direct product, non-abelian, soluble
Aliases: C2×C6.5S4, C6⋊CSU2(𝔽3), SL2(𝔽3).7D6, C6.31(C2×S4), (C2×C6).14S4, (C6×Q8).6S3, (C3×Q8).14D6, C22.4(C3⋊S4), C3⋊2(C2×CSU2(𝔽3)), (C2×SL2(𝔽3)).2S3, (C6×SL2(𝔽3)).3C2, (C3×SL2(𝔽3)).7C22, C2.5(C2×C3⋊S4), Q8.1(C2×C3⋊S3), (C2×Q8).2(C3⋊S3), SmallGroup(288,910)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C3×SL2(𝔽3) — C2×C6.5S4 |
| C3×SL2(𝔽3) — C2×C6.5S4 |
Generators and relations for C2×C6.5S4
G = < a,b,c,d,e,f | a2=b6=e3=1, c2=d2=f2=b3, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=b-1, dcd-1=b3c, ece-1=b3cd, fcf-1=cd, ede-1=c, fdf-1=b3d, fef-1=e-1 >
Subgroups: 504 in 104 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, Q8, Q8, C32, Dic3, C12, C2×C6, C2×C6, C2×C8, Q16, C2×Q8, C2×Q8, C3×C6, C3⋊C8, SL2(𝔽3), Dic6, C2×Dic3, C2×C12, C3×Q8, C3×Q8, C2×Q16, C3⋊Dic3, C62, C2×C3⋊C8, C3⋊Q16, CSU2(𝔽3), C2×SL2(𝔽3), C2×Dic6, C6×Q8, C3×SL2(𝔽3), C2×C3⋊Dic3, C2×C3⋊Q16, C2×CSU2(𝔽3), C6.5S4, C6×SL2(𝔽3), C2×C6.5S4
Quotients: C1, C2, C22, S3, D6, C3⋊S3, S4, C2×C3⋊S3, CSU2(𝔽3), C2×S4, C3⋊S4, C2×CSU2(𝔽3), C6.5S4, C2×C3⋊S4, C2×C6.5S4
Character table of C2×C6.5S4
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 8A | 8B | 8C | 8D | 12A | 12B | |
| size | 1 | 1 | 1 | 1 | 2 | 8 | 8 | 8 | 6 | 6 | 36 | 36 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 18 | 18 | 18 | 18 | 12 | 12 | |
| 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 | 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 | 1 | 1 | linear of order 2 |
| ρ3 | 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 | -1 | 1 | linear of order 2 |
| ρ4 | 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 | -1 | 1 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -2 | -2 | -1 | -1 | 2 | -1 | 2 | -2 | 0 | 0 | 1 | -1 | 1 | -1 | -2 | -2 | 1 | 1 | 1 | 1 | -1 | 2 | 0 | 0 | 0 | 0 | 1 | -1 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | -2 | -2 | -1 | -1 | -1 | 2 | 2 | -2 | 0 | 0 | 1 | -1 | 1 | 2 | 1 | 1 | 1 | 1 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | 2 | 2 | -1 | 2 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ9 | 2 | 2 | -2 | -2 | 2 | -1 | -1 | -1 | 2 | -2 | 0 | 0 | -2 | 2 | -2 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | 2 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 2 | 2 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | -2 | -2 | -1 | 2 | -1 | -1 | 2 | -2 | 0 | 0 | 1 | -1 | 1 | -1 | 1 | 1 | -2 | -2 | 1 | 1 | 2 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | orthogonal lifted from D6 |
| ρ13 | 2 | -2 | 2 | -2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | √2 | √2 | -√2 | -√2 | 0 | 0 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ14 | 2 | -2 | -2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -√2 | √2 | -√2 | √2 | 0 | 0 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ15 | 2 | -2 | 2 | -2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -√2 | -√2 | √2 | √2 | 0 | 0 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ16 | 2 | -2 | -2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | √2 | -√2 | √2 | -√2 | 0 | 0 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ17 | 3 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from S4 |
