Copied to
clipboard

G = C5⋊D16order 160 = 25·5

The semidirect product of C5 and D16 acting via D16/D8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52D16, D81D5, D403C2, C20.3D4, C10.8D8, C8.4D10, C40.2C22, (C5×D8)⋊1C2, C52C161C2, C2.4(D4⋊D5), C4.1(C5⋊D4), SmallGroup(160,33)

Series: Derived Chief Lower central Upper central

C1C40 — C5⋊D16
C1C5C10C20C40D40 — C5⋊D16
C5C10C20C40 — C5⋊D16
C1C2C4C8D8

Generators and relations for C5⋊D16
 G = < a,b,c | a5=b16=c2=1, bab-1=cac=a-1, cbc=b-1 >

8C2
40C2
4C22
20C22
8D5
8C10
2D4
10D4
4D10
4C2×C10
5C16
5D8
2D20
2C5×D4
5D16

Character table of C5⋊D16

 class 12A2B2C45A5B8A8B10A10B10C10D10E10F16A16B16C16D20A20B40A40B40C40D
 size 118402222222888810101010444444
ρ11111111111111111111111111    trivial
ρ211-111111111-1-1-1-1-1-1-1-1111111    linear of order 2
ρ3111-111111111111-1-1-1-1111111    linear of order 2
ρ411-1-11111111-1-1-1-11111111111    linear of order 2
ρ52200222-2-2220000000022-2-2-2-2    orthogonal lifted from D4
ρ62200-222002200002-2-22-2-20000    orthogonal lifted from D8
ρ722202-1+5/2-1-5/222-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/20000-1+5/2-1-5/2-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D5
ρ82-2000222-2-2-2000016716165163ζ165163ζ1671600-222-2    orthogonal lifted from D16
ρ922-202-1-5/2-1+5/222-1-5/2-1+5/21+5/21-5/21+5/21-5/20000-1-5/2-1+5/2-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D10
ρ102-200022-22-2-20000ζ16516316716ζ16716165163002-2-22    orthogonal lifted from D16
ρ1122202-1-5/2-1+5/222-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/20000-1-5/2-1+5/2-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D5
ρ1222-202-1+5/2-1-5/222-1+5/2-1-5/21-5/21+5/21-5/21+5/20000-1+5/2-1-5/2-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D10
ρ132200-22200220000-222-2-2-20000    orthogonal lifted from D8
ρ142-2000222-2-2-20000ζ16716ζ1651631651631671600-222-2    orthogonal lifted from D16
ρ152-200022-22-2-20000165163ζ1671616716ζ165163002-2-22    orthogonal lifted from D16
ρ1622002-1-5/2-1+5/2-2-2-1-5/2-1+5/2ζ5352ζ54553525450000-1-5/2-1+5/21-5/21-5/21+5/21+5/2    complex lifted from C5⋊D4
ρ1722002-1+5/2-1-5/2-2-2-1+5/2-1-5/2ζ5455352545ζ53520000-1+5/2-1-5/21+5/21+5/21-5/21-5/2    complex lifted from C5⋊D4
ρ1822002-1-5/2-1+5/2-2-2-1-5/2-1+5/25352545ζ5352ζ5450000-1-5/2-1+5/21-5/21-5/21+5/21+5/2    complex lifted from C5⋊D4
ρ1922002-1+5/2-1-5/2-2-2-1+5/2-1-5/2545ζ5352ζ54553520000-1+5/2-1-5/21+5/21+5/21-5/21-5/2    complex lifted from C5⋊D4
ρ204400-4-1-5-1+500-1-5-1+5000000001+51-50000    orthogonal lifted from D4⋊D5, Schur index 2
ρ214400-4-1+5-1-500-1+5-1-5000000001-51+50000    orthogonal lifted from D4⋊D5, Schur index 2
ρ224-4000-1-5-1+5-22221+51-50000000000ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5483ζ58ζ548ζ5ζ83ζ5383ζ528ζ538ζ5283ζ5383ζ528ζ538ζ52    orthogonal faithful, Schur index 2
ρ234-4000-1-5-1+522-221+51-50000000000ζ83ζ5483ζ58ζ548ζ5ζ87ζ5487ζ585ζ5485ζ583ζ5383ζ528ζ538ζ52ζ83ζ5383ζ528ζ538ζ52    orthogonal faithful, Schur index 2
ρ244-4000-1+5-1-5-22221-51+5000000000083ζ5383ζ528ζ538ζ52ζ83ζ5383ζ528ζ538ζ52ζ83ζ5483ζ58ζ548ζ5ζ87ζ5487ζ585ζ5485ζ5    orthogonal faithful, Schur index 2
ρ254-4000-1+5-1-522-221-51+50000000000ζ83ζ5383ζ528ζ538ζ5283ζ5383ζ528ζ538ζ52ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5483ζ58ζ548ζ5    orthogonal faithful, Schur index 2

Smallest permutation representation of C5⋊D16
On 80 points
Generators in S80
(1 78 17 48 52)(2 53 33 18 79)(3 80 19 34 54)(4 55 35 20 65)(5 66 21 36 56)(6 57 37 22 67)(7 68 23 38 58)(8 59 39 24 69)(9 70 25 40 60)(10 61 41 26 71)(11 72 27 42 62)(12 63 43 28 73)(13 74 29 44 64)(14 49 45 30 75)(15 76 31 46 50)(16 51 47 32 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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(49 65)(50 80)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 72)(59 71)(60 70)(61 69)(62 68)(63 67)(64 66)

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

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

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

C5⋊D16 is a maximal subgroup of
D5×D16  D16⋊D5  C16⋊D10  SD323D5  D8.D10  D8⋊D10  C40.30C23  C15⋊D16  C5⋊D48  C157D16
C5⋊D16 is a maximal quotient of
C40.2Q8  C40.5D4  C5⋊D32  D16.D5  C5⋊SD64  C5⋊Q64  C10.D16  C15⋊D16  C5⋊D48  C157D16

Matrix representation of C5⋊D16 in GL4(𝔽241) generated by

1000
0100
002401
0050190
,
585400
21411200
006927
0020172
,
1000
124000
00511
0051190
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,240,50,0,0,1,190],[58,214,0,0,54,112,0,0,0,0,69,20,0,0,27,172],[1,1,0,0,0,240,0,0,0,0,51,51,0,0,1,190] >;

C5⋊D16 in GAP, Magma, Sage, TeX

C_5\rtimes D_{16}
% in TeX

G:=Group("C5:D16");
// GroupNames label

G:=SmallGroup(160,33);
// by ID

G=gap.SmallGroup(160,33);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,73,218,116,122,579,297,69,4613]);
// Polycyclic

G:=Group<a,b,c|a^5=b^16=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5⋊D16 in TeX
Character table of C5⋊D16 in TeX

׿
×
𝔽