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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, (Q8×D5)⋊2S3, C10.7(C2×S4), Q8⋊D155C2, C2.10(D5×S4), (C5×Q8).7D6, Q8.7(S3×D5), C51(Q8.D6), (D5×SL2(𝔽3))⋊2C2, (C5×CSU2(𝔽3))⋊4C2, (C5×SL2(𝔽3)).6C22, SmallGroup(480,973)

Series: Derived Chief Lower central Upper central

C1C2Q8C5×SL2(𝔽3) — D10.2S4
C1C2Q8C5×Q8C5×SL2(𝔽3)D5×SL2(𝔽3) — D10.2S4
C5×SL2(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, C2×C4, D4, Q8, Q8, D5, C10, Dic3, D6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, D10, D10, SL2(𝔽3), C3⋊D4, C3×D5, D15, C30, C8.C22, C52C8, C40, Dic10, C4×D5, D20, C5×Q8, C5×Q8, CSU2(𝔽3), GL2(𝔽3), C2×SL2(𝔽3), C5×Dic3, C6×D5, D30, C8⋊D5, C40⋊C2, Q8⋊D5, C5⋊Q16, C5×Q16, Q8×D5, Q82D5, Q8.D6, C3⋊D20, C5×SL2(𝔽3), Q16⋊D5, C5×CSU2(𝔽3), Q8⋊D15, D5×SL2(𝔽3), D10.2S4
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, C2×S4, S3×D5, Q8.D6, D5×S4, 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 C2×S4
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 C2×S4
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 S3×D5
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 S3×D5
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 D5×S4
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 D5×S4
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 D5×S4
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 D5×S4
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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