Copied to
clipboard

G = SD16⋊3D5  order 160 = 25·5

The semidirect product of SD16 and D5 acting through Inn(SD16)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: SD16⋊3D5, D4.5D10, D10.6D4, C8.11D10, Q8.2D10, C20.7C23, C40.11C22, Dic5.25D4, D20.3C22, Dic10.3C22, D4⋊D5⋊4C2, (C8×D5)⋊5C2, C5⋊3(C4○D8), C40⋊C2⋊6C2, C5⋊Q16⋊2C2, C2.21(D4×D5), D4⋊2D5⋊3C2, Q8⋊2D5⋊2C2, (C5×SD16)⋊4C2, C10.33(C2×D4), C4.7(C22×D5), C5⋊2C8.6C22, (C5×D4).5C22, (C5×Q8).2C22, (C4×D5).18C22, SmallGroup(160,137)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — SD16⋊3D5
Chief series
C1 — C5 — C10 — C20 — C4×D5 — D4⋊2D5 — SD16⋊3D5
Lower central
C5 — C10 — C20 — SD16⋊3D5
Upper central
C1 — C2 — C4 — SD16

Generators and relations for SD16⋊3D5
 G = < a,b,c,d | a8=b2=c5=d2=1, bab=a3, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 216 in 62 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, C2×C8, D8, SD16, SD16, Q16, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C4○D8, C5⋊2C8, C40, Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C8×D5, C40⋊C2, D4⋊D5, C5⋊Q16, C5×SD16, D4⋊2D5, Q8⋊2D5, SD16⋊3D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4○D8, C22×D5, D4×D5, SD16⋊3D5

Character table of SD16⋊3D5

 class 12A2B2C2D4A4B4C4D4E5A5B8A8B8C8D10A10B10C10D20A20B20C20D40A40B40C40D
 size 114102024552022221010228844884444
ρ11111111111111111111111111111    trivial
ρ211-1111111-111-1-1-1-111-1-11111-1-1-1-1    linear of order 2
ρ3111-111-1-1-1-111-1-111111111-1-1-1-1-1-1    linear of order 2
ρ411-1-111-1-1-111111-1-111-1-111-1-11111    linear of order 2
ρ511-11-11-111-111111111-1-111-1-11111    linear of order 2
ρ61111-11-111111-1-1-1-1111111-1-1-1-1-1-1    linear of order 2
ρ7111-1-111-1-1-11111-1-1111111111111    linear of order 2
ρ811-1-1-111-1-1111-1-11111-1-11111-1-1-1-1    linear of order 2
ρ922020-20-2-202200002200-2-2000000    orthogonal lifted from D4
ρ10220-20-202202200002200-2-2000000    orthogonal lifted from D4
ρ1122-20022000-1+√5/2-1-√5/2-2-200-1-√5/2-1+√5/21-√5/21+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/21-√5/21-√5/21+√5/21+√5/2    orthogonal lifted from D10
ρ12222002-2000-1+√5/2-1-√5/2-2-200-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/21+√5/21-√5/21-√5/21-√5/21+√5/21+√5/2    orthogonal lifted from D10
ρ132220022000-1+√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
ρ1422-2002-2000-1+√5/2-1-√5/22200-1-√5/2-1+√5/21-√5/21+√5/2-1+√5/2-1-√5/21+√5/21-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2    orthogonal lifted from D10
ρ1522-2002-2000-1-√5/2-1+√5/22200-1+√5/2-1-√5/21+√5/21-√5/2-1-√5/2-1+√5/21-√5/21+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2    orthogonal lifted from D10
ρ162220022000-1-√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
ρ17222002-2000-1-√5/2-1+√5/2-2-200-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/21-√5/21+√5/21+√5/21+√5/21-√5/21-√5/2    orthogonal lifted from D10
ρ1822-20022000-1-√5/2-1+√5/2-2-200-1+√5/2-1-√5/21+√5/21-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/21+√5/21+√5/21-√5/21-√5/2    orthogonal lifted from D10
ρ192-200000-2i2i022√-2-√-2-√2√2-2-2000000√-2-√-2-√-2√-2    complex lifted from C4○D8
ρ202-200000-2i2i022-√-2√-2√2-√2-2-2000000-√-2√-2√-2-√-2    complex lifted from C4○D8
ρ212-2000002i-2i022-√-2√-2-√2√2-2-2000000-√-2√-2√-2-√-2    complex lifted from C4○D8
ρ222-2000002i-2i022√-2-√-2√2-√2-2-2000000√-2-√-2-√-2√-2    complex lifted from C4○D8
ρ2344000-40000-1-√5-1+√50000-1+√5-1-√5001+√51-√5000000    orthogonal lifted from D4×D5
ρ2444000-40000-1+√5-1-√50000-1-√5-1+√5001-√51+√5000000    orthogonal lifted from D4×D5
ρ254-400000000-1+√5-1-√52√-2-2√-2001+√51-√5000000ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5ζ87ζ54+ζ87ζ5+ζ85ζ54+ζ85ζ5ζ87ζ53+ζ87ζ52+ζ85ζ53+ζ85ζ52ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52    complex faithful
ρ264-400000000-1+√5-1-√5-2√-22√-2001+√51-√5000000ζ87ζ54+ζ87ζ5+ζ85ζ54+ζ85ζ5ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52ζ87ζ53+ζ87ζ52+ζ85ζ53+ζ85ζ52    complex faithful
ρ274-400000000-1-√5-1+√5-2√-22√-2001-√51+√5000000ζ87ζ53+ζ87ζ52+ζ85ζ53+ζ85ζ52ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5ζ87ζ54+ζ87ζ5+ζ85ζ54+ζ85ζ5    complex faithful
ρ284-400000000-1-√5-1+√52√-2-2√-2001-√51+√5000000ζ83ζ53+ζ83ζ52+ζ8ζ53+ζ8ζ52ζ87ζ53+ζ87ζ52+ζ85ζ53+ζ85ζ52ζ87ζ54+ζ87ζ5+ζ85ζ54+ζ85ζ5ζ83ζ54+ζ83ζ5+ζ8ζ54+ζ8ζ5    complex faithful

