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G = D10.1S4order 480 = 25·3·5

1st non-split extension by D10 of S4 acting via S4/A4=C2

non-abelian, soluble

Aliases: D10.1S4, GL2(F3):2D5, SL2(F3).5D10, (Q8xD5):1S3, C2.9(D5xS4), C10.6(C2xS4), Q8.6(S3xD5), (C5xQ8).6D6, Q8.D15:6C2, C5:2(Q8.D6), (D5xSL2(F3)):1C2, (C5xGL2(F3)):2C2, (C5xSL2(F3)).5C22, SmallGroup(480,972)

Series: Derived Chief Lower central Upper central

C1C2Q8C5xSL2(F3) — D10.1S4
C1C2Q8C5xQ8C5xSL2(F3)D5xSL2(F3) — D10.1S4
C5xSL2(F3) — D10.1S4
C1C2

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

Subgroups: 578 in 78 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C5, S3, C6, C8, C2xC4, D4, Q8, Q8, D5, C10, C10, Dic3, D6, C2xC6, C15, M4(2), SD16, Q16, C2xQ8, C4oD4, Dic5, C20, D10, C2xC10, SL2(F3), C3:D4, C5xS3, C3xD5, C30, C8.C22, C5:2C8, C40, Dic10, C4xD5, C2xDic5, C5:D4, C5xD4, C5xQ8, CSU2(F3), GL2(F3), C2xSL2(F3), Dic15, C6xD5, S3xC10, C8:D5, Dic20, D4.D5, C5:Q16, C5xSD16, D4:2D5, Q8xD5, Q8.D6, C15:D4, C5xSL2(F3), SD16:D5, C5xGL2(F3), Q8.D15, D5xSL2(F3), D10.1S4
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, C2xS4, S3xD5, Q8.D6, D5xS4, D10.1S4

Character table of D10.1S4

 class 12A2B2C34A4B4C5A5B6A6B6C8A8B10A10B10C10D15A15B20A20B30A30B40A40B40C40D
 size 1110128630602284040126022242416161212161612121212
ρ111111111111111111111111111111    trivial
ρ211-1-111-11111-1-1-1111-1-1111111-1-1-1-1    linear of order 2
ρ3111-1111-111111-1-111-1-1111111-1-1-1-1    linear of order 2
ρ411-1111-1-1111-1-11-111111111111111    linear of order 2
ρ52220-122022-1-1-1002200-1-122-1-10000    orthogonal lifted from S3
ρ622-20-12-2022-111002200-1-122-1-10000    orthogonal lifted from D6
ρ722022200-1+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
ρ822022200-1-5/2-1+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
ρ9220-22200-1+5/2-1-5/2200-20-1-5/2-1+5/21+5/21-5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1+5/21+5/21+5/21-5/21-5/2    orthogonal lifted from D10
ρ10220-22200-1-5/2-1+5/2200-20-1+5/2-1-5/21-5/21+5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/21-5/21-5/21+5/21+5/2    orthogonal lifted from D10
ρ1133310-1-1133000-1-1331100-1-100-1-1-1-1    orthogonal lifted from S4
ρ1233-310-11-133000-11331100-1-100-1-1-1-1    orthogonal lifted from C2xS4
ρ13333-10-1-1-1330001133-1-100-1-1001111    orthogonal lifted from S4
ρ1433-3-10-111330001-133-1-100-1-1001111    orthogonal lifted from C2xS4
ρ154400-2400-1-5-1+5-20000-1+5-1-5001-5/21+5/2-1-5-1+51-5/21+5/20000    orthogonal lifted from S3xD5
ρ164400-2400-1+5-1-5-20000-1-5-1+5001+5/21-5/2-1+5-1-51+5/21-5/20000    orthogonal lifted from S3xD5
ρ174-400-2000-1+5-1-5200001+51-5001+5/21-5/200-1-5/2-1+5/2ζ83ζ5383ζ528ζ538ζ5283ζ5383ζ528ζ538ζ52ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5483ζ58ζ548ζ5    symplectic faithful, Schur index 2
ρ184-400-2000-1+5-1-5200001+51-5001+5/21-5/200-1-5/2-1+5/283ζ5383ζ528ζ538ζ52ζ83ζ5383ζ528ζ538ζ52ζ83ζ5483ζ58ζ548ζ5ζ87ζ5487ζ585ζ5485ζ5    symplectic faithful, Schur index 2
ρ194-400-20004420000-4-400-2-200220000    symplectic lifted from Q8.D6, Schur index 2
ρ204-400-2000-1-5-1+5200001-51+5001-5/21+5/200-1+5/2-1-5/2ζ83ζ5483ζ58ζ548ζ5ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5383ζ528ζ538ζ5283ζ5383ζ528ζ538ζ52    symplectic faithful, Schur index 2
ρ214-400-2000-1-5-1+5200001-51+5001-5/21+5/200-1+5/2-1-5/2ζ87ζ5487ζ585ζ5485ζ5ζ83ζ5483ζ58ζ548ζ583ζ5383ζ528ζ538ζ52ζ83ζ5383ζ528ζ538ζ52    symplectic faithful, Schur index 2
ρ224-400100044-1--3-300-4-4001100-1-10000    complex lifted from Q8.D6
ρ234-400100044-1-3--300-4-4001100-1-10000    complex lifted from Q8.D6
ρ24660-20-200-3+35/2-3-35/200020-3-35/2-3+35/21+5/21-5/2001-5/21+5/200-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D5xS4
ρ2566020-200-3-35/2-3+35/2000-20-3+35/2-3-35/2-1+5/2-1-5/2001+5/21-5/2001-5/21-5/21+5/21+5/2    orthogonal lifted from D5xS4
ρ26660-20-200-3-35/2-3+35/200020-3+35/2-3-35/21-5/21+5/2001+5/21-5/200-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D5xS4
ρ2766020-200-3+35/2-3-35/2000-20-3-35/2-3+35/2-1-5/2-1+5/2001-5/21+5/2001+5/21+5/21-5/21-5/2    orthogonal lifted from D5xS4
ρ288-8002000-2+25-2-25-200002+252-2500-1-5/2-1+5/2001+5/21-5/20000    symplectic faithful, Schur index 2
ρ298-8002000-2-25-2+25-200002-252+2500-1+5/2-1-5/2001-5/21+5/20000    symplectic faithful, Schur index 2

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

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

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

Matrix representation of D10.1S4 in GL6(F241)

01900000
521890000
00240000
00024000
00002400
00000240
,
891680000
1251520000
00017170171
00700171171
00171700171
007070700
,
100000
010000
000010
000001
00240000
00024000
,
100000
010000
00000240
000010
00024000
001000
,
100000
010000
001000
000001
00024000
00002400
,
24000000
02400000
001000
000010
000100
00000240

G:=sub<GL(6,GF(241))| [0,52,0,0,0,0,190,189,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[89,125,0,0,0,0,168,152,0,0,0,0,0,0,0,70,171,70,0,0,171,0,70,70,0,0,70,171,0,70,0,0,171,171,171,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,240,0,0,0,0,1,0,0,0,0,240,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,1,0,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,240] >;

D10.1S4 in GAP, Magma, Sage, TeX

D_{10}._1S_4
% in TeX

G:=Group("D10.1S4");
// GroupNames label

G:=SmallGroup(480,972);
// by ID

G=gap.SmallGroup(480,972);
# by ID

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

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

Export

Character table of D10.1S4 in TeX

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