non-abelian, soluble, monomial
Aliases: Dic5.5S4, C5⋊(A4⋊C8), A4⋊(C5⋊C8), (C2×A4).F5, (C5×A4)⋊1C8, C23.(C3⋊F5), (C10×A4).1C4, C22⋊(C15⋊C8), C2.1(A4⋊F5), C10.3(A4⋊C4), (A4×Dic5).3C2, (C22×C10).Dic3, (C22×Dic5).2S3, (C2×C10)⋊(C3⋊C8), SmallGroup(480,963)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×A4 — Dic5.S4 |
Generators and relations for Dic5.S4
G = < a,b,c,d,e,f | a10=c2=d2=e3=1, b2=a5, f2=b, bab-1=a-1, ac=ca, ad=da, ae=ea, faf-1=a7, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf-1=cd=dc, ede-1=c, df=fd, fef-1=e-1 >
3C2
3C2
4C3
3C22
3C22
5C4
15C4
4C6
3C10
3C10
4C15
15C2×C4
15C2×C4
30C8
30C8
20C12
3Dic5
4C30
15C2×C8
15C2×C8
20C3⋊C8
15C22⋊C8
Character table of Dic5.S4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5 | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 12A | 12B | 15A | 15B | 30A | 30B | |
| size | 1 | 1 | 3 | 3 | 8 | 5 | 5 | 15 | 15 | 4 | 8 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 4 | 12 | 12 | 40 | 40 | 16 | 16 | 16 | 16 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | i | -i | -i | -i | -i | i | i | i | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | i | i | i | i | -i | -i | -i | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 1 | -1 | 1 | -1 | 1 | -i | i | i | -i | 1 | -1 | ζ8 | ζ87 | ζ87 | ζ83 | ζ83 | ζ85 | ζ85 | ζ8 | -1 | 1 | -1 | -i | i | 1 | 1 | -1 | -1 | linear of order 8 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | i | -i | -i | i | 1 | -1 | ζ87 | ζ8 | ζ8 | ζ85 | ζ85 | ζ83 | ζ83 | ζ87 | -1 | 1 | -1 | i | -i | 1 | 1 | -1 | -1 | linear of order 8 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | i | -i | -i | i | 1 | -1 | ζ83 | ζ85 | ζ85 | ζ8 | ζ8 | ζ87 | ζ87 | ζ83 | -1 | 1 | -1 | i | -i | 1 | 1 | -1 | -1 | linear of order 8 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | -i | i | i | -i | 1 | -1 | ζ85 | ζ83 | ζ83 | ζ87 | ζ87 | ζ8 | ζ8 | ζ85 | -1 | 1 | -1 | -i | i | 1 | 1 | -1 | -1 | linear of order 8 |
| ρ9 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 1 | 1 | -1 | -1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ11 | 2 | -2 | 2 | -2 | -1 | -2i | 2i | 2i | -2i | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | i | -i | -1 | -1 | 1 | 1 | complex lifted from C3⋊C8 |
| ρ12 | 2 | -2 | 2 | -2 | -1 | 2i | -2i | -2i | 2i | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -i | i | -1 | -1 | 1 | 1 | complex lifted from C3⋊C8 |
| ρ13 | 3 | 3 | -1 | -1 | 0 | 3 | 3 | -1 | -1 | 3 | 0 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ14 | 3 | 3 | -1 | -1 | 0 | 3 | 3 | -1 | -1 | 3 | 0 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ15 | 3 | 3 | -1 | -1 | 0 | -3 | -3 | 1 | 1 | 3 | 0 | -i | -i | i | -i | i | i | -i | i | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from A4⋊C4 |
| ρ16 | 3 | 3 | -1 | -1 | 0 | -3 | -3 | 1 | 1 | 3 | 0 | i | i | -i | i | -i | -i | i | -i | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from A4⋊C4 |
| ρ17 | 3 | -3 | -1 | 1 | 0 | 3i | -3i | i | -i | 3 | 0 | ζ83 | ζ8 | ζ85 | ζ85 | ζ8 | ζ83 | ζ87 | ζ87 | -3 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from A4⋊C8 |
| ρ18 | 3 | -3 | -1 | 1 | 0 | -3i | 3i | -i | i | 3 | 0 | ζ85 | ζ87 | ζ83 | ζ83 | ζ87 | ζ85 | ζ8 | ζ8 | -3 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from A4⋊C8 |
| ρ19 | 3 | -3 | -1 | 1 | 0 | 3i | -3i | i | -i | 3 | 0 | ζ87 | ζ85 | ζ8 | ζ8 | ζ85 | ζ87 | ζ83 | ζ83 | -3 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from A4⋊C8 |
