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## G = D10.1S4order 480 = 25·3·5

### 1st non-split extension by D10 of S4 acting via S4/A4=C2

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.D156C2, C52(Q8.D6), (D5×SL2(𝔽3))⋊1C2, (C5×GL2(𝔽3))⋊2C2, (C5×SL2(𝔽3)).5C22, SmallGroup(480,972)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C5×SL2(𝔽3) — D10.1S4
 Chief series C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — D5×SL2(𝔽3) — D10.1S4
 Lower central C5×SL2(𝔽3) — D10.1S4
 Upper central C1 — C2

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, C52C8, 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, D42D5, 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

Smallest permutation representation of D10.1S4
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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