Aliases: Dic5.7S4, CSU2(𝔽3)⋊3D5, SL2(𝔽3).3D10, C2.6(D5×S4), C10.3(C2×S4), Q8⋊D15⋊3C2, Q8.3(S3×D5), (C5×Q8).3D6, C5⋊1(C4.6S4), Q8⋊2D5⋊2S3, Dic5.A4⋊2C2, (C5×CSU2(𝔽3))⋊1C2, (C5×SL2(𝔽3)).3C22, SmallGroup(480,969)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C5×SL2(𝔽3) — Dic5.7S4 |
| C5×SL2(𝔽3) — Dic5.7S4 |
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, C2×C4, D4, Q8, Q8, D5, C10, Dic3, C12, D6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, D10, SL2(𝔽3), C4×S3, D15, C30, C4○D8, C5⋊2C8, C40, C4×D5, D20, C5×Q8, C5×Q8, CSU2(𝔽3), GL2(𝔽3), C4.A4, C5×Dic3, C3×Dic5, D30, C8×D5, D40, Q8⋊D5, C5×Q16, Q8⋊2D5, Q8⋊2D5, C4.6S4, D30.C2, C5×SL2(𝔽3), Q8.D10, C5×CSU2(𝔽3), Q8⋊D15, Dic5.A4, Dic5.7S4
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, C2×S4, S3×D5, C4.6S4, D5×S4, 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 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6 | 8A | 8B | 8C | 8D | 10A | 10B | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 30A | 30B | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 30 | 60 | 8 | 5 | 5 | 6 | 12 | 2 | 2 | 8 | 6 | 6 | 30 | 30 | 2 | 2 | 40 | 40 | 16 | 16 | 12 | 12 | 24 | 24 | 16 | 16 | 12 | 12 | 12 | 12 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | S3 | D5 | D6 | D10 | C4.6S4 | S4 | C2×S4 | S3×D5 | C4.6S4 | Dic5.7S4 | D5×S4 | Dic5.7S4 |
| kernel | Dic5.7S4 | C5×CSU2(𝔽3) | Q8⋊D15 | Dic5.A4 | Q8⋊2D5 | CSU2(𝔽3) | C5×Q8 | SL2(𝔽3) | C5 | Dic5 | C10 | Q8 | C5 | C1 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of Dic5.7S4 ►in GL4(𝔽241) generated by
| 240 | 1 | 0 | 0 |
| 50 | 190 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 51 | 1 | 0 | 0 |
| 51 | 190 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 68 | 174 |
| 0 | 0 | 105 | 173 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 173 | 105 |
| 0 | 0 | 174 | 68 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 67 | 135 |
| 0 | 0 | 68 | 173 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 177 | 0 |
| 0 | 0 | 64 | 64 |
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