| ρ20 | 3 | -3 | -1 | 1 | 0 | -3i | 3i | -i | i | 3 | 0 | ζ8 | ζ83 | ζ87 | ζ87 | ζ83 | ζ8 | ζ85 | ζ85 | -3 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from A4⋊C8 |
| ρ21 | 4 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ22 | 4 | -4 | 4 | -4 | 4 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | -1 | -1 | 1 | 1 | symplectic lifted from C5⋊C8, Schur index 2 |
| ρ23 | 4 | -4 | 4 | -4 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 1-√-15/2 | 1+√-15/2 | -1+√-15/2 | -1-√-15/2 | complex lifted from C15⋊C8 |
| ρ24 | 4 | -4 | 4 | -4 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 1+√-15/2 | 1-√-15/2 | -1-√-15/2 | -1+√-15/2 | complex lifted from C15⋊C8 |
| ρ25 | 4 | 4 | 4 | 4 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 1+√-15/2 | 1-√-15/2 | 1+√-15/2 | 1-√-15/2 | complex lifted from C3⋊F5 |
| ρ26 | 4 | 4 | 4 | 4 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 1-√-15/2 | 1+√-15/2 | 1-√-15/2 | 1+√-15/2 | complex lifted from C3⋊F5 |
| ρ27 | 12 | 12 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4⋊F5 |
| ρ28 | 12 | -12 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 120 points
Generators in S120
(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)(81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120)
(1 36 6 31)(2 35 7 40)(3 34 8 39)(4 33 9 38)(5 32 10 37)(11 44 16 49)(12 43 17 48)(13 42 18 47)(14 41 19 46)(15 50 20 45)(21 54 26 59)(22 53 27 58)(23 52 28 57)(24 51 29 56)(25 60 30 55)(61 94 66 99)(62 93 67 98)(63 92 68 97)(64 91 69 96)(65 100 70 95)(71 104 76 109)(72 103 77 108)(73 102 78 107)(74 101 79 106)(75 110 80 105)(81 114 86 119)(82 113 87 118)(83 112 88 117)(84 111 89 116)(85 120 90 115)
(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)(91 96)(92 97)(93 98)(94 99)(95 100)(101 106)(102 107)(103 108)(104 109)(105 110)
(1 6)(2 7)(3 8)(4 9)(5 10)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(81 86)(82 87)(83 88)(84 89)(85 90)(91 96)(92 97)(93 98)(94 99)(95 100)(111 116)(112 117)(113 118)(114 119)(115 120)
(1 19 29)(2 20 30)(3 11 21)(4 12 22)(5 13 23)(6 14 24)(7 15 25)(8 16 26)(9 17 27)(10 18 28)(31 41 51)(32 42 52)(33 43 53)(34 44 54)(35 45 55)(36 46 56)(37 47 57)(38 48 58)(39 49 59)(40 50 60)(61 81 71)(62 82 72)(63 83 73)(64 84 74)(65 85 75)(66 86 76)(67 87 77)(68 88 78)(69 89 79)(70 90 80)(91 111 101)(92 112 102)(93 113 103)(94 114 104)(95 115 105)(96 116 106)(97 117 107)(98 118 108)(99 119 109)(100 120 110)
(1 92 36 68 6 97 31 63)(2 95 35 65 7 100 40 70)(3 98 34 62 8 93 39 67)(4 91 33 69 9 96 38 64)(5 94 32 66 10 99 37 61)(11 108 44 72 16 103 49 77)(12 101 43 79 17 106 48 74)(13 104 42 76 18 109 47 71)(14 107 41 73 19 102 46 78)(15 110 50 80 20 105 45 75)(21 118 54 82 26 113 59 87)(22 111 53 89 27 116 58 84)(23 114 52 86 28 119 57 81)(24 117 51 83 29 112 56 88)(25 120 60 90 30 115 55 85)
G:=sub<Sym(120)| (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)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120), (1,36,6,31)(2,35,7,40)(3,34,8,39)(4,33,9,38)(5,32,10,37)(11,44,16,49)(12,43,17,48)(13,42,18,47)(14,41,19,46)(15,50,20,45)(21,54,26,59)(22,53,27,58)(23,52,28,57)(24,51,29,56)(25,60,30,55)(61,94,66,99)(62,93,67,98)(63,92,68,97)(64,91,69,96)(65,100,70,95)(71,104,76,109)(72,103,77,108)(73,102,78,107)(74,101,79,106)(75,110,80,105)(81,114,86,119)(82,113,87,118)(83,112,88,117)(84,111,89,116)(85,120,90,115), (11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80)(91,96)(92,97)(93,98)(94,99)(95,100)(101,106)(102,107)(103,108)(104,109)(105,110), (1,6)(2,7)(3,8)(4,9)(5,10)