Aliases: D10.2S4, CSU2(𝔽3)⋊2D5, SL2(𝔽3).6D10, (Q8×D5)⋊2S3, C10.7(C2×S4), Q8⋊D15⋊5C2, C2.10(D5×S4), (C5×Q8).7D6, Q8.7(S3×D5), C5⋊1(Q8.D6), (D5×SL2(𝔽3))⋊2C2, (C5×CSU2(𝔽3))⋊4C2, (C5×SL2(𝔽3)).6C22, SmallGroup(480,973)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C5×SL2(𝔽3) — D10.2S4 |
| C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — D5×SL2(𝔽3) — D10.2S4 |
| C5×SL2(𝔽3) — D10.2S4 |
Generators and relations for D10.2S4
G = < a,b,c,d,e,f | a10=b2=e3=1, c2=d2=f2=a5, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=a5b, dcd-1=a5c, ece-1=a5cd, fcf-1=cd, ede-1=c, fdf-1=a5d, fef-1=e-1 >
Subgroups: 658 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, D6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, D10, D10, SL2(𝔽3), C3⋊D4, C3×D5, D15, C30, C8.C22, C5⋊2C8, C40, Dic10, C4×D5, D20, C5×Q8, C5×Q8, CSU2(𝔽3), GL2(𝔽3), C2×SL2(𝔽3), C5×Dic3, C6×D5, D30, C8⋊D5, C40⋊C2, Q8⋊D5, C5⋊Q16, C5×Q16, Q8×D5, Q8⋊2D5, Q8.D6, C3⋊D20, C5×SL2(𝔽3), Q16⋊D5, C5×CSU2(𝔽3), Q8⋊D15, D5×SL2(𝔽3), D10.2S4
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, C2×S4, S3×D5, Q8.D6, D5×S4, D10.2S4
Character table of D10.2S4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 15A | 15B | 20A | 20B | 20C | 20D | 30A | 30B | 40A | 40B | 40C | 40D | |
| size | 1 | 1 | 10 | 60 | 8 | 6 | 12 | 30 | 2 | 2 | 8 | 40 | 40 | 12 | 60 | 2 | 2 | 16 | 16 | 12 | 12 | 24 | 24 | 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 | 0 | -2 | 2 | 2 | -1 | 1 | 1 | 0 | 0 | 2 | 2 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 2 | 0 | -1 | 2 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | 0 | 0 | 2 | 2 | -2 | 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 |
| ρ8 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 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 | 0 | 2 | 2 | -2 | 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 | 0 | 2 | 2 | 2 | 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 |
| ρ11 | 3 | 3 | -3 | -1 | 0 | -1 | 1 | 1 | 3 | 3 | 0 | 0 | 0 | -1 | 1 | 3 | 3 | 0 | 0 | -1 | -1 | 1 | 1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from C2×S4 |
| ρ12 | 3 | 3 | 3 | 1 | 0 | -1 | 1 | -1 | 3 | 3 | 0 | 0 | 0 | -1 | -1 | 3 | 3 | 0 | 0 | -1 | -1 | 1 | 1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ13 | 3 | 3 | 3 | -1 | 0 | -1 | -1 | -1 | 3 | 3 | 0 | 0 | 0 | 1 | 1 | 3 | 3 | 0 | 0 | -1 | -1 | -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 | 0 | 0 | -1 | -1 | -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 | 1-√5/2 | 1+√5/2 | -1-√5 | -1+√5 | 0 | 0 | 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 | 1+√5/2 | 1-√5/2 | -1+√5 | -1-√5 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D5 |
| ρ17 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 4 | 4 | 2 | 0 | 0 | 0 | 0 | -4 | -4 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8.D6, Schur index 2 |
| ρ18 | 4 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | 4 | 4 | -1 | -√-3 | √-3 | 0 | 0 | -4 | -4 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | complex lifted from Q8.D6 |