| ρ18 | 3 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ19 | 3 | 3 | -3 | -3 | 3 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | -3 | 3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 1 | 1 | -1 | orthogonal lifted from C2×S4 |
| ρ20 | 3 | 3 | -3 | -3 | 3 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | -3 | 3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | 1 | -1 | orthogonal lifted from C2×S4 |
| ρ21 | 4 | -4 | 4 | -4 | -2 | 1 | 1 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 2 | -1 | 1 | -1 | 1 | 2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C6.5S4, Schur index 2 |
| ρ22 | 4 | -4 | -4 | 4 | -2 | 1 | -2 | 1 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -1 | -2 | 2 | 1 | -1 | 1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C6.5S4, Schur index 2 |
| ρ23 | 4 | -4 | -4 | 4 | 4 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 4 | -4 | -4 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ24 | 4 | -4 | 4 | -4 | -2 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -1 | -1 | 1 | 2 | -2 | -1 | 1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C6.5S4, Schur index 2 |
| ρ25 | 4 | -4 | -4 | 4 | -2 | 1 | 1 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 2 | 1 | -1 | 1 | -1 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C6.5S4, Schur index 2 |
| ρ26 | 4 | -4 | 4 | -4 | 4 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | -4 | -4 | 4 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ27 | 4 | -4 | -4 | 4 | -2 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -1 | 1 | -1 | -2 | 2 | 1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C6.5S4, Schur index 2 |
| ρ28 | 4 | -4 | 4 | -4 | -2 | 1 | -2 | 1 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -1 | 2 | -2 | -1 | 1 | -1 | 1 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C6.5S4, Schur index 2 |
| ρ29 | 6 | 6 | 6 | 6 | -3 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | -3 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from C3⋊S4 |
| ρ30 | 6 | 6 | -6 | -6 | -3 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | orthogonal lifted from C2×C3⋊S4 |
►On 96 points
Generators in S96
(1 51)(2 52)(3 53)(4 54)(5 49)(6 50)(7 83)(8 84)(9 79)(10 80)(11 81)(12 82)(13 76)(14 77)(15 78)(16 73)(17 74)(18 75)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 33)(26 34)(27 35)(28 36)(29 31)(30 32)(37 59)(38 60)(39 55)(40 56)(41 57)(42 58)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)
(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)
(1 29 4 26)(2 30 5 27)(3 25 6 28)(7 85 10 88)(8 86 11 89)(9 87 12 90)(13 91 16 94)(14 92 17 95)(15 93 18 96)(19 40 22 37)(20 41 23 38)(21 42 24 39)(31 54 34 51)(32 49 35 52)(33 50 36 53)(43 56 46 59)(44 57 47 60)(45 58 48 55)(61 80 64 83)(62 81 65 84)(63 82 66 79)(67 73 70 76)(68 74 71 77)(69 75 72 78)
(1 41 4 38)(2 42 5 39)(3 37 6 40)(7 14 10 17)(8 15 11 18)(9 16 12 13)(19 25 22 28)(20 26 23 29)(21 27 24 30)(31 44 34 47)(32 45 35 48)(33 46 36 43)(49 55 52 58)(50 56 53 59)(51 57 54 60)(61 71 64 68)(62 72 65 69)(63 67 66 70)(73 82 76 79)(74 83 77 80)(75 84 78 81)(85 95 88 92)(86 96 89 93)(87 91 90 94)
(7 85 95)(8 86 96)(9 87 91)(10 88 92)(11 89 93)(12 90 94)(19 25 37)(20 26 38)(21 27 39)(22 28 40)(23 29 41)(24 30 42)(31 57 47)(32 58 48)(33 59 43)(34 60 44)(35 55 45)(36 56 46)(61 71 83)(62 72 84)(63 67 79)(64 68 80)(65 69 81)(66 70 82)
