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G = D10.1Q16order 320 = 26·5

1st non-split extension by D10 of Q16 acting via Q16/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.1Q16, D10.4SD16, C4⋊C42F5, C10.1C4≀C2, (C2×Dic10)⋊5C4, D10⋊C8.1C2, C10.8(C23⋊C4), C2.4(D4⋊F5), D102Q8.1C2, C2.4(Q8⋊F5), (C2×Dic5).93D4, (C22×D5).55D4, C10.1(Q8⋊C4), C51(C23.31D4), D10.3Q8.1C2, C22.56(C22⋊F5), C2.11(D10.D4), (C5×C4⋊C4)⋊2C4, (C2×C4).9(C2×F5), (C2×C20).6(C2×C4), (C2×C4×D5).2C22, (C2×C10).19(C22⋊C4), SmallGroup(320,207)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D10.1Q16
C1C5C10C2×C10C22×D5C2×C4×D5D10.3Q8 — D10.1Q16
C5C2×C10C2×C20 — D10.1Q16
C1C22C2×C4C4⋊C4

Generators and relations for D10.1Q16
 G = < a,b,c,d | a10=b2=c8=1, d2=a5c4, bab=a-1, cac-1=a3, ad=da, cbc-1=a7b, dbd-1=a5b, dcd-1=a4bc-1 >

Subgroups: 426 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, C4⋊C4, 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, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C2×F5 [×4], C22×D5, C23.31D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C5⋊C8, C2×Dic10, C2×C4×D5, C22×F5, D10⋊C8, D10.3Q8, D102Q8, D10.1Q16
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, D10.D4, D4⋊F5, Q8⋊F5, D10.1Q16

Character table of D10.1Q16

 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
ρ112-2-22-220000000002-2-222-22-2000000    symplectic lifted from Q16, Schur index 2
ρ122-2-22-22000000000222-2-2-22-2000000    symplectic lifted from Q16, Schur index 2
ρ132-22-200002i-2i1-i-1-i1+i-1+i0200002-2-2000000    complex lifted from C4≀C2
ρ142-22-20000-2i2i1+i-1+i1-i-1-i0200002-2-2000000    complex lifted from C4≀C2
ρ152-22-200002i-2i-1+i1+i-1-i1-i0200002-2-2000000    complex lifted from C4≀C2
ρ162-22-20000-2i2i-1-i1-i-1+i1+i0200002-2-2000000    complex lifted from C4≀C2
ρ172-2-222-20000000002-2--2-2--2-22-2000000    complex lifted from SD16
ρ182-2-222-20000000002--2-2--2-2-22-2000000    complex lifted from SD16
ρ19444400440000000-10000-1-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ204444004-40000000-10000-1-1-11-1-1111    orthogonal lifted from C2×F5
ρ2144-4-40000000000040000-4-44000000    orthogonal lifted from C23⋊C4
ρ22444400-400000000-10000-1-1-1-511-555    orthogonal lifted from C22⋊F5
ρ23444400-400000000-10000-1-1-15115-5-5    orthogonal lifted from C22⋊F5
ρ2444-4-400000000000-1000011-14ζ52+2ζ4ζ54-554ζ54+2ζ4ζ53443ζ54+2ζ43ζ524343ζ53+2ζ43ζ543    orthogonal lifted from D10.D4
ρ2544-4-400000000000-1000011-143ζ54+2ζ43ζ52435-543ζ53+2ζ43ζ5434ζ54+2ζ4ζ5344ζ52+2ζ4ζ54    orthogonal lifted from D10.D4
ρ2644-4-400000000000-1000011-143ζ53+2ζ43ζ5435-543ζ54+2ζ43ζ52434ζ52+2ζ4ζ544ζ54+2ζ4ζ534    orthogonal lifted from D10.D4
ρ2744-4-400000000000-1000011-14ζ54+2ζ4ζ534-554ζ52+2ζ4ζ5443ζ53+2ζ43ζ54343ζ54+2ζ43ζ5243    orthogonal lifted from D10.D4
ρ288-8-8800000000000-200002-22000000    symplectic lifted from Q8⋊F5, Schur index 2
ρ298-88-800000000000-20000-222000000    symplectic lifted from D4⋊F5, Schur index 2

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

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

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

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

Matrix representation of D10.1Q16 in GL6(𝔽41)

100000
010000
0000040
001111
0040000
0004000
,
100000
010000
000001
000010
000100
001000
,
29120000
29290000
0037334034
007184
0073406
00373384
,
15150000
15260000
0036740
001478
0033343740
001343538

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,40,0,0,0,0,1,0,40,0,0,0,1,0,0,0,0,40,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[29,29,0,0,0,0,12,29,0,0,0,0,0,0,37,7,7,37,0,0,33,1,3,3,0,0,40,8,40,38,0,0,34,4,6,4],[15,15,0,0,0,0,15,26,0,0,0,0,0,0,3,1,33,1,0,0,6,4,34,34,0,0,7,7,37,35,0,0,40,8,40,38] >;

D10.1Q16 in GAP, Magma, Sage, TeX

D_{10}._1Q_{16}
% in TeX

G:=Group("D10.1Q16");
// GroupNames label

G:=SmallGroup(320,207);
// by ID

G=gap.SmallGroup(320,207);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,387,675,794,192,6278,3156]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=a^5*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 D10.1Q16 in TeX

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