metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q16⋊1F5, D5.2Q32, Dic20⋊4C4, D10.20D8, D5.3SD32, Dic5.5SD16, C5⋊(C2.Q32), (C5×Q16)⋊4C4, C8.12(C2×F5), C40.10(C2×C4), D5⋊C16.1C2, (C4×D5).25D4, C5⋊2C8.16D4, D5.D8.1C2, (D5×Q16).4C2, C4.6(C22⋊F5), C20.6(C22⋊C4), (C8×D5).19C22, C2.11(D20⋊C4), C10.10(D4⋊C4), SmallGroup(320,246)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5.Q32
G = < a,b,c,d | a5=b2=c16=1, d2=c8, bab=a-1, cac-1=a3, ad=da, cbc-1=a2b, bd=db, dcd-1=a-1bc-1 >
Subgroups: 298 in 58 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, Q8, D5, C10, C16, C4⋊C4, C2×C8, Q16, Q16, C2×Q8, Dic5, Dic5, C20, C20, F5, D10, C2.D8, C2×C16, C2×Q16, C5⋊2C8, C40, Dic10, C4×D5, C4×D5, C5×Q8, C2×F5, C2.Q32, C5⋊C16, C8×D5, Dic20, C5⋊Q16, C5×Q16, C4⋊F5, Q8×D5, D5⋊C16, D5.D8, D5×Q16, D5.Q32
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, D8, SD16, F5, D4⋊C4, SD32, Q32, C2×F5, C2.Q32, C22⋊F5, D20⋊C4, D5.Q32
Character table of D5.Q32
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | 8B | 8C | 8D | 10 | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20A | 20B | 20C | 40A | 40B | |
| size | 1 | 1 | 5 | 5 | 2 | 8 | 10 | 40 | 40 | 40 | 4 | 2 | 2 | 10 | 10 | 4 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 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 | -i | 1 | i | 1 | 1 | 1 | -1 | -1 | 1 | i | -i | -i | -i | -i | i | i | i | 1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | -1 | i | 1 | 1 | 1 | -1 | -1 | 1 | -i | i | i | i | i | -i | -i | -i | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | i | 1 | -i | 1 | 1 | 1 | -1 | -1 | 1 | -i | i | i | i | i | -i | -i | -i | 1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | i | -1 | -i | 1 | 1 | 1 | -1 | -1 | 1 | i | -i | -i | -i | -i | i | i | i | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -√2 | √2 | √2 | -√2 | -√2 | √2 | √2 | -√2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ12 | 2 | 2 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | √2 | -√2 | -√2 | √2 | √2 | -√2 | -√2 | √2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | √2 | -√2 | -√2 | √2 | -2 | ζ167-ζ16 | -ζ165+ζ163 | ζ165-ζ163 | ζ167-ζ16 | -ζ167+ζ16 | ζ165-ζ163 | -ζ165+ζ163 | -ζ167+ζ16 | 0 | 0 | 0 | √2 | -√2 | symplectic lifted from Q32, Schur index 2 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | √2 | -√2 | -√2 | √2 | -2 | -ζ167+ζ16 | ζ165-ζ163 | -ζ165+ζ163 | -ζ167+ζ16 | ζ167-ζ16 | -ζ165+ζ163 | ζ165-ζ163 | ζ167-ζ16 | 0 | 0 | 0 | √2 | -√2 | symplectic lifted from Q32, Schur index 2 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -√2 | √2 | √2 | -√2 | -2 | ζ165-ζ163 | ζ167-ζ16 | -ζ167+ζ16 | ζ165-ζ163 | -ζ165+ζ163 | -ζ167+ζ16 | ζ167-ζ16 | -ζ165+ζ163 | 0 | 0 | 0 | -√2 | √2 | symplectic lifted from Q32, Schur index 2 |
| ρ16 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -√2 | √2 | √2 | -√2 | -2 | -ζ165+ζ163 | -ζ167+ζ16 | ζ167-ζ16 | -ζ165+ζ163 | ζ165-ζ163 | ζ167-ζ16 | -ζ167+ζ16 | ζ165-ζ163 | 0 | 0 | 0 | -√2 | √2 | symplectic lifted from Q32, Schur index 2 |
| ρ17 | 2 | 2 | -2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | √-2 | √-2 | √-2 | -√-2 | -√-2 | -√-2 | -√-2 | √-2 | -2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -√2 | √2 | -√2 | √2 | -2 | ζ167+ζ16 | ζ1613+ζ1611 | ζ165+ζ163 | ζ1615+ζ169 | ζ167+ζ16 | ζ1613+ζ1611 | ζ165+ζ163 | ζ1615+ζ169 | 0 | 0 | 0 | -√2 | √2 | complex lifted from SD32 |
