C5xC4.S4

direct product, non-abelian, soluble

Aliases: C₅×C₄.S₄, C₂₀.₈S₄, CSU₂(𝔽₃)⋊₂C₁₀, C₄.₂(C₅×S₄), C₂.₈(C₁₀×S₄), C₁₀.₃₃(C₂×S₄), C₄.A₄.₁C₁₀, Q₈.₃(S₃×C₁₀), (C₅×Q₈).₁₅D₆, (C₅×CSU₂(𝔽₃))⋊₆C₂, SL₂(𝔽₃).₃(C₂×C₁₀), (C₅×SL₂(𝔽₃)).₁₅C₂², (C₅×C₄○D₄).₄S₃, (C₅×C₄.A₄).₃C₂, C₄○D₄.₂(C₅×S₃), SmallGroup(480,1019)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — Q₈ — SL₂(𝔽₃) — C₅×C₄.S₄

Chief series

C₁ — C₂ — Q₈ — SL₂(𝔽₃) — C₅×SL₂(𝔽₃) — C₅×CSU₂(𝔽₃) — C₅×C₄.S₄

Lower central

SL₂(𝔽₃) — C₅×C₄.S₄

Upper central

C₁ — C₁₀ — C₂₀

Subgroups: 218 in 72 conjugacy classes, 20 normal (16 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄^[×3], C₂², C₅, C₆, C₈^[×2], C₂×C₄^[×2], D₄, Q₈, Q₈^[×3], C₁₀, C₁₀, Dic₃^[×2], C₁₂, C₁₅, M₄(2), SD₁₆^[×2], Q₁₆^[×2], C₂×Q₈, C₄○D₄, C₂₀, C₂₀^[×3], C₂×C₁₀, SL₂(𝔽₃), Dic₆, C₃₀, C₈.C₂², C₄₀^[×2], C₂×C₂₀^[×2], C₅×D₄, C₅×Q₈, C₅×Q₈^[×3], CSU₂(𝔽₃)^[×2], C₄.A₄, C₅×Dic₃^[×2], C₆₀, C₅×M₄(2), C₅×SD₁₆^[×2], C₅×Q₁₆^[×2], Q₈×C₁₀, C₅×C₄○D₄, C₄.S₄, C₅×SL₂(𝔽₃), C₅×Dic₆, C₅×C₈.C₂², C₅×CSU₂(𝔽₃)^[×2], C₅×C₄.A₄, C₅×C₄.S₄

Quotients:
C₁, C₂^[×3], C₂², C₅, S₃, C₁₀^[×3], D₆, C₂×C₁₀, S₄, C₅×S₃, C₂×S₄, S₃×C₁₀, C₄.S₄, C₅×S₄, C₁₀×S₄, C₅×C₄.S₄

Generators and relations
G = < a,b,c,d,e,f | a⁵=b⁴=e³=1, c²=d²=f²=b², ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf^-1=b^-1, dcd^-1=b²c, ece^-1=b²cd, fcf^-1=cd, ede^-1=c, fdf^-1=b²d, fef^-1=e^-1 >

Smallest permutation representation
►On 160 points

Generators in S₁₆₀

(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)(121 122 123 124 125)(126 127 128 129 130)(131 132 133 134 135)(136 137 138 139 140)(141 142 143 144 145)(146 147 148 149 150)(151 152 153 154 155)(156 157 158 159 160)

(1 57 29 81)(2 58 30 82)(3 59 26 83)(4 60 27 84)(5 56 28 85)(6 96 136 125)(7 97 137 121)(8 98 138 122)(9 99 139 123)(10 100 140 124)(11 93 53 63)(12 94 54 64)(13 95 55 65)(14 91 51 61)(15 92 52 62)(16 117 157 146)(17 118 158 147)(18 119 159 148)(19 120 160 149)(20 116 156 150)(21 102 142 132)(22 103 143 133)(23 104 144 134)(24 105 145 135)(25 101 141 131)(31 112 152 126)(32 113 153 127)(33 114 154 128)(34 115 155 129)(35 111 151 130)(36 110 70 76)(37 106 66 77)(38 107 67 78)(39 108 68 79)(40 109 69 80)(41 86 46 72)(42 87 47 73)(43 88 48 74)(44 89 49 75)(45 90 50 71)

