Aliases: D10.1S4, GL2(𝔽3)⋊2D5, SL2(𝔽3).5D10, (Q8×D5)⋊1S3, C2.9(D5×S4), C10.6(C2×S4), Q8.6(S3×D5), (C5×Q8).6D6, Q8.D15⋊6C2, C5⋊2(Q8.D6), (D5×SL2(𝔽3))⋊1C2, (C5×GL2(𝔽3))⋊2C2, (C5×SL2(𝔽3)).5C22, SmallGroup(480,972)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C5×SL2(𝔽3) — D10.1S4 |
| C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — D5×SL2(𝔽3) — D10.1S4 |
| C5×SL2(𝔽3) — D10.1S4 |
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, C2×C4, D4, Q8, Q8, D5, C10, C10, Dic3, D6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, SL2(𝔽3), C3⋊D4, C5×S3, C3×D5, C30, C8.C22, C5⋊2C8, C40, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, CSU2(𝔽3), GL2(𝔽3), C2×SL2(𝔽3), Dic15, C6×D5, S3×C10, C8⋊D5, Dic20, D4.D5, C5⋊Q16, C5×SD16, D4⋊2D5, Q8×D5, Q8.D6, C15⋊D4, C5×SL2(𝔽3), SD16⋊D5, C5×GL2(𝔽3), Q8.D15, D5×SL2(𝔽3), D10.1S4
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, C2×S4, S3×D5, Q8.D6, D5×S4, D10.1S4
Character table of D10.1S4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 10D | 15A | 15B | 20A | 20B | 30A | 30B | 40A | 40B | 40C | 40D | |
| size | 1 | 1 | 10 | 12 | 8 | 6 | 30 | 60 | 2 | 2 | 8 | 40 | 40 | 12 | 60 | 2 | 2 | 24 | 24 | 16 | 16 | 12 | 12 | 16 | 16 | 12 | 12 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -2 | 0 | -1 | 2 | -2 | 0 | 2 | 2 | -1 | 1 | 1 | 0 | 0 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | 0 | 0 | 2 | 0 | -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 |
| ρ8 | 2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | 0 | 0 | 2 | 0 | -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 |
| ρ9 | 2 | 2 | 0 | -2 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | 0 | 0 | -2 | 0 | -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 D10 |
| ρ10 | 2 | 2 | 0 | -2 | 2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | 0 | 0 | -2 | 0 | -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 D10 |
| ρ11 | 3 | 3 | 3 | 1 | 0 | -1 | -1 | 1 | 3 | 3 | 0 | 0 | 0 | -1 | -1 | 3 | 3 | 1 | 1 | 0 | 0 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ12 | 3 | 3 | -3 | 1 | 0 | -1 | 1 | -1 | 3 | 3 | 0 | 0 | 0 | -1 | 1 | 3 | 3 | 1 | 1 | 0 | 0 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from C2×S4 |
| ρ13 | 3 | 3 | 3 | -1 | 0 | -1 | -1 | -1 | 3 | 3 | 0 | 0 | 0 | 1 | 1 | 3 | 3 | -1 | -1 | 0 | 0 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from S4 |
| ρ14 | 3 | 3 | -3 | -1 | 0 | -1 | 1 | 1 | 3 | 3 | 0 | 0 | 0 | 1 | -1 | 3 | 3 | -1 | -1 | 0 | 0 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×S4 |
| ρ15 | 4 | 4 | 0 | 0 | -2 | 4 | 0 | 0 | -1-√5 | -1+√5 | -2 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | 1-√5/2 | 1+√5/2 | -1-√5 | -1+√5 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D5 |
| ρ16 | 4 | 4 | 0 | 0 | -2 | 4 | 0 | 0 | -1+√5 | -1-√5 | -2 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | 1+√5/2 | 1-√5/2 | -1+√5 | -1-√5 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D5 |
| ρ17 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | -1+√5 | -1-√5 | 2 | 0 | 0 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | ζ83ζ53-ζ83ζ52+ζ8ζ53-ζ8ζ52 | -ζ83ζ53+ζ83ζ52-ζ8ζ53+ζ8ζ52 | ζ87ζ54-ζ87ζ5+ζ85ζ54-ζ85ζ5 | ζ83ζ54-ζ83ζ5+ζ8ζ54-ζ8ζ5 | symplectic faithful, Schur index 2 |