| ρ19 | 2 | 2 | -2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -√-2 | -√-2 | -√-2 | √-2 | √-2 | √-2 | √-2 | -√-2 | -2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | √2 | -√2 | √2 | -√2 | -2 | ζ165+ζ163 | ζ167+ζ16 | ζ1615+ζ169 | ζ1613+ζ1611 | ζ165+ζ163 | ζ167+ζ16 | ζ1615+ζ169 | ζ1613+ζ1611 | 0 | 0 | 0 | √2 | -√2 | complex lifted from SD32 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -√2 | √2 | -√2 | √2 | -2 | ζ1615+ζ169 | ζ165+ζ163 | ζ1613+ζ1611 | ζ167+ζ16 | ζ1615+ζ169 | ζ165+ζ163 | ζ1613+ζ1611 | ζ167+ζ16 | 0 | 0 | 0 | -√2 | √2 | complex lifted from SD32 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | √2 | -√2 | √2 | -√2 | -2 | ζ1613+ζ1611 | ζ1615+ζ169 | ζ167+ζ16 | ζ165+ζ163 | ζ1613+ζ1611 | ζ1615+ζ169 | ζ167+ζ16 | ζ165+ζ163 | 0 | 0 | 0 | √2 | -√2 | complex lifted from SD32 |
| ρ23 | 4 | 4 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | -1 | 4 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | -1 | orthogonal lifted from C2×F5 |
| ρ24 | 4 | 4 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | 4 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ25 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | -4 | -4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | √5 | -√5 | 1 | 1 | orthogonal lifted from C22⋊F5 |
| ρ26 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | -4 | -4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -√5 | √5 | 1 | 1 | orthogonal lifted from C22⋊F5 |
| ρ27 | 8 | 8 | 0 | 0 | -8 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D20⋊C4, Schur index 2 |
| ρ28 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -4√2 | 4√2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | symplectic faithful, Schur index 2 |
| ρ29 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 4√2 | -4√2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | symplectic faithful, Schur index 2 |
►On 80 points
Generators in S80
(1 61 69 42 17)(2 43 62 18 70)(3 19 44 71 63)(4 72 20 64 45)(5 49 73 46 21)(6 47 50 22 74)(7 23 48 75 51)(8 76 24 52 33)(9 53 77 34 25)(10 35 54 26 78)(11 27 36 79 55)(12 80 28 56 37)(13 57 65 38 29)(14 39 58 30 66)(15 31 40 67 59)(16 68 32 60 41)
(1 25)(2 78)(3 55)(4 37)(5 29)(6 66)(7 59)(8 41)(9 17)(10 70)(11 63)(12 45)(13 21)(14 74)(15 51)(16 33)(18 35)(19 79)(20 28)(22 39)(23 67)(24 32)(26 43)(27 71)(30 47)(31 75)(34 61)(36 44)(38 49)(40 48)(42 53)(46 57)(50 58)(52 68)(54 62)(56 72)(60 76)(64 80)(65 73)(69 77)
(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 2 9 10)(3 16 11 8)(4 7 12 15)(5 14 13 6)(17 70 25 78)(18 34 26 42)(19 68 27 76)(20 48 28 40)(21 66 29 74)(22 46 30 38)(23 80 31 72)(24 44 32 36)(33 63 41 55)(35 61 43 53)(37 59 45 51)(39 57 47 49)(50 73 58 65)(52 71 60 79)(54 69 62 77)(56 67 64 75)
G:=sub<Sym(80)| (1,61,69,42,17)(2,43,62,18,70)(3,19,44,71,63)(4,72,20,64,45)(5,49,73,46,21)(6,47,50,22,74)(7,23,48,75,51)(8,76,24,52,33)(9,53,77,34,25)(10,35,54,26,78)(11,27,36,79,55)(12,80,28,56,37)(13,57,65,38,29)(14,39,58,30,66)(15,31,40,67,59)(16,68,32,60,41), (1,25)(2,78)(3,55)(4,37)(5,29)(6,66)(7,59)(8,41)(9,17)(10,70)(11,63)(12,45)(13,21)(14,74)(15,51)(16,33)(18,35)(19,79)(20,28)(22,39)(23,67)(24,32)(26,43)(27,71)(30,47)(31,75)(34,61)(36,44)(38,49)(40,48)(42,53)(46,57)(50,58)(52,68)(54,62)(56,72)(60,76)(64,80)(65,73)(69,77), (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), (1,2,9,10)(3,16,11,8)(4,7,12,15)(5,14,13,6)(17,70,25,78)(18,34,26,42)(19,68,27,76)(20,48,28,40)(21,66,29,74)(22,46,30,38)(23,80,31,72)(24,44,32,36)(33,63,41,55)(35,61,43,53)(37,59,45,51)(39,57,47,49)(50,73,58,65)(52,71,60,79)(54,69,62,77)(56,67,64,75)>;
