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G = D10.2S4order 480 = 25·3·5

2nd non-split extension by D10 of S4 acting via S4/A4=C2

non-abelian, soluble

Aliases: D10.2S4, CSU2(F3):2D5, SL2(F3).6D10, (Q8xD5):2S3, C10.7(C2xS4), Q8:D15:5C2, C2.10(D5xS4), (C5xQ8).7D6, Q8.7(S3xD5), C5:1(Q8.D6), (D5xSL2(F3)):2C2, (C5xCSU2(F3)):4C2, (C5xSL2(F3)).6C22, SmallGroup(480,973)

Series: Derived Chief Lower central Upper central

C1C2Q8C5xSL2(F3) — D10.2S4
C1C2Q8C5xQ8C5xSL2(F3)D5xSL2(F3) — D10.2S4
C5xSL2(F3) — D10.2S4
C1C2

Generators and relations for D10.2S4
 G = < a,b,c,d,e,f | a10=b2=e3=1, c2=d2=f2=a5, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=a5b, dcd-1=a5c, ece-1=a5cd, fcf-1=cd, ede-1=c, fdf-1=a5d, fef-1=e-1 >

Subgroups: 658 in 78 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C5, S3, C6, C8, C2xC4, D4, Q8, Q8, D5, C10, Dic3, D6, C2xC6, C15, M4(2), SD16, Q16, C2xQ8, C4oD4, Dic5, C20, D10, D10, SL2(F3), C3:D4, C3xD5, D15, C30, C8.C22, C5:2C8, C40, Dic10, C4xD5, D20, C5xQ8, C5xQ8, CSU2(F3), GL2(F3), C2xSL2(F3), C5xDic3, C6xD5, D30, C8:D5, C40:C2, Q8:D5, C5:Q16, C5xQ16, Q8xD5, Q8:2D5, Q8.D6, C3:D20, C5xSL2(F3), Q16:D5, C5xCSU2(F3), Q8:D15, D5xSL2(F3), D10.2S4
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, C2xS4, S3xD5, Q8.D6, D5xS4, D10.2S4

Character table of D10.2S4

 class 12A2B2C34A4B4C5A5B6A6B6C8A8B10A10B15A15B20A20B20C20D30A30B40A40B40C40D
 size 1110608612302284040126022161612122424161612121212
ρ111111111111111111111111111111    trivial
ρ2111-111-1111111-1-1111111-1-111-1-1-1-1    linear of order 2
ρ311-1-1111-1111-1-11-111111111111111    linear of order 2
ρ411-1111-1-1111-1-1-11111111-1-111-1-1-1-1    linear of order 2
ρ522-20-120-222-1110022-1-12200-1-10000    orthogonal lifted from D6
ρ62220-120222-1-1-10022-1-12200-1-10000    orthogonal lifted from S3
ρ7220022-20-1-5/2-1+5/2200-20-1-5/2-1+5/2-1-5/2-1+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
ρ822002220-1+5/2-1-5/220020-1+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+5/2-1-5/2-1-5/2    orthogonal lifted from D5
ρ9220022-20-1+5/2-1-5/2200-20-1+5/2-1-5/2-1+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
ρ1022002220-1-5/2-1+5/220020-1-5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D5
ρ1133-3-10-11133000-113300-1-11100-1-1-1-1    orthogonal lifted from C2xS4
ρ1233310-11-133000-1-13300-1-11100-1-1-1-1    orthogonal lifted from S4
ρ13333-10-1-1-133000113300-1-1-1-1001111    orthogonal lifted from S4
ρ1433-310-1-11330001-13300-1-1-1-1001111    orthogonal lifted from C2xS4
ρ154400-2400-1+5-1-5-20000-1+5-1-51-5/21+5/2-1-5-1+5001-5/21+5/20000    orthogonal lifted from S3xD5
ρ164400-2400-1-5-1+5-20000-1-5-1+51+5/21-5/2-1+5-1-5001+5/21-5/20000    orthogonal lifted from S3xD5
ρ174-400-20004420000-4-4-2-20000220000    symplectic lifted from Q8.D6, Schur index 2
ρ184-400100044-1--3-300-4-4110000-1-10000    complex lifted from Q8.D6
ρ194-400100044-1-3--300-4-4110000-1-10000    complex lifted from Q8.D6
ρ204-400-2000-1-5-1+5200001+51-51+5/21-5/20000-1-5/2-1+5/283ζ5383ζ528ζ538ζ52ζ83ζ5383ζ528ζ538ζ52ζ83ζ5483ζ58ζ548ζ5ζ87ζ5487ζ585ζ5485ζ5    complex faithful
ρ214-400-2000-1+5-1-5200001-51+51-5/21+5/20000-1+5/2-1-5/2ζ83ζ5483ζ58ζ548ζ5ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5383ζ528ζ538ζ5283ζ5383ζ528ζ538ζ52    complex faithful
ρ224-400-2000-1-5-1+5200001+51-51+5/21-5/20000-1-5/2-1+5/2ζ83ζ5383ζ528ζ538ζ5283ζ5383ζ528ζ538ζ52ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5483ζ58ζ548ζ5    complex faithful
ρ234-400-2000-1+5-1-5200001-51+51-5/21+5/20000-1+5/2-1-5/2ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5483ζ58ζ548ζ583ζ5383ζ528ζ538ζ52ζ83ζ5383ζ528ζ538ζ52    complex faithful
ρ2466000-2-20-3+35/2-3-35/200020-3+35/2-3-35/2001+5/21-5/21+5/21-5/200-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D5xS4
ρ2566000-220-3+35/2-3-35/2000-20-3+35/2-3-35/2001+5/21-5/2-1-5/2-1+5/2001-5/21-5/21+5/21+5/2    orthogonal lifted from D5xS4
ρ2666000-2-20-3-35/2-3+35/200020-3-35/2-3+35/2001-5/21+5/21-5/21+5/200-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D5xS4
ρ2766000-220-3-35/2-3+35/2000-20-3-35/2-3+35/2001-5/21+5/2-1+5/2-1-5/2001+5/21+5/21-5/21-5/2    orthogonal lifted from D5xS4
ρ288-8002000-2+25-2-25-200002-252+25-1+5/2-1-5/200001-5/21+5/20000    orthogonal faithful, Schur index 2
ρ298-8002000-2-25-2+25-200002+252-25-1-5/2-1+5/200001+5/21-5/20000    orthogonal faithful, Schur index 2