(1 76 4 73)(2 75 5 78)(3 74 6 77)(7 59 10 56)(8 58 11 55)(9 57 12 60)(13 54 16 51)(14 53 17 50)(15 52 18 49)(19 68 22 71)(20 67 23 70)(21 72 24 69)(25 64 28 61)(26 63 29 66)(27 62 30 65)(31 90 34 87)(32 89 35 86)(33 88 36 85)(37 80 40 83)(38 79 41 82)(39 84 42 81)(43 92 46 95)(44 91 47 94)(45 96 48 93)
G:=sub<Sym(96)| (1,51)(2,52)(3,53)(4,54)(5,49)(6,50)(7,83)(8,84)(9,79)(10,80)(11,81)(12,82)(13,76)(14,77)(15,78)(16,73)(17,74)(18,75)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,33)(26,34)(27,35)(28,36)(29,31)(30,32)(37,59)(38,60)(39,55)(40,56)(41,57)(42,58)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96), (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), (1,29,4,26)(2,30,5,27)(3,25,6,28)(7,85,10,88)(8,86,11,89)(9,87,12,90)(13,91,16,94)(14,92,17,95)(15,93,18,96)(19,40,22,37)(20,41,23,38)(21,42,24,39)(31,54,34,51)(32,49,35,52)(33,50,36,53)(43,56,46,59)(44,57,47,60)(45,58,48,55)(61,80,64,83)(62,81,65,84)(63,82,66,79)(67,73,70,76)(68,74,71,77)(69,75,72,78), (1,41,4,38)(2,42,5,39)(3,37,6,40)(7,14,10,17)(8,15,11,18)(9,16,12,13)(19,25,22,28)(20,26,23,29)(21,27,24,30)(31,44,34,47)(32,45,35,48)(33,46,36,43)(49,55,52,58)(50,56,53,59)(51,57,54,60)(61,71,64,68)(62,72,65,69)(63,67,66,70)(73,82,76,79)(74,83,77,80)(75,84,78,81)(85,95,88,92)(86,96,89,93)(87,91,90,94), (7,85,95)(8,86,96)(9,87,91)(10,88,92)(11,89,93)(12,90,94)(19,25,37)(20,26,38)(21,27,39)(22,28,40)(23,29,41)(24,30,42)(31,57,47)(32,58,48)(33,59,43)(34,60,44)(35,55,45)(36,56,46)(61,71,83)(62,72,84)(63,67,79)(64,68,80)(65,69,81)(66,70,82), (1,76,4,73)(2,75,5,78)(3,74,6,77)(7,59,10,56)(8,58,11,55)(9,57,12,60)(13,54,16,51)(14,53,17,50)(15,52,18,49)(19,68,22,71)(20,67,23,70)(21,72,24,69)(25,64,28,61)(26,63,29,66)(27,62,30,65)(31,90,34,87)(32,89,35,86)(33,88,36,85)(37,80,40,83)(38,79,41,82)(39,84,42,81)(43,92,46,95)(44,91,47,94)(45,96,48,93)>;
G:=Group( (1,51)(2,52)(3,53)(4,54)(5,49)(6,50)(7,83)(8,84)(9,79)(10,80)(11,81)(12,82)(13,76)(14,77)(15,78)(16,73)(17,74)(18,75)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,33)(26,34)(27,35)(28,36)(29,31)(30,32)(37,59)(38,60)(39,55)(40,56)(41,57)(42,58)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96), (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), (1,29,4,26)(2,30,5,27)(3,25,6,28)(7,85,10,88)(8,86,11,89)(9,87,12,90)(13,91,16,94)(14,92,17,95)(15,93,18,96)(19,40,22,37)(20,41,23,38)(21,42,24,39)(31,54,34,51)(32,49,35,52)(33,50,36,53)(43,56,46,59)(44,57,47,60)(45,58,48,55)(61,80,64,83)(62,81,65,84)(63,82,66,79)(67,73,70,76)(68,74,71,77)(69,75,72,78), (1,41,4,38)(2,42,5,39)(3,37,6,40)(7,14,10,17)(8,15,11,18)(9,16,12,13)(19,25,22,28)(20,26,23,29)(21,27,24,30)(31,44,34,47)(32,45,35,48)(33,46,36,43)(49,55,52,58)(50,56,53,59)(51,57,54,60)(61,71,64,68)(62,72,65,69)(63,67,66,70)(73,82,76,79)(74,83,77,80)(75,84,78,81)(85,95,88,92)(86,96,89,93)(87,91,90,94), (7,85,95)(8,86,96)(9,87,91)(10,88,92)(11,89,93)(12,90,94)(19,25,37)(20,26,38)(21,27,39)(22,28,40)(23,29,41)(24,30,42)(31,57,47)(32,58,48)(33,59,43)(34,60,44)(35,55,45)(36,56,46)(61,71,83)(62,72,84)(63,67,79)(64,68,80)(65,69,81)(66,70,82), (1,76,4,73)(2,75,5,78)(3,74,6,77)(7,59,10,56)(8,58,11,55)(9,57,12,60)(13,54,16,51)(14,53,17,50)(15,52,18,49)(19,68,22,71)(20,67,23,70)(21,72,24,69)(25,64,28,61)(26,63,29,66)(27,62,30,65)(31,90,34,87)(32,89,35,86)(33,88,36,85)(37,80,40,83)(38,79,41,82)(39,84,42,81)(43,92,46,95)(44,91,47,94)(45,96,48,93) );