| ρ18 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | -1+√5 | -1-√5 | 2 | 0 | 0 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -ζ83ζ53+ζ83ζ52-ζ8ζ53+ζ8ζ52 | ζ83ζ53-ζ83ζ52+ζ8ζ53-ζ8ζ52 | ζ83ζ54-ζ83ζ5+ζ8ζ54-ζ8ζ5 | ζ87ζ54-ζ87ζ5+ζ85ζ54-ζ85ζ5 | symplectic faithful, Schur index 2 |
| ρ19 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 4 | 4 | 2 | 0 | 0 | 0 | 0 | -4 | -4 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8.D6, Schur index 2 |
| ρ20 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | -1-√5 | -1+√5 | 2 | 0 | 0 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ83ζ54-ζ83ζ5+ζ8ζ54-ζ8ζ5 | ζ87ζ54-ζ87ζ5+ζ85ζ54-ζ85ζ5 | ζ83ζ53-ζ83ζ52+ζ8ζ53-ζ8ζ52 | -ζ83ζ53+ζ83ζ52-ζ8ζ53+ζ8ζ52 | symplectic faithful, Schur index 2 |
| ρ21 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | -1-√5 | -1+√5 | 2 | 0 | 0 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ87ζ54-ζ87ζ5+ζ85ζ54-ζ85ζ5 | ζ83ζ54-ζ83ζ5+ζ8ζ54-ζ8ζ5 | -ζ83ζ53+ζ83ζ52-ζ8ζ53+ζ8ζ52 | ζ83ζ53-ζ83ζ52+ζ8ζ53-ζ8ζ52 | symplectic faithful, Schur index 2 |
| ρ22 | 4 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | 4 | 4 | -1 | -√-3 | √-3 | 0 | 0 | -4 | -4 | 0 | 0 | 1 | 1 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | complex lifted from Q8.D6 |
| ρ23 | 4 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | 4 | 4 | -1 | √-3 | -√-3 | 0 | 0 | -4 | -4 | 0 | 0 | 1 | 1 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | complex lifted from Q8.D6 |
| ρ24 | 6 | 6 | 0 | -2 | 0 | -2 | 0 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | 0 | 2 | 0 | -3-3√5/2 | -3+3√5/2 | 1+√5/2 | 1-√5/2 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D5×S4 |
| ρ25 | 6 | 6 | 0 | 2 | 0 | -2 | 0 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | 0 | -2 | 0 | -3+3√5/2 | -3-3√5/2 | -1+√5/2 | -1-√5/2 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | orthogonal lifted from D5×S4 |
| ρ26 | 6 | 6 | 0 | -2 | 0 | -2 | 0 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | 0 | 2 | 0 | -3+3√5/2 | -3-3√5/2 | 1-√5/2 | 1+√5/2 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D5×S4 |
| ρ27 | 6 | 6 | 0 | 2 | 0 | -2 | 0 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | 0 | -2 | 0 | -3-3√5/2 | -3+3√5/2 | -1-√5/2 | -1+√5/2 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | orthogonal lifted from D5×S4 |
| ρ28 | 8 | -8 | 0 | 0 | 2 | 0 | 0 | 0 | -2+2√5 | -2-2√5 | -2 | 0 | 0 | 0 | 0 | 2+2√5 | 2-2√5 | 0 | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ29 | 8 | -8 | 0 | 0 | 2 | 0 | 0 | 0 | -2-2√5 | -2+2√5 | -2 | 0 | 0 | 0 | 0 | 2-2√5 | 2+2√5 | 0 | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►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)
| 0 | 190 | 0 | 0 | 0 | 0 |
| 52 | 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 | 168 | 0 | 0 | 0 | 0 |
| 125 | 152 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 171 | 70 | 171 |
| 0 | 0 | 70 | 0 | 171 | 171 |
| 0 | 0 | 171 | 70 | 0 | 171 |
| 0 | 0 | 70 | 70 | 70 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 1 | 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 | 0 | 1 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 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 |
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
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