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G = Q8.D10order 160 = 25·5

5th non-split extension by Q8 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D405C2, Q163D5, D10.7D4, C8.10D10, Q8.5D10, C40.8C22, C20.10C23, Dic5.26D4, D20.5C22, (C8×D5)⋊3C2, C54(C4○D8), Q8⋊D54C2, (C5×Q16)⋊3C2, C2.24(D4×D5), Q82D53C2, C10.36(C2×D4), C52C8.8C22, C4.10(C22×D5), (C5×Q8).5C22, (C4×D5).20C22, SmallGroup(160,140)

Series: Derived Chief Lower central Upper central

C1C20 — Q8.D10
C1C5C10C20C4×D5Q82D5 — Q8.D10
C5C10C20 — Q8.D10
C1C2C4Q16

Generators and relations for Q8.D10
 G = < a,b,c,d | a4=d2=1, b2=c10=a2, bab-1=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=a2c9 >

Subgroups: 232 in 62 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×3], C5, C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], D5 [×3], C10, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5, C20, C20 [×2], D10, D10 [×2], C4○D8, C52C8, C40, C4×D5, C4×D5 [×2], D20 [×2], D20 [×2], C5×Q8 [×2], C8×D5, D40, Q8⋊D5 [×2], C5×Q16, Q82D5 [×2], Q8.D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, D10 [×3], C4○D8, C22×D5, D4×D5, Q8.D10

Character table of Q8.D10

 class 12A2B2C2D4A4B4C4D4E5A5B8A8B8C8D10A10B20A20B20C20D20E20F40A40B40C40D
 size 111020202445522221010224488884444
ρ11111111111111111111111111111    trivial
ρ211-1111-1-1-1-11111-1-11111-1-1-1-11111    linear of order 2
ρ31111-11-111111-1-1-1-11111-111-1-1-1-1-1    linear of order 2
ρ411-11-111-1-1-111-1-11111111-1-11-1-1-1-1    linear of order 2
ρ5111-1111-11111-1-1-1-111111-1-11-1-1-1-1    linear of order 2
ρ611-1-1-1111-1-11111-1-1111111111111    linear of order 2
ρ711-1-111-11-1-111-1-1111111-111-1-1-1-1-1    linear of order 2
ρ8111-1-11-1-1111111111111-1-1-1-11111    linear of order 2
ρ922200-200-2-222000022-2-200000000    orthogonal lifted from D4
ρ1022-200-2002222000022-2-200000000    orthogonal lifted from D4
ρ112200022200-1-5/2-1+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
ρ12220002-2-200-1-5/2-1+5/22200-1+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
ρ132200022-200-1+5/2-1-5/2-2-200-1-5/2-1+5/2-1+5/2-1-5/2-1+5/21-5/21+5/2-1-5/21+5/21+5/21-5/21-5/2    orthogonal lifted from D10
ρ14220002-2-200-1+5/2-1-5/22200-1-5/2-1+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
ρ15220002-2200-1-5/2-1+5/2-2-200-1+5/2-1-5/2-1-5/2-1+5/21+5/2-1-5/2-1+5/21-5/21-5/21-5/21+5/21+5/2    orthogonal lifted from D10
ρ16220002-2200-1+5/2-1-5/2-2-200-1-5/2-1+5/2-1+5/2-1-5/21-5/2-1+5/2-1-5/21+5/21+5/21+5/21-5/21-5/2    orthogonal lifted from D10
ρ172200022200-1+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
ρ182200022-200-1-5/2-1+5/2-2-200-1+5/2-1-5/2-1-5/2-1+5/2-1-5/21+5/21-5/2-1+5/21-5/21-5/21+5/21+5/2    orthogonal lifted from D10
ρ192-20000002i-2i222-2-2--2-2-20000002-2-22    complex lifted from C4○D8
ρ202-2000000-2i2i222-2--2-2-2-20000002-2-22    complex lifted from C4○D8
ρ212-2000000-2i2i22-22-2--2-2-2000000-222-2    complex lifted from C4○D8
ρ222-20000002i-2i22-22--2-2-2-2000000-222-2    complex lifted from C4○D8
ρ2344000-40000-1+5-1-50000-1-5-1+51-51+500000000    orthogonal lifted from D4×D5
ρ2444000-40000-1-5-1+50000-1+5-1-51+51-500000000    orthogonal lifted from D4×D5
ρ254-400000000-1-5-1+5-2222001-51+500000087ζ5487ζ585ζ5485ζ583ζ5483ζ58ζ548ζ583ζ5383ζ528ζ538ζ52ζ83ζ5383ζ528ζ538ζ52    orthogonal faithful
ρ264-400000000-1-5-1+522-22001-51+500000083ζ5483ζ58ζ548ζ587ζ5487ζ585ζ5485ζ5ζ83ζ5383ζ528ζ538ζ5283ζ5383ζ528ζ538ζ52    orthogonal faithful
ρ274-400000000-1+5-1-5-2222001+51-5000000ζ83ζ5383ζ528ζ538ζ5283ζ5383ζ528ζ538ζ5283ζ5483ζ58ζ548ζ587ζ5487ζ585ζ5485ζ5    orthogonal faithful
ρ284-400000000-1+5-1-522-22001+51-500000083ζ5383ζ528ζ538ζ52ζ83ζ5383ζ528ζ538ζ5287ζ5487ζ585ζ5485ζ583ζ5483ζ58ζ548ζ5    orthogonal faithful