(1 37 29 66)(2 38 30 67)(3 39 26 68)(4 40 27 69)(5 36 28 70)(6 22 136 143)(7 23 137 144)(8 24 138 145)(9 25 139 141)(10 21 140 142)(11 41 53 46)(12 42 54 47)(13 43 55 48)(14 44 51 49)(15 45 52 50)(16 32 157 153)(17 33 158 154)(18 34 159 155)(19 35 160 151)(20 31 156 152)(56 110 85 76)(57 106 81 77)(58 107 82 78)(59 108 83 79)(60 109 84 80)(61 75 91 89)(62 71 92 90)(63 72 93 86)(64 73 94 87)(65 74 95 88)(96 103 125 133)(97 104 121 134)(98 105 122 135)(99 101 123 131)(100 102 124 132)(111 149 130 120)(112 150 126 116)(113 146 127 117)(114 147 128 118)(115 148 129 119)

(1 47 29 42)(2 48 30 43)(3 49 26 44)(4 50 27 45)(5 46 28 41)(6 152 136 31)(7 153 137 32)(8 154 138 33)(9 155 139 34)(10 151 140 35)(11 36 53 70)(12 37 54 66)(13 38 55 67)(14 39 51 68)(15 40 52 69)(16 23 157 144)(17 24 158 145)(18 25 159 141)(19 21 160 142)(20 22 156 143)(56 72 85 86)(57 73 81 87)(58 74 82 88)(59 75 83 89)(60 71 84 90)(61 79 91 108)(62 80 92 109)(63 76 93 110)(64 77 94 106)(65 78 95 107)(96 126 125 112)(97 127 121 113)(98 128 122 114)(99 129 123 115)(100 130 124 111)(101 148 131 119)(102 149 132 120)(103 150 133 116)(104 146 134 117)(105 147 135 118)

(11 36 46)(12 37 47)(13 38 48)(14 39 49)(15 40 50)(16 23 153)(17 24 154)(18 25 155)(19 21 151)(20 22 152)(31 156 143)(32 157 144)(33 158 145)(34 159 141)(35 160 142)(41 53 70)(42 54 66)(43 55 67)(44 51 68)(45 52 69)(61 79 89)(62 80 90)(63 76 86)(64 77 87)(65 78 88)(71 92 109)(72 93 110)(73 94 106)(74 95 107)(75 91 108)(101 129 119)(102 130 120)(103 126 116)(104 127 117)(105 128 118)(111 149 132)(112 150 133)(113 146 134)(114 147 135)(115 148 131)

(1 97 29 121)(2 98 30 122)(3 99 26 123)(4 100 27 124)(5 96 28 125)(6 85 136 56)(7 81 137 57)(8 82 138 58)(9 83 139 59)(10 84 140 60)(11 133 53 103)(12 134 54 104)(13 135 55 105)(14 131 51 101)(15 132 52 102)(16 106 157 77)(17 107 158 78)(18 108 159 79)(19 109 160 80)(20 110 156 76)(21 92 142 62)(22 93 143 63)(23 94 144 64)(24 95 145 65)(25 91 141 61)(31 86 152 72)(32 87 153 73)(33 88 154 74)(34 89 155 75)(35 90 151 71)(36 150 70 116)(37 146 66 117)(38 147 67 118)(39 148 68 119)(40 149 69 120)(41 126 46 112)(42 127 47 113)(43 128 48 114)(44 129 49 115)(45 130 50 111)