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(81,86)(82,87)(83,88)(84,89)(85,90)(91,96)(92,97)(93,98)(94,99)(95,100)(111,116)(112,117)(113,118)(114,119)(115,120), (1,19,29)(2,20,30)(3,11,21)(4,12,22)(5,13,23)(6,14,24)(7,15,25)(8,16,26)(9,17,27)(10,18,28)(31,41,51)(32,42,52)(33,43,53)(34,44,54)(35,45,55)(36,46,56)(37,47,57)(38,48,58)(39,49,59)(40,50,60)(61,81,71)(62,82,72)(63,83,73)(64,84,74)(65,85,75)(66,86,76)(67,87,77)(68,88,78)(69,89,79)(70,90,80)(91,111,101)(92,112,102)(93,113,103)(94,114,104)(95,115,105)(96,116,106)(97,117,107)(98,118,108)(99,119,109)(100,120,110), (1,92,36,68,6,97,31,63)(2,95,35,65,7,100,40,70)(3,98,34,62,8,93,39,67)(4,91,33,69,9,96,38,64)(5,94,32,66,10,99,37,61)(11,108,44,72,16,103,49,77)(12,101,43,79,17,106,48,74)(13,104,42,76,18,109,47,71)(14,107,41,73,19,102,46,78)(15,110,50,80,20,105,45,75)(21,118,54,82,26,113,59,87)(22,111,53,89,27,116,58,84)(23,114,52,86,28,119,57,81)(24,117,51,83,29,112,56,88)(25,120,60,90,30,115,55,85)>;
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)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120), (1,36,6,31)(2,35,7,40)(3,34,8,39)(4,33,9,38)(5,32,10,37)(11,44,16,49)(12,43,17,48)(13,42,18,47)(14,41,19,46)(15,50,20,45)(21,54,26,59)(22,53,27,58)(23,52,28,57)(24,51,29,56)(25,60,30,55)(61,94,66,99)(62,93,67,98)(63,92,68,97)(64,91,69,96)(65,100,70,95)(71,104,76,109)(72,103,77,108)(73,102,78,107)(74,101,79,106)(75,110,80,105)(81,114,86,119)(82,113,87,118)(83,112,88,117)(84,111,89,116)(85,120,90,115), (11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80)(91,96)(92,97)(93,98)(94,99)(95,100)(101,106)(102,107)(103,108)(104,109)(105,110), (1,6)(2,7)(3,8)(4,9)(5,10)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(81,86)(82,87)(83,88)(84,89)(85,90)(91,96)(92,97)(93,98)(94,99)(95,100)(111,116)(112,117)(113,118)(114,119)(115,120), (1,19,29)(2,20,30)(3,11,21)(4,12,22)(5,13,23)(6,14,24)(7,15,25)(8,16,26)(9,17,27)(10,18,28)(31,41,51)(32,42,52)(33,43,53)(34,44,54)(35,45,55)(36,46,56)(37,47,57)(38,48,58)(39,49,59)(40,50,60)(61,81,71)(62,82,72)(63,83,73)(64,84,74)(65,85,75)(66,86,76)(67,87,77)(68,88,78)(69,89,79)(70,90,80)(91,111,101)(92,112,102)(93,113,103)(94,114,104)(95,115,105)(96,116,106)(97,117,107)(98,118,108)(99,119,109)(100,120,110), (1,92,36,68,6,97,31,63)(2,95,35,65,7,100,40,70)(3,98,34,62,8,93,39,67)(4,91,33,69,9,96,38,64)(5,94,32,66,10,99,37,61)(11,108,44,72,16,103,49,77)(12,101,43,79,17,106,48,74)(13,104,42,76,18,109,47,71)(14,107,41,73,19,102,46,78)(15,110,50,80,20,105,45,75)(21,118,54,82,26,113,59,87)(22,111,53,89,27,116,58,84)(23,114,52,86,28,119,57,81)(24,117,51,83,29,112,56,88)(25,120,60,90,30,115,55,85) );
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),(81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120)], [(1,36,6,31),(2,35,7,40),(3,34,8,39),(4,33,9,38),(5,32,10,37),(11,44,16,49),(12,43,17,48),(13,42,18,47),(14,41,19,46),(15,50,20,45),(21,54,26,59),(22,53,27,58),(23,52,28,57),(24,51,29,56),(25,60,30,55),(61,94,66,99),(62,93,67,98),(63,92,68,97),(64,91,69,96),(65,100,70,95),(71,104,76,109),(72,103,77,108),(73,102,78,107),(74,101,79,106),(75,110,80,105),(81,114,86,119),(82,113,87,118),(83,112,88,117),(84,111,89,116),(85,120,90,115)], [(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(41,46),(42,47),(43,48),(44,49),(45,50),(51,56),(52,57),(53,58),(54,59),(55,60),(61,66),(62,67),(63,68),(64,69),(65,70),(71,76),(72,77),(73,78),(74,79),(75,80),(91,96),(92,97),(93,98),(94,99),(95,100),(101,106),(102,107),(103,108),(