Smallest permutation representation of Q8.D10
On 80 points
Generators in S80
(1 41 11 51)(2 52 12 42)(3 43 13 53)(4 54 14 44)(5 45 15 55)(6 56 16 46)(7 47 17 57)(8 58 18 48)(9 49 19 59)(10 60 20 50)(21 73 31 63)(22 64 32 74)(23 75 33 65)(24 66 34 76)(25 77 35 67)(26 68 36 78)(27 79 37 69)(28 70 38 80)(29 61 39 71)(30 72 40 62)
(1 37 11 27)(2 70 12 80)(3 39 13 29)(4 72 14 62)(5 21 15 31)(6 74 16 64)(7 23 17 33)(8 76 18 66)(9 25 19 35)(10 78 20 68)(22 56 32 46)(24 58 34 48)(26 60 36 50)(28 42 38 52)(30 44 40 54)(41 79 51 69)(43 61 53 71)(45 63 55 73)(47 65 57 75)(49 67 59 77)
(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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 40)(18 39)(19 38)(20 37)(41 68)(42 67)(43 66)(44 65)(45 64)(46 63)(47 62)(48 61)(49 80)(50 79)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 70)(60 69)

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

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

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

Q8.D10 is a maximal subgroup of
D401C4  Q16.F5  C16⋊D10  SD323D5  Q32⋊D5  D805C2  Q165F5  Q16⋊F5  D20.30D4  D5×C4○D8  D815D10  D40⋊C22  C40.C23  D120⋊C2  D405S3  D20.D6  D20.16D6  D1208C2  Dic5.7S4
Q8.D10 is a maximal quotient of
Dic57SD16  Q8.Dic10  Q8⋊Dic5⋊C2  Q82D5⋊C4  D102SD16  D204D4  (C2×C8).D10  D20.12D4  D4012C4  C8.6Dic10  C8.27(C4×D5)  C87D20  C2.D8⋊D5  D20.2Q8  Q16×Dic5  (C2×Q16)⋊D5  D20.17D4  D103Q16  C40.28D4  D120⋊C2  D405S3  D20.D6  D20.16D6  D1208C2

Matrix representation of Q8.D10 in GL4(𝔽41) generated by

1000
0100
00137
002140
,
1000
0100
003236
0009
,
1600
35600
001119
001330
,
1000
354000
00034
00350
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,21,0,0,37,40],[1,0,0,0,0,1,0,0,0,0,32,0,0,0,36,9],[1,35,0,0,6,6,0,0,0,0,11,13,0,0,19,30],[1,35,0,0,0,40,0,0,0,0,0,35,0,0,34,0] >;

Q8.D10 in GAP, Magma, Sage, TeX

Q_8.D_{10}
% in TeX

G:=Group("Q8.D10");
// GroupNames label

G:=SmallGroup(160,140);
// by ID

G=gap.SmallGroup(160,140);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,103,362,116,86,297,159,69,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^4=d^2=1,b^2=c^10=a^2,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=a^2*c^9>;
// generators/relations

Export

Character table of Q8.D10 in TeX

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