );
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)], [(1,60,5,64),(2,59,6,63),(3,58,7,62),(4,57,8,61),(9,30,13,26),(10,29,14,25),(11,28,15,32),(12,27,16,31),(17,33,21,37),(18,40,22,36),(19,39,23,35),(20,38,24,34),(41,74,45,78),(42,73,46,77),(43,80,47,76),(44,79,48,75),(49,71,53,67),(50,70,54,66),(51,69,55,65),(52,68,56,72)], [(1,10,37,77,66),(2,11,38,78,67),(3,12,39,79,68),(4,13,40,80,69),(5,14,33,73,70),(6,15,34,74,71),(7,16,35,75,72),(8,9,36,76,65),(17,42,50,60,29),(18,43,51,61,30),(19,44,52,62,31),(20,45,53,63,32),(21,46,54,64,25),(22,47,55,57,26),(23,48,56,58,27),(24,41,49,59,28)], [(2,6),(4,8),(9,40,65,80),(10,37,66,77),(11,34,67,74),(12,39,68,79),(13,36,69,76),(14,33,70,73),(15,38,71,78),(16,35,72,75),(17,56,46,31),(18,53,47,28),(19,50,48,25),(20,55,41,30),(21,52,42,27),(22,49,43,32),(23,54,44,29),(24,51,45,26),(57,59,61,63),(58,64,62,60)]])
Matrix representation of Q16⋊F5 ►in GL8(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 33 |
| 0 | 0 | 0 | 0 | 0 | 20 | 31 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 21 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 36 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 37 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 33 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,20,4,0,0,0,0,0,0,31,21,0,0,0,0,0,33,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,32,0,0,0,0,0,37,32,0],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,33,32,0,0,0,0,0,0,0,0,40] >;
Q16⋊F5 in GAP, Magma, Sage, TeX
Q_{16}\rtimes F_5 % in TeX
G:=Group("Q16:F5"); // GroupNames label
G:=SmallGroup(320,1079);
// by ID
G=gap.SmallGroup(320,1079);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,758,219,184,136,851,438,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^8=c^5=d^4=1,b^2=a^4,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=c^3>;
// generators/relations
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