G:=sub<Sym(160)| (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)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135)(136,137,138,139,140)(141,142,143,144,145)(146,147,148,149,150)(151,152,153,154,155)(156,157,158,159,160), (1,57,29,81)(2,58,30,82)(3,59,26,83)(4,60,27,84)(5,56,28,85)(6,96,136,125)(7,97,137,121)(8,98,138,122)(9,99,139,123)(10,100,140,124)(11,93,53,63)(12,94,54,64)(13,95,55,65)(14,91,51,61)(15,92,52,62)(16,117,157,146)(17,118,158,147)(18,119,159,148)(19,120,160,149)(20,116,156,150)(21,102,142,132)(22,103,143,133)(23,104,144,134)(24,105,145,135)(25,101,141,131)(31,112,152,126)(32,113,153,127)(33,114,154,128)(34,115,155,129)(35,111,151,130)(36,110,70,76)(37,106,66,77)(38,107,67,78)(39,108,68,79)(40,109,69,80)(41,86,46,72)(42,87,47,73)(43,88,48,74)(44,89,49,75)(45,90,50,71), (1,37,29,66)(2,38,30,67)(3,39,26,68)(4,40,27,69)(5,36,28,70)(6,22,136,143)(7,23,137,144)(8,24,138,145)(9,25,139,141)(10,21,140,142)(11,41,53,46)(12,42,54,47)(13,43,55,48)(14,44,51,49)(15,45,52,50)(16,32,157,153)(17,33,158,154)(18,34,159,155)(19,35,160,151)(20,31,156,152)(56,110,85,76)(57,106,81,77)(58,107,82,78)(59,108,83,79)(60,109,84,80)(61,75,91,89)(62,71,92,90)(63,72,93,86)(64,73,94,87)(65,74,95,88)(96,103,125,133)(97,104,121,134)(98,105,122,135)(99,101,123,131)(100,102,124,132)(111,149,130,120)(112,150,126,116)(113,146,127,117)(114,147,128,118)(115,148,129,119), (1,47,29,42)(2,48,30,43)(3,49,26,44)(4,50,27,45)(5,46,28,41)(6,152,136,31)(7,153,137,32)(8,154,138,33)(9,155,139,34)(10,151,140,35)(11,36,53,70)(12,37,54,66)(13,38,55,67)(14,39,51,68)(15,40,52,69)(16,23,157,144)(17,24,158,145)(18,25,159,141)(19,21,160,142)(20,22,156,143)(56,72,85,86)(57,73,81,87)(58,74,82,88)(59,75,83,89)(60,71,84,90)(61,79,91,108)(62,80,92,109)(63,76,93,110)(64,77,94,106)(65,78,95,107)(96,126,125,112)(97,127,121,113)(98,128,122,114)(99,129,123,115)(100,130,124,111)(101,148,131,119)(102,149,132,120)(103,150,133,116)(104,146,134,117)(105,147,135,118), (11,36,46)(12,37,47)(13,38,48)(14,39,49)(15,40,50)(16,23,153)(17,24,154)(18,25,155)(19,21,151)(20,22,152)(31,156,143)(32,157,144)(33,158,145)(34,159,141)(35,160,142)(41,53,70)(42,54,66)(43,55,67)(44,51,68)(45,52,69)(61,79,89)(62,80,90)(63,76,86)(64,77,87)(65,78,88)(71,92,109)(72,93,110)(73,94,106)(74,95,107)(75,91,108)(101,129,119)(102,130,120)(103,126,116)(104,127,117)(105,128,118)(111,149,132)(112,150,133)(113,146,134)(114,147,135)(115,148,131), (1,97,29,121)(2,98,30,122)(3,99,26,123)(4,100,27,124)(5,96,28,125)(6,85,136,56)(7,81,137,57)(8,82,138,58)(9,83,139,59)(10,84,140,60)(11,133,53,103)(12,134,54,104)(13,135,55,105)(14,131,51,101)(15,132,52,102)(16,106,157,77)(17,107,158,78)(18,108,159,79)(19,109,160,80)(20,110,156,76)(21,92,142,62)(22,93,143,63)(23,94,144,64)(24,95,145,65)(25,91,141,61)(31,86,152,72)(32,87,153,73)(33,88,154,74)(34,89,155,75)(35,90,151,71)(36,150,70,116)(37,146,66,117)(38,147,67,118)(39,148,68,119)(40,149,69,120)(41,126,46,112)(42,127,47,113)(43,128,48,114)(44,129,49,115)(45,130,50,111)>;

