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G = Q16⋊D5order 160 = 25·5

2nd semidirect product of Q16 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q162D5, C8.3D10, Q8.4D10, D10.17D4, C20.9C23, C40.10C22, Dic5.19D4, D20.4C22, Dic10.5C22, Q8⋊D53C2, (Q8×D5)⋊3C2, C8⋊D54C2, C40⋊C24C2, (C5×Q16)⋊4C2, C5⋊Q164C2, C2.23(D4×D5), C10.35(C2×D4), C53(C8.C22), C4.9(C22×D5), C52C8.2C22, Q82D5.1C2, (C4×D5).4C22, (C5×Q8).4C22, SmallGroup(160,139)

Series: Derived Chief Lower central Upper central

C1C20 — Q16⋊D5
C1C5C10C20C4×D5Q8×D5 — Q16⋊D5
C5C10C20 — Q16⋊D5
C1C2C4Q16

Generators and relations for Q16⋊D5
 G = < a,b,c,d | a8=c5=d2=1, b2=a4, bab-1=a-1, ac=ca, dad=a5, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 208 in 60 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2 [×2], C4, C4 [×4], C22 [×2], C5, C8, C8, C2×C4 [×3], D4 [×2], Q8 [×2], Q8 [×2], D5 [×2], C10, M4(2), SD16 [×2], Q16, Q16, C2×Q8, C4○D4, Dic5, Dic5, C20, C20 [×2], D10, D10, C8.C22, C52C8, C40, Dic10, Dic10, C4×D5, C4×D5 [×2], D20, D20, C5×Q8 [×2], C8⋊D5, C40⋊C2, Q8⋊D5, C5⋊Q16, C5×Q16, Q8×D5, Q82D5, Q16⋊D5
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, D10 [×3], C8.C22, C22×D5, D4×D5, Q16⋊D5

Character table of Q16⋊D5

 class 12A2B2C4A4B4C4D4E5A5B8A8B10A10B20A20B20C20D20E20F40A40B40C40D
 size 111020244102022420224488884444
ρ11111111111111111111111111    trivial
ρ211-1-1111-1-1111-1111111111111    linear of order 2
ρ3111-111-11111-1-11111-1-111-1-1-1-1    linear of order 2
ρ411-1111-1-1-111-111111-1-111-1-1-1-1    linear of order 2
ρ511111-111-111-1-1111111-1-1-1-1-1-1    linear of order 2
ρ611-1-11-11-1111-11111111-1-1-1-1-1-1    linear of order 2
ρ7111-11-1-11-111111111-1-1-1-11111    linear of order 2
ρ811-111-1-1-11111-11111-1-1-1-11111    linear of order 2
ρ92220-200-20220022-2-200000000    orthogonal lifted from D4
ρ1022-20-20020220022-2-200000000    orthogonal lifted from D4
ρ1122002-2-200-1+5/2-1-5/220-1-5/2-1+5/2-1+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    orthogonal lifted from D10
ρ12220022200-1+5/2-1-5/220-1-5/2-1+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
ρ13220022-200-1+5/2-1-5/2-20-1-5/2-1+5/2-1+5/2-1-5/21+5/21-5/2-1+5/2-1-5/21+5/21+5/21-5/21-5/2    orthogonal lifted from D10
ρ1422002-2200-1+5/2-1-5/2-20-1-5/2-1+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
ρ1522002-2200-1-5/2-1+5/2-20-1+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
ρ16220022-200-1-5/2-1+5/2-20-1+5/2-1-5/2-1-5/2-1+5/21-5/21+5/2-1-5/2-1+5/21-5/21-5/21+5/21+5/2    orthogonal lifted from D10
ρ1722002-2-200-1-5/2-1+5/220-1+5/2-1-5/2-1-5/2-1+5/21-5/21+5/21+5/21-5/2-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D10
ρ18220022200-1-5/2-1+5/220-1+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
ρ194400-40000-1-5-1+500-1+5-1-51+51-500000000    orthogonal lifted from D4×D5
ρ204400-40000-1+5-1-500-1-5-1+51-51+500000000    orthogonal lifted from D4×D5
ρ214-400000004400-4-40000000000    symplectic lifted from C8.C22, Schur index 2
ρ224-40000000-1-5-1+5001-51+5000000ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5483ζ58ζ548ζ583ζ5383ζ528ζ538ζ52ζ83ζ5383ζ528ζ538ζ52    complex faithful
ρ234-40000000-1-5-1+5001-51+5000000ζ83ζ5483ζ58ζ548ζ5ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5383ζ528ζ538ζ5283ζ5383ζ528ζ538ζ52    complex faithful
ρ244-40000000-1+5-1-5001+51-5000000ζ83ζ5383ζ528ζ538ζ5283ζ5383ζ528ζ538ζ52ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5483ζ58ζ548ζ5    complex faithful
ρ254-40000000-1+5-1-5001+51-500000083ζ5383ζ528ζ538ζ52ζ83ζ5383ζ528ζ538ζ52ζ83ζ5483ζ58ζ548ζ5ζ87ζ5487ζ585ζ5485ζ5    complex faithful