| ρ19 | 4 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | 4 | 4 | -1 | √-3 | -√-3 | 0 | 0 | -4 | -4 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | complex lifted from Q8.D6 |
| ρ20 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | -1-√5 | -1+√5 | 2 | 0 | 0 | 0 | 0 | 1+√5 | 1-√5 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -ζ83ζ53+ζ83ζ52+ζ8ζ53-ζ8ζ52 | ζ83ζ53-ζ83ζ52-ζ8ζ53+ζ8ζ52 | ζ83ζ54-ζ83ζ5-ζ8ζ54+ζ8ζ5 | ζ87ζ54-ζ87ζ5-ζ85ζ54+ζ85ζ5 | complex faithful |
| ρ21 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | -1+√5 | -1-√5 | 2 | 0 | 0 | 0 | 0 | 1-√5 | 1+√5 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ83ζ54-ζ83ζ5-ζ8ζ54+ζ8ζ5 | ζ87ζ54-ζ87ζ5-ζ85ζ54+ζ85ζ5 | ζ83ζ53-ζ83ζ52-ζ8ζ53+ζ8ζ52 | -ζ83ζ53+ζ83ζ52+ζ8ζ53-ζ8ζ52 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | -1-√5 | -1+√5 | 2 | 0 | 0 | 0 | 0 | 1+√5 | 1-√5 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | ζ83ζ53-ζ83ζ52-ζ8ζ53+ζ8ζ52 | -ζ83ζ53+ζ83ζ52+ζ8ζ53-ζ8ζ52 | ζ87ζ54-ζ87ζ5-ζ85ζ54+ζ85ζ5 | ζ83ζ54-ζ83ζ5-ζ8ζ54+ζ8ζ5 | complex faithful |
| ρ23 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | -1+√5 | -1-√5 | 2 | 0 | 0 | 0 | 0 | 1-√5 | 1+√5 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ87ζ54-ζ87ζ5-ζ85ζ54+ζ85ζ5 | ζ83ζ54-ζ83ζ5-ζ8ζ54+ζ8ζ5 | -ζ83ζ53+ζ83ζ52+ζ8ζ53-ζ8ζ52 | ζ83ζ53-ζ83ζ52-ζ8ζ53+ζ8ζ52 | complex faithful |
| ρ24 | 6 | 6 | 0 | 0 | 0 | -2 | -2 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | 0 | 2 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | 1+√5/2 | 1-√5/2 | 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 | 0 | 0 | -2 | 2 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | 0 | -2 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | 1+√5/2 | 1-√5/2 | -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 | 0 | 0 | -2 | -2 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | 0 | 2 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | 1-√5/2 | 1+√5/2 | 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 | 0 | 0 | -2 | 2 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | 0 | -2 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | 1-√5/2 | 1+√5/2 | -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 | -1+√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | orthogonal 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 | -1-√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | orthogonal 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 76)(2 75)(3 74)(4 73)(5 72)(6 71)(7 80)(8 79)(9 78)(10 77)(11 52)(12 51)(13 60)(14 59)(15 58)(16 57)(17 56)(18 55)(19 54)(20 53)(21 42)(22 41)(23 50)(24 49)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 64)(32 63)(33 62)(34 61)(35 70)(36 69)(37 68)(38 67)(39 66)(40 65)
(1 69 6 64)(2 70 7 65)(3 61 8 66)(4 62 9 67)(5 63 10 68)(11 22 16 27)(12 23 17 28)(13 24 18 29)(14 25 19 30)(15 26 20 21)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)(41 57 46 52)(42 58 47 53)(43 59 48 54)(44 60 49 55)(45 51 50 56)
(1 58 6 53)(2 59 7 54)(3 60 8 55)(4 51 9 56)(5 52 10 57)(11 77 16 72)(12 78 17 73)(13 79 18 74)(14 80 19 75)(15 71 20 76)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 63 46 68)(42 64 47 69)(43 65 48 70)(44 66 49 61)(45 67 50 62)
