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G = Dic5.7S4order 480 = 25·3·5

2nd non-split extension by Dic5 of S4 acting through Inn(Dic5)

non-abelian, soluble

Aliases: Dic5.7S4, CSU2(F3):3D5, SL2(F3).3D10, C2.6(D5xS4), C10.3(C2xS4), Q8:D15:3C2, Q8.3(S3xD5), (C5xQ8).3D6, C5:1(C4.6S4), Q8:2D5:2S3, Dic5.A4:2C2, (C5xCSU2(F3)):1C2, (C5xSL2(F3)).3C22, SmallGroup(480,969)

Series: Derived Chief Lower central Upper central

C1C2Q8C5xSL2(F3) — Dic5.7S4
C1C2Q8C5xQ8C5xSL2(F3)Dic5.A4 — Dic5.7S4
C5xSL2(F3) — Dic5.7S4
C1C2

Generators and relations for Dic5.7S4
 G = < a,b,c,d,e,f | a10=e3=1, b2=c2=d2=f2=a5, bab-1=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=a5c, ece-1=a5cd, fcf-1=cd, ede-1=c, fdf-1=a5d, fef-1=e-1 >

Subgroups: 634 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, C12, D6, C15, C2xC8, D8, SD16, Q16, C4oD4, Dic5, C20, D10, SL2(F3), C4xS3, D15, C30, C4oD8, C5:2C8, C40, C4xD5, D20, C5xQ8, C5xQ8, CSU2(F3), GL2(F3), C4.A4, C5xDic3, C3xDic5, D30, C8xD5, D40, Q8:D5, C5xQ16, Q8:2D5, Q8:2D5, C4.6S4, D30.C2, C5xSL2(F3), Q8.D10, C5xCSU2(F3), Q8:D15, Dic5.A4, Dic5.7S4
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, C2xS4, S3xD5, C4.6S4, D5xS4, Dic5.7S4

Smallest permutation representation of Dic5.7S4
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 6 71)(2 75 7 80)(3 74 8 79)(4 73 9 78)(5 72 10 77)(11 52 16 57)(12 51 17 56)(13 60 18 55)(14 59 19 54)(15 58 20 53)(21 43 26 48)(22 42 27 47)(23 41 28 46)(24 50 29 45)(25 49 30 44)(31 69 36 64)(32 68 37 63)(33 67 38 62)(34 66 39 61)(35 65 40 70)
(1 54 6 59)(2 55 7 60)(3 56 8 51)(4 57 9 52)(5 58 10 53)(11 78 16 73)(12 79 17 74)(13 80 18 75)(14 71 19 76)(15 72 20 77)(21 35 26 40)(22 36 27 31)(23 37 28 32)(24 38 29 33)(25 39 30 34)(41 63 46 68)(42 64 47 69)(43 65 48 70)(44 66 49 61)(45 67 50 62)
(1 62 6 67)(2 63 7 68)(3 64 8 69)(4 65 9 70)(5 66 10 61)(11 26 16 21)(12 27 17 22)(13 28 18 23)(14 29 19 24)(15 30 20 25)(31 79 36 74)(32 80 37 75)(33 71 38 76)(34 72 39 77)(35 73 40 78)(41 60 46 55)(42 51 47 56)(43 52 48 57)(44 53 49 58)(45 54 50 59)
(11 40 21)(12 31 22)(13 32 23)(14 33 24)(15 34 25)(16 35 26)(17 36 27)(18 37 28)(19 38 29)(20 39 30)(41 60 68)(42 51 69)(43 52 70)(44 53 61)(45 54 62)(46 55 63)(47 56 64)(48 57 65)(49 58 66)(50 59 67)
(1 76 6 71)(2 77 7 72)(3 78 8 73)(4 79 9 74)(5 80 10 75)(11 42 16 47)(12 43 17 48)(13 44 18 49)(14 45 19 50)(15 46 20 41)(21 51 26 56)(22 52 27 57)(23 53 28 58)(24 54 29 59)(25 55 30 60)(31 70 36 65)(32 61 37 66)(33 62 38 67)(34 63 39 68)(35 64 40 69)

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

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

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

32 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D5A5B 6 8A8B8C8D10A10B12A12B15A15B20A20B20C20D30A30B40A40B40C40D
order122234444556888810101212151520202020303040404040
size113060855612228663030224040161612122424161612121212

32 irreducible representations

dim1111222223344468
type++++++++++++++
imageC1C2C2C2S3D5D6D10C4.6S4S4C2xS4S3xD5C4.6S4Dic5.7S4D5xS4Dic5.7S4
kernelDic5.7S4C5xCSU2(F3)Q8:D15Dic5.A4Q8:2D5CSU2(F3)C5xQ8SL2(F3)C5Dic5C10Q8C5C1C2C1
# reps1111121242222442

Matrix representation of Dic5.7S4 in GL4(F241) generated by

240100
5019000
002400
000240
,
51100
5119000
00640
00064
,
1000
0100
0068174
00105173
,
1000
0100
00173105
0017468
,
1000
0100
0067135
0068173
,
1000
0100
001770
006464
G:=sub<GL(4,GF(241))| [240,50,0,0,1,190,0,0,0,0,240,0,0,0,0,240],[51,51,0,0,1,190,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,68,105,0,0,174,173],[1,0,0,0,0,1,0,0,0,0,173,174,0,0,105,68],[1,0,0,0,0,1,0,0,0,0,67,68,0,0,135,173],[1,0,0,0,0,1,0,0,0,0,177,64,0,0,0,64] >;

Dic5.7S4 in GAP, Magma, Sage, TeX

{\rm Dic}_5._7S_4
% in TeX

G:=Group("Dic5.7S4");
// GroupNames label

G:=SmallGroup(480,969);
// by ID

G=gap.SmallGroup(480,969);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,1680,1688,93,1347,2111,3168,172,1272,1909,285,124]);
// Polycyclic

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

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