G:=Group( (1,61,69,42,17)(2,43,62,18,70)(3,19,44,71,63)(4,72,20,64,45)(5,49,73,46,21)(6,47,50,22,74)(7,23,48,75,51)(8,76,24,52,33)(9,53,77,34,25)(10,35,54,26,78)(11,27,36,79,55)(12,80,28,56,37)(13,57,65,38,29)(14,39,58,30,66)(15,31,40,67,59)(16,68,32,60,41), (1,25)(2,78)(3,55)(4,37)(5,29)(6,66)(7,59)(8,41)(9,17)(10,70)(11,63)(12,45)(13,21)(14,74)(15,51)(16,33)(18,35)(19,79)(20,28)(22,39)(23,67)(24,32)(26,43)(27,71)(30,47)(31,75)(34,61)(36,44)(38,49)(40,48)(42,53)(46,57)(50,58)(52,68)(54,62)(56,72)(60,76)(64,80)(65,73)(69,77), (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), (1,2,9,10)(3,16,11,8)(4,7,12,15)(5,14,13,6)(17,70,25,78)(18,34,26,42)(19,68,27,76)(20,48,28,40)(21,66,29,74)(22,46,30,38)(23,80,31,72)(24,44,32,36)(33,63,41,55)(35,61,43,53)(37,59,45,51)(39,57,47,49)(50,73,58,65)(52,71,60,79)(54,69,62,77)(56,67,64,75) );
G=PermutationGroup([[(1,61,69,42,17),(2,43,62,18,70),(3,19,44,71,63),(4,72,20,64,45),(5,49,73,46,21),(6,47,50,22,74),(7,23,48,75,51),(8,76,24,52,33),(9,53,77,34,25),(10,35,54,26,78),(11,27,36,79,55),(12,80,28,56,37),(13,57,65,38,29),(14,39,58,30,66),(15,31,40,67,59),(16,68,32,60,41)], [(1,25),(2,78),(3,55),(4,37),(5,29),(6,66),(7,59),(8,41),(9,17),(10,70),(11,63),(12,45),(13,21),(14,74),(15,51),(16,33),(18,35),(19,79),(20,28),(22,39),(23,67),(24,32),(26,43),(27,71),(30,47),(31,75),(34,61),(36,44),(38,49),(40,48),(42,53),(46,57),(50,58),(52,68),(54,62),(56,72),(60,76),(64,80),(65,73),(69,77)], [(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)], [(1,2,9,10),(3,16,11,8),(4,7,12,15),(5,14,13,6),(17,70,25,78),(18,34,26,42),(19,68,27,76),(20,48,28,40),(21,66,29,74),(22,46,30,38),(23,80,31,72),(24,44,32,36),(33,63,41,55),(35,61,43,53),(37,59,45,51),(39,57,47,49),(50,73,58,65),(52,71,60,79),(54,69,62,77),(56,67,64,75)]])
Matrix representation of D5.Q32 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 | 1 | 0 |
| 0 | 0 | 240 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 | 1 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 |
| 85 | 214 | 0 | 0 | 0 | 0 |
| 27 | 85 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 138 | 41 | 0 | 0 | 0 | 0 |
| 41 | 103 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,240,240,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,240,240,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0],[85,27,0,0,0,0,214,85,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0],[138,41,0,0,0,0,41,103,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
D5.Q32 in GAP, Magma, Sage, TeX
D_5.Q_{32} % in TeX
G:=Group("D5.Q32"); // GroupNames label
G:=SmallGroup(320,246);
// by ID
G=gap.SmallGroup(320,246);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,675,346,192,1684,851,102,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^16=1,d^2=c^8,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=a^-1*b*c^-1>;
// generators/relations
Export