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G = D10.1D8order 320 = 26·5

1st non-split extension by D10 of D8 acting via D8/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.1D8, D10.3SD16, C4⋊C41F5, C10.7C4≀C2, (C2×D20)⋊5C4, D10⋊C81C2, C4⋊D20.1C2, C10.7(C23⋊C4), C2.4(D20⋊C4), D10.3Q81C2, C2.4(Q82F5), (C2×Dic5).92D4, (C22×D5).54D4, C51(C22.SD16), C10.1(D4⋊C4), C22.55(C22⋊F5), C2.10(D10.D4), (C5×C4⋊C4)⋊1C4, (C2×C4).8(C2×F5), (C2×C20).5(C2×C4), (C2×C4×D5).1C22, (C2×C10).18(C22⋊C4), SmallGroup(320,206)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D10.1D8
C1C5C10C2×C10C22×D5C2×C4×D5D10.3Q8 — D10.1D8
C5C2×C10C2×C20 — D10.1D8
C1C22C2×C4C4⋊C4

Generators and relations for D10.1D8
 G = < a,b,c,d | a10=b2=c8=1, d2=a-1b, bab=a-1, cac-1=dad-1=a3, cbc-1=a7b, dbd-1=a2b, dcd-1=a4bc-1 >

Subgroups: 570 in 90 conjugacy classes, 24 normal (all characteristic)
C1, C2 [×3], C2 [×3], C4 [×5], C22, C22 [×7], C5, C8, C2×C4, C2×C4 [×7], D4 [×3], C23 [×2], D5 [×3], C10 [×3], C22⋊C4, C4⋊C4, C2×C8, C22×C4 [×2], C2×D4 [×2], Dic5, C20 [×2], F5 [×2], D10 [×2], D10 [×5], C2×C10, C2.C42, C22⋊C8, C4⋊D4, C5⋊C8, C4×D5, D20 [×3], C2×Dic5, C2×C20, C2×C20, C2×F5 [×4], C22×D5, C22×D5, C22.SD16, D10⋊C4, C5×C4⋊C4, C2×C5⋊C8, C2×C4×D5, C2×D20, C2×D20, C22×F5, D10⋊C8, D10.3Q8, C4⋊D20, D10.1D8
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, D8, SD16, F5, C23⋊C4, D4⋊C4, C4≀C2, C2×F5, C22.SD16, C22⋊F5, D10.D4, D20⋊C4, Q82F5, D10.1D8

Character table of D10.1D8

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H58A8B8C8D10A10B10C20A20B20C20D20E20F
 size 111110104048101020202020420202020444888888
ρ111111111111111111111111111111    trivial
ρ211111111111-1-1-1-11-1-1-1-1111111111    linear of order 2
ρ3111111-11-11111111-1-1-1-1111-1-1-111-1    linear of order 2
ρ4111111-11-111-1-1-1-111111111-1-1-111-1    linear of order 2
ρ51111-1-111-1-1-1-iii-i1i-i-ii111-1-1-111-1    linear of order 4
ρ61111-1-111-1-1-1i-i-ii1-iii-i111-1-1-111-1    linear of order 4
ρ71111-1-1-111-1-1-iii-i1-iii-i111111111    linear of order 4
ρ81111-1-1-111-1-1i-i-ii1i-i-ii111111111    linear of order 4
ρ92222220-20-2-2000020000222000-2-20    orthogonal lifted from D4
ρ102222-2-20-2022000020000222000-2-20    orthogonal lifted from D4
ρ112-22-22-20000000002-22-222-2-2000000    orthogonal lifted from D8
ρ122-22-22-200000000022-22-22-2-2000000    orthogonal lifted from D8
ρ1322-2-200000-2i2i-1-i1-i-1+i1+i20000-22-2000000    complex lifted from C4≀C2
ρ1422-2-200000-2i2i1+i-1+i1-i-1-i20000-22-2000000    complex lifted from C4≀C2
ρ152-22-2-220000000002--2--2-2-22-2-2000000    complex lifted from SD16
ρ1622-2-2000002i-2i-1+i1+i-1-i1-i20000-22-2000000    complex lifted from C4≀C2
ρ1722-2-2000002i-2i1-i-1-i1+i-1+i20000-22-2000000    complex lifted from C4≀C2
ρ182-22-2-220000000002-2-2--2--22-2-2000000    complex lifted from SD16
ρ19444400044000000-10000-1-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ204-4-440000000000040000-4-44000000    orthogonal lifted from C23⋊C4
ρ2144440004-4000000-10000-1-1-1111-1-11    orthogonal lifted from C2×F5
ρ224444000-40000000-10000-1-1-1-5-55115    orthogonal lifted from C22⋊F5
ρ234444000-40000000-10000-1-1-155-511-5    orthogonal lifted from C22⋊F5
ρ244-4-4400000000000-1000011-14ζ53+2ζ4ζ544ζ54+2ζ4ζ52443ζ54+2ζ43ζ5343-5543ζ52+2ζ43ζ543    orthogonal lifted from D10.D4
ρ254-4-4400000000000-1000011-143ζ54+2ζ43ζ534343ζ52+2ζ43ζ5434ζ54+2ζ4ζ5245-54ζ53+2ζ4ζ54    orthogonal lifted from D10.D4
ρ264-4-4400000000000-1000011-14ζ54+2ζ4ζ5244ζ53+2ζ4ζ5443ζ52+2ζ43ζ543-5543ζ54+2ζ43ζ5343    orthogonal lifted from D10.D4
ρ274-4-4400000000000-1000011-143ζ52+2ζ43ζ54343ζ54+2ζ43ζ53434ζ53+2ζ4ζ545-54ζ54+2ζ4ζ524    orthogonal lifted from D10.D4
ρ2888-8-800000000000-200002-22000000    orthogonal lifted from Q82F5
ρ298-88-800000000000-20000-222000000    orthogonal lifted from D20⋊C4, Schur index 2

