Copied to
clipboard

G = D8:3D5  order 160 = 25·5

The semidirect product of D8 and D5 acting through Inn(D8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8:3D5, C8.8D10, D4.1D10, D10.5D4, Dic20:4C2, C20.3C23, C40.6C22, Dic5.24D4, Dic10.1C22, (C8xD5):2C2, (C5xD8):3C2, C5:2(C4oD8), D4.D5:2C2, C2.17(D4xD5), D4:2D5:2C2, C10.29(C2xD4), C4.3(C22xD5), C5:2C8.5C22, (C5xD4).1C22, (C4xD5).16C22, SmallGroup(160,133)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — D8:3D5
Chief series
C1 — C5 — C10 — C20 — C4xD5 — D4:2D5 — D8:3D5
Lower central
C5 — C10 — C20 — D8:3D5
Upper central
C1 — C2 — C4 — D8

Generators and relations for D8:3D5
 G = < a,b,c,d | a8=b2=c5=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 200 in 62 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2xC4, D4, D4, Q8, D5, C10, C10, C2xC8, D8, SD16, Q16, C4oD4, Dic5, Dic5, C20, D10, C2xC10, C4oD8, C5:2C8, C40, Dic10, C4xD5, C2xDic5, C5:D4, C5xD4, C8xD5, Dic20, D4.D5, C5xD8, D4:2D5, D8:3D5
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, D10, C4oD8, C22xD5, D4xD5, D8:3D5

Character table of D8:3D5

 class 12A2B2C2D4A4B4C4D4E5A5B8A8B8C8D10A10B10C10D10E10F20A20B40A40B40C40D
 size 114410255202022221010228888444444
ρ11111111111111111111111111111    trivial
ρ211-1-11111-1-111111111-1-1-1-1111111    linear of order 2
ρ311-11-11-1-11-111-1-11111-111-111-1-1-1-1    linear of order 2
ρ4111-1-11-1-1-1111-1-111111-1-1111-1-1-1-1    linear of order 2
ρ5111-111111-111-1-1-1-1111-1-1111-1-1-1-1    linear of order 2
ρ611-111111-1111-1-1-1-111-111-111-1-1-1-1    linear of order 2
ρ711-1-1-11-1-1111111-1-111-1-1-1-1111111    linear of order 2
ρ81111-11-1-1-1-11111-1-1111111111111    linear of order 2
ρ922002-2-2-200220000220000-2-20000    orthogonal lifted from D4
ρ102200-2-22200220000220000-2-20000    orthogonal lifted from D4
ρ1122-22020000-1+√5/2-1-√5/2-2-200-1-√5/2-1+√5/21-√5/2-1+√5/2-1-√5/21+√5/2-1-√5/2-1+√5/21-√5/21+√5/21-√5/21+√5/2    orthogonal lifted from D10
ρ1222-22020000-1-√5/2-1+√5/2-2-200-1+√5/2-1-√5/21+√5/2-1-√5/2-1+√5/21-√5/2-1+√5/2-1-√5/21+√5/21-√5/21+√5/21-√5/2    orthogonal lifted from D10
ρ1322-2-2020000-1+√5/2-1-√5/22200-1-√5/2-1+√5/21-√5/21-√5/21+√5/21+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2    orthogonal lifted from D10
ρ142222020000-1-√5/2-1+√5/22200-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ152222020000-1+√5/2-1-√5/22200-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ1622-2-2020000-1-√5/2-1+√5/22200-1+√5/2-1-√5/21+√5/21+√5/21-√5/21-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2    orthogonal lifted from D10
ρ17222-2020000-1+√5/2-1-√5/2-2-200-1-√5/2-1+√5/2-1+√5/21-√5/21+√5/2-1-√5/2-1-√5/2-1+√5/21-√5/21+√5/21-√5/21+√5/2    orthogonal lifted from D10
ρ18222-2020000-1-√5/2-1+√5/2-2-200-1+√5/2-1-√5/2-1-√5/21+√5/21-√5/2-1+√5/2-1+√5/2-1-√5/21+√5/21-√5/21+√5/21-√5/2    orthogonal lifted from D10
ρ192-20000-2i2i0022-√2√2√-2-√-2-2-2000000√2-√2-√2√2    complex lifted from C4oD8
ρ202-200002i-2i0022√2-√2√-2-√-2-2-2000000-√2√2√2-√2    complex lifted from C4oD8
ρ212-200002i-2i0022-√2√2-√-2√-2-2-2000000√2-√2-√2√2    complex lifted from C4oD8
ρ222-20000-2i2i0022√2-√2-√-2√-2-2-2000000-√2√2√2-√2    complex lifted from C4oD8
ρ2344000-40000-1+√5-1-√50000-1-√5-1+√500001+√51-√50000    orthogonal lifted from D4xD5
ρ2444000-40000-1-√5-1+√50000-1+√5-1-√500001-√51+√50000    orthogonal lifted from D4xD5
ρ254-400000000-1+√5-1-√5-2√22√2001+√51-√5000000ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5-ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52    symplectic faithful, Schur index 2
ρ264-400000000-1-√5-1+√52√2-2√2001-√51+√5000000ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5-ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5    symplectic faithful, Schur index 2
ρ274-400000000-1-√5-1+√5-2√22√2001-√51+√5000000-ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5    symplectic faithful, Schur index 2
ρ284-400000000-1+√5-1-√52√2-2√2001+√51-√5000000ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5-ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52    symplectic faithful, Schur index 2

Smallest permutation representation of D8:3D5
►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 74)(2 73)(3 80)(4 79)(5 78)(6 77)(7 76)(8 75)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 27)(18 26)(19 25)(20 32)(21 31)(22 30)(23 29)(24 28)(33 64)(34 63)(35 62)(36 61)(37 60)(38 59)(39 58)(40 57)(49 70)(50 69)(51 68)(52 67)(53 66)(54 65)(55 72)(56 71)

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

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

D8:3D5 is a maximal subgroup of
 D10.D8  D8.F5  D16:D5  D16:3D5  SD32:D5  SD32:3D5  D8:5F5  D8:F5  D8:13D10  D5xC4oD8  D20.47D4  SD16:D10  D8:6D10  D24:7D5  D24:5D5  D12.24D10  D30.11D4  D8:3D15
D8:3D5 is a maximal quotient of
 Dic5:6SD16  D4.2Dic10  Dic10.D4  (C8xDic5):C2  D4:2D5:C4  D10:SD16  D4.D20  C40:5C4:C2  Dic5:5Q16  Dic10.2Q8  C8.6Dic10  C8.27(C4xD5)  D10:2Q16  C2.D8:7D5  D8xDic5  (C2xD8).D5  C40.22D4  C40:6D4  Dic10:D4  D24:7D5  D24:5D5  D12.24D10  D30.11D4  D8:3D15

Matrix representation of D8:3D5 ►in GL4(F41) generated by

27000
03800
00400
00040
,
03800
27000
0010
0001
,
1000
0100
0061
00400
,
1000
04000
0016
00040
G:=sub<GL(4,GF(41))| [27,0,0,0,0,38,0,0,0,0,40,0,0,0,0,40],[0,27,0,0,38,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,6,40,0,0,1,0],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,6,40] >;
 

D8:3D5 in GAP, Magma, Sage, TeX 

D_8\rtimes_3D_5
 
% in TeX
 
G:=Group("D8:3D5");
 
// GroupNames label
 
G:=SmallGroup(160,133);
 
// by ID
 
G=gap.SmallGroup(160,133);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,362,116,297,159,69,4613]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^5=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of D8:3D5 in TeX

׿
x
:
Z
F
o
wr
Q
<