Smallest permutation representation of D10.2S4
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 76)(2 75)(3 74)(4 73)(5 72)(6 71)(7 80)(8 79)(9 78)(10 77)(11 52)(12 51)(13 60)(14 59)(15 58)(16 57)(17 56)(18 55)(19 54)(20 53)(21 42)(22 41)(23 50)(24 49)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 64)(32 63)(33 62)(34 61)(35 70)(36 69)(37 68)(38 67)(39 66)(40 65)
(1 69 6 64)(2 70 7 65)(3 61 8 66)(4 62 9 67)(5 63 10 68)(11 22 16 27)(12 23 17 28)(13 24 18 29)(14 25 19 30)(15 26 20 21)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)(41 57 46 52)(42 58 47 53)(43 59 48 54)(44 60 49 55)(45 51 50 56)
(1 58 6 53)(2 59 7 54)(3 60 8 55)(4 51 9 56)(5 52 10 57)(11 77 16 72)(12 78 17 73)(13 79 18 74)(14 80 19 75)(15 71 20 76)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 63 46 68)(42 64 47 69)(43 65 48 70)(44 66 49 61)(45 67 50 62)
(11 22 32)(12 23 33)(13 24 34)(14 25 35)(15 26 36)(16 27 37)(17 28 38)(18 29 39)(19 30 40)(20 21 31)(41 63 52)(42 64 53)(43 65 54)(44 66 55)(45 67 56)(46 68 57)(47 69 58)(48 70 59)(49 61 60)(50 62 51)
(1 72 6 77)(2 73 7 78)(3 74 8 79)(4 75 9 80)(5 76 10 71)(11 58 16 53)(12 59 17 54)(13 60 18 55)(14 51 19 56)(15 52 20 57)(21 68 26 63)(22 69 27 64)(23 70 28 65)(24 61 29 66)(25 62 30 67)(31 46 36 41)(32 47 37 42)(33 48 38 43)(34 49 39 44)(35 50 40 45)

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

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

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

Matrix representation of D10.2S4 in GL8(F241)

5205200000
0520520000
189024000000
018902400000
0000240000
0000024000
0000002400
0000000240
,
5205200000
0520520000
240018900000
024001890000
0000017170171
0000700171171
0000171700171
00007070700
,
10000000
01000000
00100000
00010000
00000010
00000001
0000240000
0000024000
,
10000000
01000000
00100000
00010000
0000000240
00000010
0000024000
00001000
,
2401000000
2400000000
0024010000
0024000000
00001000
00000001
0000024000
0000002400
,
01000000
10000000
00010000
00100000
000007017170
000070171070
000017107070
00007070700

G:=sub<GL(8,GF(241))| [52,0,189,0,0,0,0,0,0,52,0,189,0,0,0,0,52,0,240,0,0,0,0,0,0,52,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[52,0,240,0,0,0,0,0,0,52,0,240,0,0,0,0,52,0,189,0,0,0,0,0,0,52,0,189,0,0,0,0,0,0,0,0,0,70,171,70,0,0,0,0,171,0,70,70,0,0,0,0,70,171,0,70,0,0,0,0,171,171,171,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0],[240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,70,171,70,0,0,0,0,70,171,0,70,0,0,0,0,171,0,70,70,0,0,0,0,70,70,70,0] >;

D10.2S4 in GAP, Magma, Sage, TeX

D_{10}._2S_4
% in TeX

G:=Group("D10.2S4");
// GroupNames label

G:=SmallGroup(480,973);
// by ID

G=gap.SmallGroup(480,973);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,3389,1688,93,1347,2111,3168,172,1272,1909,285,124]);
// Polycyclic

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

Export

Character table of D10.2S4 in TeX

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