Smallest permutation representation of SD16⋊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 59)(2 62)(3 57)(4 60)(5 63)(6 58)(7 61)(8 64)(9 28)(10 31)(11 26)(12 29)(13 32)(14 27)(15 30)(16 25)(17 36)(18 39)(19 34)(20 37)(21 40)(22 35)(23 38)(24 33)(41 77)(42 80)(43 75)(44 78)(45 73)(46 76)(47 79)(48 74)(49 69)(50 72)(51 67)(52 70)(53 65)(54 68)(55 71)(56 66)

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

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

SD16⋊3D5 is a maximal subgroup of
 SD16⋊3F5  SD16⋊2F5  D20.29D4  D5×C4○D8  D8⋊11D10  SD16⋊D10  D8⋊5D10  D40⋊C22  D20.44D4  C40.31D6  Dic6.D10  C60.16C23  D20.10D6  D20.14D6  D12.D10  D4.5D30  Dic5.6S4
SD16⋊3D5 is a maximal quotient of
 Dic5⋊4D8  D4.Dic10  (C8×Dic5)⋊C2  D4⋊2D5⋊C4  D10⋊D8  C40⋊6C4⋊C2  D4⋊3D20  D20.D4  Dic5⋊4Q16  Dic10.11D4  Q8.2Dic10  Q8⋊Dic5⋊C2  Q8⋊2D5⋊C4  Q8.D20  D10⋊Q16  D10⋊1C8.C2  Dic5⋊8SD16  Dic10.Q8  C8.8Dic10  (C8×D5)⋊C4  C8⋊8D20  C4.Q8⋊D5  C20.(C4○D4)  D20.Q8  SD16×Dic5  (C5×D4).D4  (C5×Q8).D4  C40.43D4  C40⋊14D4  D20⋊7D4  Dic10.16D4  C40.31D6  Dic6.D10  C60.16C23  D20.10D6  D20.14D6  D12.D10  D4.5D30

Matrix representation of SD16⋊3D5 ►in GL4(𝔽41) generated by

1000
0100
001129
00170
,
40000
04000
002426
001117
,
0100
403400
0010
0001
,
04000
40000
00940
003932
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,11,17,0,0,29,0],[40,0,0,0,0,40,0,0,0,0,24,11,0,0,26,17],[0,40,0,0,1,34,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,40,0,0,0,0,0,9,39,0,0,40,32] >;
 

SD16⋊3D5 in GAP, Magma, Sage, TeX 

{\rm SD}_{16}\rtimes_3D_5
 
% in TeX
 
G:=Group("SD16:3D5");
 
// GroupNames label
 
G:=SmallGroup(160,137);
 
// by ID
 
G=gap.SmallGroup(160,137);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,362,116,86,297,159,69,4613]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^5=d^2=1,b*a*b=a^3,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 SD16⋊3D5 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