(11 22 32)(12 23 33)(13 24 34)(14 25 35)(15 26 36)(16 27 37)(17 28 38)(18 29 39)(19 30 40)(20 21 31)(41 63 52)(42 64 53)(43 65 54)(44 66 55)(45 67 56)(46 68 57)(47 69 58)(48 70 59)(49 61 60)(50 62 51)
(1 72 6 77)(2 73 7 78)(3 74 8 79)(4 75 9 80)(5 76 10 71)(11 58 16 53)(12 59 17 54)(13 60 18 55)(14 51 19 56)(15 52 20 57)(21 68 26 63)(22 69 27 64)(23 70 28 65)(24 61 29 66)(25 62 30 67)(31 46 36 41)(32 47 37 42)(33 48 38 43)(34 49 39 44)(35 50 40 45)
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)(2,75)(3,74)(4,73)(5,72)(6,71)(7,80)(8,79)(9,78)(10,77)(11,52)(12,51)(13,60)(14,59)(15,58)(16,57)(17,56)(18,55)(19,54)(20,53)(21,42)(22,41)(23,50)(24,49)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,64)(32,63)(33,62)(34,61)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65), (1,69,6,64)(2,70,7,65)(3,61,8,66)(4,62,9,67)(5,63,10,68)(11,22,16,27)(12,23,17,28)(13,24,18,29)(14,25,19,30)(15,26,20,21)(31,76,36,71)(32,77,37,72)(33,78,38,73)(34,79,39,74)(35,80,40,75)(41,57,46,52)(42,58,47,53)(43,59,48,54)(44,60,49,55)(45,51,50,56), (1,58,6,53)(2,59,7,54)(3,60,8,55)(4,51,9,56)(5,52,10,57)(11,77,16,72)(12,78,17,73)(13,79,18,74)(14,80,19,75)(15,71,20,76)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,63,46,68)(42,64,47,69)(43,65,48,70)(44,66,49,61)(45,67,50,62), (11,22,32)(12,23,33)(13,24,34)(14,25,35)(15,26,36)(16,27,37)(17,28,38)(18,29,39)(19,30,40)(20,21,31)(41,63,52)(42,64,53)(43,65,54)(44,66,55)(45,67,56)(46,68,57)(47,69,58)(48,70,59)(49,61,60)(50,62,51), (1,72,6,77)(2,73,7,78)(3,74,8,79)(4,75,9,80)(5,76,10,71)(11,58,16,53)(12,59,17,54)(13,60,18,55)(14,51,19,56)(15,52,20,57)(21,68,26,63)(22,69,27,64)(23,70,28,65)(24,61,29,66)(25,62,30,67)(31,46,36,41)(32,47,37,42)(33,48,38,43)(34,49,39,44)(35,50,40,45)>;
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)(2,75)(3,74)(4,73)(5,72)(6,71)(7,80)(8,79)(9,78)(10,77)(11,52)(12,51)(13,60)(14,59)(15,58)(16,57)(17,56)(18,55)(19,54)(20,53)(21,42)(22,41)(23,50)(24,49)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,64)(32,63)(33,62)(34,61)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65), (1,69,6,64)(2,70,7,65)(3,61,8,66)(4,62,9,67)(5,63,10,68)(11,22,16,27)(12,23,17,28)(13,24,18,29)(14,25,19,30)(15,26,20,21)(31,76,36,71)(32,77,37,72)(33,78,38,73)(34,79,39,74)(35,80,40,75)(41,57,46,52)(42,58,47,53)(43,59,48,54)(44,60,49,55)(45,51,50,56), (1,58,6,53)(2,59,7,54)(3,60,8,55)(4,51,9,56)(5,52,10,57)(11,77,16,72)(12,78,17,73)(13,79,18,74)(14,80,19,75)(15,71,20,76)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,63,46,68)(42,64,47,69)(43,65,48,70)(44,66,49,61)(45,67,50,62), (11,22,32)(12,23,33)(13,24,34)(14,25,35)(15,26,36)(16,27,37)(17,28,38)(18,29,39)(19,30,40)(20,21,31)(41,63,52)(42,64,53)(43,65,54)(44,66,55)(45,67,56)(46,68,57)(47,69,58)(48,70,59)(49,61,60)(50,62,51), (1,72,6,77)(2,73,7,78)(3,74,8,79)(4,75,9,80)(5,76,10,71)(11,58,16,53)(12,59,17,54)(13,60,18,55)(14,51,19,56)(15,52,20,57)(21,68,26,63)(22,69,27,64)(23,70,28,65)(24,61,29,66)(25,62,30,67)(31,46,36,41)(32,47,37,42)(33,48,38,43)(34,49,39,44)(35,50,40,45) );
