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G = D10.Q16order 320 = 26·5

2nd non-split extension by D10 of Q16 acting via Q16/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.2Q16, D10.8SD16, (C2×Q8)⋊1F5, C4⋊Dic56C4, (Q8×C10)⋊1C4, C10.10C4≀C2, C2.8(C23⋊F5), D10⋊C8.2C2, D103Q8.1C2, C2.6(Q8⋊F5), C2.6(Q82F5), (C22×D5).61D4, C10.17(C23⋊C4), C10.5(Q8⋊C4), (C2×Dic5).109D4, C52(C23.31D4), D10.3Q8.2C2, C22.62(C22⋊F5), (C2×C4).17(C2×F5), (C2×C4×D5).4C22, (C2×C20).14(C2×C4), (C2×C10).39(C22⋊C4), SmallGroup(320,264)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D10.Q16
C1C5C10C2×C10C22×D5C2×C4×D5D10.3Q8 — D10.Q16
C5C2×C10C2×C20 — D10.Q16
C1C22C2×C4C2×Q8

Generators and relations for D10.Q16
 G = < a,b,c,d | a10=b2=c8=1, d2=c4, bab=a-1, cac-1=a3, ad=da, cbc-1=a7b, dbd-1=a5b, dcd-1=a4bc-1 >

Subgroups: 410 in 80 conjugacy classes, 24 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C22, C22 [×4], C5, C8, C2×C4, C2×C4 [×8], Q8, C23, D5 [×2], C10 [×3], C22⋊C4, C4⋊C4 [×2], C2×C8, C22×C4 [×2], C2×Q8, Dic5 [×2], C20 [×2], F5 [×2], D10 [×2], D10 [×2], C2×C10, C2.C42, C22⋊C8, C22⋊Q8, C5⋊C8, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C2×F5 [×4], C22×D5, C23.31D4, C10.D4, C4⋊Dic5, D10⋊C4, C2×C5⋊C8, C2×C4×D5, Q8×C10, C22×F5, D10⋊C8, D10.3Q8, D103Q8, D10.Q16
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, SD16, Q16, F5, C23⋊C4, Q8⋊C4, C4≀C2, C2×F5, C23.31D4, C22⋊F5, Q8⋊F5, Q82F5, C23⋊F5, D10.Q16

Character table of D10.Q16

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I58A8B8C8D10A10B10C20A20B20C20D20E20F
 size 111110104810102020202040420202020444888888
ρ111111111111111111111111111111    trivial
ρ21111111111-1-1-1-111-1-1-1-1111111111    linear of order 2
ρ31111111-1111111-11-1-1-1-1111-111-1-1-1    linear of order 2
ρ41111111-111-1-1-1-1-111111111-111-1-1-1    linear of order 2
ρ51111-1-111-1-1-iii-i-11-iii-i111111111    linear of order 4
ρ61111-1-111-1-1i-i-ii-11i-i-ii111111111    linear of order 4
ρ71111-1-11-1-1-1i-i-ii11-iii-i111-111-1-1-1    linear of order 4
ρ81111-1-11-1-1-1-iii-i11i-i-ii111-111-1-1-1    linear of order 4
ρ9222222-20-2-200000200002220-2-2000    orthogonal lifted from D4
ρ102222-2-2-202200000200002220-2-2000    orthogonal lifted from D4
ρ112-22-2-220000000002-2-2222-2-2000000    symplectic lifted from Q16, Schur index 2
ρ122-22-2-22000000000222-2-22-2-2000000    symplectic lifted from Q16, Schur index 2
ρ1322-2-20000-2i2i-1-i1-i-1+i1+i020000-2-22000000    complex lifted from C4≀C2
ρ1422-2-200002i-2i-1+i1+i-1-i1-i020000-2-22000000    complex lifted from C4≀C2
ρ1522-2-20000-2i2i1+i-1+i1-i-1-i020000-2-22000000    complex lifted from C4≀C2
ρ1622-2-200002i-2i1-i-1-i1+i-1+i020000-2-22000000    complex lifted from C4≀C2
ρ172-22-22-20000000002-2--2-2--22-2-2000000    complex lifted from SD16
ρ182-22-22-20000000002--2-2--2-22-2-2000000    complex lifted from SD16
ρ194-4-440000000000040000-44-4000000    orthogonal lifted from C23⋊C4
ρ20444400440000000-10000-1-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ214444004-40000000-10000-1-1-11-1-1111    orthogonal lifted from C2×F5
ρ22444400-400000000-10000-1-1-15115-5-5    orthogonal lifted from C22⋊F5
ρ23444400-400000000-10000-1-1-1-511-555    orthogonal lifted from C22⋊F5
ρ244-4-4400000000000-100001-1153+2ζ5+15-554+2ζ52+154+2ζ53+152+2ζ5+1    complex lifted from C23⋊F5
ρ254-4-4400000000000-100001-1154+2ζ52+15-553+2ζ5+152+2ζ5+154+2ζ53+1    complex lifted from C23⋊F5
ρ264-4-4400000000000-100001-1152+2ζ5+1-5554+2ζ53+153+2ζ5+154+2ζ52+1    complex lifted from C23⋊F5
ρ274-4-4400000000000-100001-1154+2ζ53+1-5552+2ζ5+154+2ζ52+153+2ζ5+1    complex lifted from C23⋊F5
ρ2888-8-800000000000-2000022-2000000    orthogonal lifted from Q82F5
ρ298-88-800000000000-20000-222000000    symplectic lifted from Q8⋊F5, Schur index 2

