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G = Dic5.S4order 480 = 25·3·5

5th non-split extension by Dic5 of S4 acting via S4/A4=C2

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

 Derived series C1 — C22 — C5×A4 — Dic5.S4
 Chief series C1 — C22 — C2×C10 — C5×A4 — C10×A4 — A4×Dic5 — Dic5.S4
 Lower central C5×A4 — Dic5.S4
 Upper central C1 — C2

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 >

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

Smallest permutation representation of Dic5.S4
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

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