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),(2,75),(3,74),(4,73),(5,72),(6,71),(7,80),(8,79),(9,78),(10,77),(11,52),(12,51),(13,60),(14,59),(15,58),(16,57),(17,56),(18,55),(19,54),(20,53),(21,42),(22,41),(23,50),(24,49),(25,48),(26,47),(27,46),(28,45),(29,44),(30,43),(31,64),(32,63),(33,62),(34,61),(35,70),(36,69),(37,68),(38,67),(39,66),(40,65)], [(1,69,6,64),(2,70,7,65),(3,61,8,66),(4,62,9,67),(5,63,10,68),(11,22,16,27),(12,23,17,28),(13,24,18,29),(14,25,19,30),(15,26,20,21),(31,76,36,71),(32,77,37,72),(33,78,38,73),(34,79,39,74),(35,80,40,75),(41,57,46,52),(42,58,47,53),(43,59,48,54),(44,60,49,55),(45,51,50,56)], [(1,58,6,53),(2,59,7,54),(3,60,8,55),(4,51,9,56),(5,52,10,57),(11,77,16,72),(12,78,17,73),(13,79,18,74),(14,80,19,75),(15,71,20,76),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40),(41,63,46,68),(42,64,47,69),(43,65,48,70),(44,66,49,61),(45,67,50,62)], [(11,22,32),(12,23,33),(13,24,34),(14,25,35),(15,26,36),(16,27,37),(17,28,38),(18,29,39),(19,30,40),(20,21,31),(41,63,52),(42,64,53),(43,65,54),(44,66,55),(45,67,56),(46,68,57),(47,69,58),(48,70,59),(49,61,60),(50,62,51)], [(1,72,6,77),(2,73,7,78),(3,74,8,79),(4,75,9,80),(5,76,10,71),(11,58,16,53),(12,59,17,54),(13,60,18,55),(14,51,19,56),(15,52,20,57),(21,68,26,63),(22,69,27,64),(23,70,28,65),(24,61,29,66),(25,62,30,67),(31,46,36,41),(32,47,37,42),(33,48,38,43),(34,49,39,44),(35,50,40,45)]])
Matrix representation of D10.2S4 ►in GL8(𝔽241)
| 52 | 0 | 52 | 0 | 0 | 0 | 0 | 0 |
| 0 | 52 | 0 | 52 | 0 | 0 | 0 | 0 |
| 189 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 189 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 52 | 0 | 52 | 0 | 0 | 0 | 0 | 0 |
| 0 | 52 | 0 | 52 | 0 | 0 | 0 | 0 |
| 240 | 0 | 189 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 189 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 171 | 70 | 171 |
| 0 | 0 | 0 | 0 | 70 | 0 | 171 | 171 |
| 0 | 0 | 0 | 0 | 171 | 70 | 0 | 171 |
| 0 | 0 | 0 | 0 | 70 | 70 | 70 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 240 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 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 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 70 | 171 | 70 |
| 0 | 0 | 0 | 0 | 70 | 171 | 0 | 70 |
| 0 | 0 | 0 | 0 | 171 | 0 | 70 | 70 |
| 0 | 0 | 0 | 0 | 70 | 70 | 70 | 0 |
G:=sub<GL(8,GF(241))| [52,0,189,0,0,0,0,0,0,52,0,189,0,0,0,0,52,0,240,0,0,0,0,0,0,52,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[52,0,240,0,0,0,0,0,0,52,0,240,0,0,0,0,52,0,189,0,0,0,0,0,0,52,0,189,0,0,0,0,0,0,0,0,0,70,171,70,0,0,0,0,171,0,70,70,0,0,0,0,70,171,0,70,0,0,0,0,171,171,171,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0],[240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,70,171,70,0,0,0,0,70,171,0,70,0,0,0,0,171,0,70,70,0,0,0,0,70,70,70,0] >;
D10.2S4 in GAP, Magma, Sage, TeX
D_{10}._2S_4 % in TeX
G:=Group("D10.2S4"); // GroupNames label
G:=SmallGroup(480,973);
// by ID
G=gap.SmallGroup(480,973);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,3389,1688,93,1347,2111,3168,172,1272,1909,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^10=b^2=e^3=1,c^2=d^2=f^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^-1=a^5*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
Export