Smallest permutation representation of D10.Q16
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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 30)(11 71)(12 80)(13 79)(14 78)(15 77)(16 76)(17 75)(18 74)(19 73)(20 72)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(51 65)(52 64)(53 63)(54 62)(55 61)(56 70)(57 69)(58 68)(59 67)(60 66)
(1 70 50 11 30 52 32 77)(2 67 49 14 21 59 31 80)(3 64 48 17 22 56 40 73)(4 61 47 20 23 53 39 76)(5 68 46 13 24 60 38 79)(6 65 45 16 25 57 37 72)(7 62 44 19 26 54 36 75)(8 69 43 12 27 51 35 78)(9 66 42 15 28 58 34 71)(10 63 41 18 29 55 33 74)
(1 70 30 52)(2 61 21 53)(3 62 22 54)(4 63 23 55)(5 64 24 56)(6 65 25 57)(7 66 26 58)(8 67 27 59)(9 68 28 60)(10 69 29 51)(11 45 77 37)(12 46 78 38)(13 47 79 39)(14 48 80 40)(15 49 71 31)(16 50 72 32)(17 41 73 33)(18 42 74 34)(19 43 75 35)(20 44 76 36)

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

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

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

Matrix representation of D10.Q16 in GL8(𝔽41)

400000000
040000000
00100000
00010000
00000001
000040404040
00001000
00000100
,
10000000
3940000000
00100000
00010000
00000001
00000010
00000100
00001000
,
44000000
437000000
0012290000
0012120000
000022033
000033022
00003819380
000019222219
,
55000000
336000000
0015150000
0015260000
00001903838
000032230
000003223
00003838019

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,0],[1,39,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[4,4,0,0,0,0,0,0,4,37,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,29,12,0,0,0,0,0,0,0,0,22,3,38,19,0,0,0,0,0,3,19,22,0,0,0,0,3,0,38,22,0,0,0,0,3,22,0,19],[5,3,0,0,0,0,0,0,5,36,0,0,0,0,0,0,0,0,15,15,0,0,0,0,0,0,15,26,0,0,0,0,0,0,0,0,19,3,0,38,0,0,0,0,0,22,3,38,0,0,0,0,38,3,22,0,0,0,0,0,38,0,3,19] >;

D10.Q16 in GAP, Magma, Sage, TeX

D_{10}.Q_{16}
% in TeX

G:=Group("D10.Q16");
// GroupNames label

G:=SmallGroup(320,264);
// by ID

G=gap.SmallGroup(320,264);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,219,1571,570,136,6278,3156]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=c^4,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^7*b,d*b*d^-1=a^5*b,d*c*d^-1=a^4*b*c^-1>;
// generators/relations

Export

Character table of D10.Q16 in TeX

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