Smallest permutation representation of Q16⋊D5
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 75 5 79)(2 74 6 78)(3 73 7 77)(4 80 8 76)(9 27 13 31)(10 26 14 30)(11 25 15 29)(12 32 16 28)(17 35 21 39)(18 34 22 38)(19 33 23 37)(20 40 24 36)(41 62 45 58)(42 61 46 57)(43 60 47 64)(44 59 48 63)(49 71 53 67)(50 70 54 66)(51 69 55 65)(52 68 56 72)
(1 66 62 39 15)(2 67 63 40 16)(3 68 64 33 9)(4 69 57 34 10)(5 70 58 35 11)(6 71 59 36 12)(7 72 60 37 13)(8 65 61 38 14)(17 29 75 50 45)(18 30 76 51 46)(19 31 77 52 47)(20 32 78 53 48)(21 25 79 54 41)(22 26 80 55 42)(23 27 73 56 43)(24 28 74 49 44)
(1 25)(2 30)(3 27)(4 32)(5 29)(6 26)(7 31)(8 28)(9 73)(10 78)(11 75)(12 80)(13 77)(14 74)(15 79)(16 76)(17 70)(18 67)(19 72)(20 69)(21 66)(22 71)(23 68)(24 65)(33 56)(34 53)(35 50)(36 55)(37 52)(38 49)(39 54)(40 51)(41 62)(42 59)(43 64)(44 61)(45 58)(46 63)(47 60)(48 57)

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

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

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

Q16⋊D5 is a maximal subgroup of
D20.30D4  Q16⋊D10  D811D10  D5×C8.C22  D40⋊C22  C40.C23  D20.44D4  C24.2D10  D30.3D4  D20.13D6  C60.C23  C60.39C23  D20.17D6  Q16⋊D15  CSU2(𝔽3)⋊D5  D10.2S4
Q16⋊D5 is a maximal quotient of
C5⋊Q165C4  Q8⋊Dic10  Q8⋊C4⋊D5  C408C4.C2  Dic10.11D4  Q8.2Dic10  (Q8×D5)⋊C4  Q8⋊(C4×D5)  D10.11SD16  Q82D20  Q8.D20  C5⋊(C8⋊D4)  D101C8.C2  C52C8.D4  Q8⋊D56C4  Dic5⋊SD16  C404Q8  Dic10.2Q8  C4020(C2×C4)  D10.13D8  C83D20  C2.D87D5  C4021(C2×C4)  D202Q8  Dic53Q16  Q16⋊Dic5  (C2×Q16)⋊D5  D105Q16  D20.17D4  C40.36D4  C40.37D4  C24.2D10  D30.3D4  D20.13D6  C60.C23  C60.39C23  D20.17D6  Q16⋊D15

Matrix representation of Q16⋊D5 in GL4(𝔽41) generated by

003917
00242
1123917
2940242
,
171120
302202
18112430
30231119
,
0100
403400
0001
004034
,
302202
171120
0403024
4001911
G:=sub<GL(4,GF(41))| [0,0,1,29,0,0,12,40,39,24,39,24,17,2,17,2],[17,30,18,30,11,22,11,23,2,0,24,11,0,2,30,19],[0,40,0,0,1,34,0,0,0,0,0,40,0,0,1,34],[30,17,0,40,22,11,40,0,0,2,30,19,2,0,24,11] >;

Q16⋊D5 in GAP, Magma, Sage, TeX

Q_{16}\rtimes D_5
% in TeX

G:=Group("Q16:D5");
// GroupNames label

G:=SmallGroup(160,139);
// by ID

G=gap.SmallGroup(160,139);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,362,116,86,297,159,69,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^8=c^5=d^2=1,b^2=a^4,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of Q16⋊D5 in TeX

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