104,109),(105,110)], [(1,6),(2,7),(3,8),(4,9),(5,10),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(51,56),(52,57),(53,58),(54,59),(55,60),(61,66),(62,67),(63,68),(64,69),(65,70),(81,86),(82,87),(83,88),(84,89),(85,90),(91,96),(92,97),(93,98),(94,99),(95,100),(111,116),(112,117),(113,118),(114,119),(115,120)], [(1,19,29),(2,20,30),(3,11,21),(4,12,22),(5,13,23),(6,14,24),(7,15,25),(8,16,26),(9,17,27),(10,18,28),(31,41,51),(32,42,52),(33,43,53),(34,44,54),(35,45,55),(36,46,56),(37,47,57),(38,48,58),(39,49,59),(40,50,60),(61,81,71),(62,82,72),(63,83,73),(64,84,74),(65,85,75),(66,86,76),(67,87,77),(68,88,78),(69,89,79),(70,90,80),(91,111,101),(92,112,102),(93,113,103),(94,114,104),(95,115,105),(96,116,106),(97,117,107),(98,118,108),(99,119,109),(100,120,110)], [(1,92,36,68,6,97,31,63),(2,95,35,65,7,100,40,70),(3,98,34,62,8,93,39,67),(4,91,33,69,9,96,38,64),(5,94,32,66,10,99,37,61),(11,108,44,72,16,103,49,77),(12,101,43,79,17,106,48,74),(13,104,42,76,18,109,47,71),(14,107,41,73,19,102,46,78),(15,110,50,80,20,105,45,75),(21,118,54,82,26,113,59,87),(22,111,53,89,27,116,58,84),(23,114,52,86,28,119,57,81),(24,117,51,83,29,112,56,88),(25,120,60,90,30,115,55,85)]])
Matrix representation of Dic5.S4 ►in GL7(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 240 | 240 | 240 | 240 |
| 177 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 171 | 173 | 141 | 121 |
| 0 | 0 | 0 | 2 | 211 | 191 | 70 |
| 0 | 0 | 0 | 209 | 189 | 68 | 239 |
| 0 | 0 | 0 | 221 | 100 | 30 | 32 |
| 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 91 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 151 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 90 | 239 | 0 | 0 | 0 | 0 | 0 |
| 119 | 151 | 1 | 0 | 0 | 0 | 0 |
| 122 | 91 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 114 | 0 | 229 | 229 |
| 0 | 0 | 0 | 12 | 126 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 | 126 | 12 |
| 0 | 0 | 0 | 229 | 229 | 0 | 114 |
| 79 | 0 | 60 | 0 | 0 | 0 | 0 |
| 85 | 211 | 49 | 0 | 0 | 0 | 0 |
| 156 | 0 | 162 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 103 | 86 | 187 | 27 |
| 0 | 0 | 0 | 81 | 54 | 157 | 140 |
| 0 | 0 | 0 | 224 | 84 | 165 | 138 |
| 0 | 0 | 0 | 214 | 76 | 59 | 160 |
G:=sub<GL(7,GF(241))| [240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,240,0,0,0,1,0,0,240,0,0,0,0,1,0,240,0,0,0,0,0,1,240],[177,0,0,0,0,0,0,0,177,0,0,0,0,0,0,0,177,0,0,0,0,0,0,0,171,2,209,221,0,0,0,173,211,189,100,0,0,0,141,191,68,30,0,0,0,121,70,239,32],[240,0,91,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[240,151,0,0,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[90,119,122,0,0,0,0,239,151,91,0,0,0,0,0,1,0,0,0,0,0,0,0,0,114,12,0,229,0,0,0,0,126,12,229,0,0,0,229,12,126,0,0,0,0,229,0,12,114],[79,85,156,0,0,0,0,0,211,0,0,0,0,0,60,49,162,0,0,0,0,0,0,0,103,81,224,214,0,0,0,86,54,84,76,0,0,0,187,157,165,59,0,0,0,27,140,138,160] >;
Dic5.S4 in GAP, Magma, Sage, TeX
{\rm Dic}_5.S_4 % in TeX
G:=Group("Dic5.S4"); // GroupNames label
G:=SmallGroup(480,963);
// by ID
G=gap.SmallGroup(480,963);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,14,36,451,2524,1691,10085,1286,5886,2232]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^10=c^2=d^2=e^3=1,b^2=a^5,f^2=b,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a^7,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,e*c*e^-1=f*c*f^-1=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations
Export
Subgroup lattice of Dic5.S4 in TeX
Character table of Dic5.S4 in TeX