G=PermutationGroup([[(1,51),(2,52),(3,53),(4,54),(5,49),(6,50),(7,83),(8,84),(9,79),(10,80),(11,81),(12,82),(13,76),(14,77),(15,78),(16,73),(17,74),(18,75),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,33),(26,34),(27,35),(28,36),(29,31),(30,32),(37,59),(38,60),(39,55),(40,56),(41,57),(42,58),(61,85),(62,86),(63,87),(64,88),(65,89),(66,90),(67,91),(68,92),(69,93),(70,94),(71,95),(72,96)], [(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)], [(1,29,4,26),(2,30,5,27),(3,25,6,28),(7,85,10,88),(8,86,11,89),(9,87,12,90),(13,91,16,94),(14,92,17,95),(15,93,18,96),(19,40,22,37),(20,41,23,38),(21,42,24,39),(31,54,34,51),(32,49,35,52),(33,50,36,53),(43,56,46,59),(44,57,47,60),(45,58,48,55),(61,80,64,83),(62,81,65,84),(63,82,66,79),(67,73,70,76),(68,74,71,77),(69,75,72,78)], [(1,41,4,38),(2,42,5,39),(3,37,6,40),(7,14,10,17),(8,15,11,18),(9,16,12,13),(19,25,22,28),(20,26,23,29),(21,27,24,30),(31,44,34,47),(32,45,35,48),(33,46,36,43),(49,55,52,58),(50,56,53,59),(51,57,54,60),(61,71,64,68),(62,72,65,69),(63,67,66,70),(73,82,76,79),(74,83,77,80),(75,84,78,81),(85,95,88,92),(86,96,89,93),(87,91,90,94)], [(7,85,95),(8,86,96),(9,87,91),(10,88,92),(11,89,93),(12,90,94),(19,25,37),(20,26,38),(21,27,39),(22,28,40),(23,29,41),(24,30,42),(31,57,47),(32,58,48),(33,59,43),(34,60,44),(35,55,45),(36,56,46),(61,71,83),(62,72,84),(63,67,79),(64,68,80),(65,69,81),(66,70,82)], [(1,76,4,73),(2,75,5,78),(3,74,6,77),(7,59,10,56),(8,58,11,55),(9,57,12,60),(13,54,16,51),(14,53,17,50),(15,52,18,49),(19,68,22,71),(20,67,23,70),(21,72,24,69),(25,64,28,61),(26,63,29,66),(27,62,30,65),(31,90,34,87),(32,89,35,86),(33,88,36,85),(37,80,40,83),(38,79,41,82),(39,84,42,81),(43,92,46,95),(44,91,47,94),(45,96,48,93)]])
Matrix representation of C2×C6.5S4 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 72 | 0 | 0 |
| 1 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 21 | 23 |
| 0 | 0 | 3 | 52 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 71 | 22 |
| 0 | 0 | 23 | 2 |
| 72 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 2 | 52 |
| 0 | 0 | 49 | 70 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 10 | 22 |
| 0 | 0 | 12 | 63 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,72,72,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,21,3,0,0,23,52],[1,0,0,0,0,1,0,0,0,0,71,23,0,0,22,2],[72,72,0,0,1,0,0,0,0,0,2,49,0,0,52,70],[0,1,0,0,1,0,0,0,0,0,10,12,0,0,22,63] >;
C2×C6.5S4 in GAP, Magma, Sage, TeX
C_2\times C_6._5S_4
% in TeX
G:=Group("C2xC6.5S4"); // GroupNames label
G:=SmallGroup(288,910);
// by ID
G=gap.SmallGroup(288,910);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1008,170,675,2524,1908,172,1517,1153,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^6=e^3=1,c^2=d^2=f^2=b^3,a*b=b*a,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=b^-1,d*c*d^-1=b^3*c,e*c*e^-1=b^3*c*d,f*c*f^-1=c*d,e*d*e^-1=c,f*d*f^-1=b^3*d,f*e*f^-1=e^-1>;
// generators/relations
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