Smallest permutation representation of D10.1D8
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 10)(2 9)(3 8)(4 7)(5 6)(11 15)(12 14)(16 20)(17 19)(21 28)(22 27)(23 26)(24 25)(29 30)(31 36)(32 35)(33 34)(37 40)(38 39)(41 48)(42 47)(43 46)(44 45)(49 50)(52 60)(53 59)(54 58)(55 57)(61 63)(64 70)(65 69)(66 68)(72 80)(73 79)(74 78)(75 77)
(1 65 39 79 30 59 45 11)(2 62 38 72 21 56 44 14)(3 69 37 75 22 53 43 17)(4 66 36 78 23 60 42 20)(5 63 35 71 24 57 41 13)(6 70 34 74 25 54 50 16)(7 67 33 77 26 51 49 19)(8 64 32 80 27 58 48 12)(9 61 31 73 28 55 47 15)(10 68 40 76 29 52 46 18)
(1 30)(2 27 10 23)(3 24 9 26)(4 21 8 29)(5 28 7 22)(6 25)(11 54 16 59)(12 51 15 52)(13 58 14 55)(17 56 20 57)(18 53 19 60)(31 38 37 40)(32 35 36 33)(34 39)(41 42 49 48)(43 46 47 44)(45 50)(61 71 64 72)(62 78 63 75)(65 79 70 74)(66 76 69 77)(67 73 68 80)

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

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

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

Matrix representation of D10.1D8 in GL6(𝔽41)

100000
010000
0000140
000010
0040010
0004010
,
4000000
0400000
0001400
0010400
0000400
0000401
,
26150000
26260000
00343446
003840840
0013313
00353777
,
900000
0320000
00338190
00223803
00303822
00019383

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,1,1,1,0,0,40,0,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,40,40,40,40,0,0,0,0,0,1],[26,26,0,0,0,0,15,26,0,0,0,0,0,0,34,38,1,35,0,0,34,40,33,37,0,0,4,8,1,7,0,0,6,40,3,7],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,3,22,3,0,0,0,38,38,0,19,0,0,19,0,38,38,0,0,0,3,22,3] >;

D10.1D8 in GAP, Magma, Sage, TeX

D_{10}._1D_8
% in TeX

G:=Group("D10.1D8");
// GroupNames label

G:=SmallGroup(320,206);
// by ID

G=gap.SmallGroup(320,206);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,219,1571,570,136,6278,3156]);
// Polycyclic

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

Export

Character table of D10.1D8 in TeX

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