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)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135)(136,137,138,139,140)(141,142,143,144,145)(146,147,148,149,150)(151,152,153,154,155)(156,157,158,159,160), (1,57,29,81)(2,58,30,82)(3,59,26,83)(4,60,27,84)(5,56,28,85)(6,96,136,125)(7,97,137,121)(8,98,138,122)(9,99,139,123)(10,100,140,124)(11,93,53,63)(12,94,54,64)(13,95,55,65)(14,91,51,61)(15,92,52,62)(16,117,157,146)(17,118,158,147)(18,119,159,148)(19,120,160,149)(20,116,156,150)(21,102,142,132)(22,103,143,133)(23,104,144,134)(24,105,145,135)(25,101,141,131)(31,112,152,126)(32,113,153,127)(33,114,154,128)(34,115,155,129)(35,111,151,130)(36,110,70,76)(37,106,66,77)(38,107,67,78)(39,108,68,79)(40,109,69,80)(41,86,46,72)(42,87,47,73)(43,88,48,74)(44,89,49,75)(45,90,50,71), (1,37,29,66)(2,38,30,67)(3,39,26,68)(4,40,27,69)(5,36,28,70)(6,22,136,143)(7,23,137,144)(8,24,138,145)(9,25,139,141)(10,21,140,142)(11,41,53,46)(12,42,54,47)(13,43,55,48)(14,44,51,49)(15,45,52,50)(16,32,157,153)(17,33,158,154)(18,34,159,155)(19,35,160,151)(20,31,156,152)(56,110,85,76)(57,106,81,77)(58,107,82,78)(59,108,83,79)(60,109,84,80)(61,75,91,89)(62,71,92,90)(63,72,93,86)(64,73,94,87)(65,74,95,88)(96,103,125,133)(97,104,121,134)(98,105,122,135)(99,101,123,131)(100,102,124,132)(111,149,130,120)(112,150,126,116)(113,146,127,117)(114,147,128,118)(115,148,129,119), (1,47,29,42)(2,48,30,43)(3,49,26,44)(4,50,27,45)(5,46,28,41)(6,152,136,31)(7,153,137,32)(8,154,138,33)(9,155,139,34)(10,151,140,35)(11,36,53,70)(12,37,54,66)(13,38,55,67)(14,39,51,68)(15,40,52,69)(16,23,157,144)(17,24,158,145)(18,25,159,141)(19,21,160,142)(20,22,156,143)(56,72,85,86)(57,73,81,87)(58,74,82,88)(59,75,83,89)(60,71,84,90)(61,79,91,108)(62,80,92,109)(63,76,93,110)(64,77,94,106)(65,78,95,107)(96,126,125,112)(97,127,121,113)(98,128,122,114)(99,129,123,115)(100,130,124,111)(101,148,131,119)(102,149,132,120)(103,150,133,116)(104,146,134,117)(105,147,135,118), (11,36,46)(12,37,47)(13,38,48)(14,39,49)(15,40,50)(16,23,153)(17,24,154)(18,25,155)(19,21,151)(20,22,152)(31,156,143)(32,157,144)(33,158,145)(34,159,141)(35,160,142)(41,53,70)(42,54,66)(43,55,67)(44,51,68)(45,52,69)(61,79,89)(62,80,90)(63,76,86)(64,77,87)(65,78,88)(71,92,109)(72,93,110)(73,94,106)(74,95,107)(75,91,108)(101,129,119)(102,130,120)(103,126,116)(104,127,117)(105,128,118)(111,149,132)(112,150,133)(113,146,134)(114,147,135)(115,148,131), (1,97,29,121)(2,98,30,122)(3,99,26,123)(4,100,27,124)(5,96,28,125)(6,85,136,56)(7,81,137,57)(8,82,138,58)(9,83,139,59)(10,84,140,60)(11,133,53,103)(12,134,54,104)(13,135,55,105)(14,131,51,101)(15,132,52,102)(16,106,157,77)(17,107,158,78)(18,108,159,79)(19,109,160,80)(20,110,156,76)(21,92,142,62)(22,93,143,63)(23,94,144,64)(24,95,145,65)(25,91,141,61)(31,86,152,72)(32,87,153,73)(33,88,154,74)(34,89,155,75)(35,90,151,71)(36,150,70,116)(37,146,66,117)(38,147,67,118)(39,148,68,119)(40,149,69,120)(41,126,46,112)(42,127,47,113)(43,128,48,114)(44,129,49,115)(45,130,50,111) );

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),(121,122,123,124,125),(126,127,128,129,130),(131,132,133,134,135),(136,137,138,139,140),(141,142,143,144,145),(146,147,148,149,150),(151,152,153,154,155),(156,157,158,159,160)], [(1,57,29,81),(2,58,30,82),(3,59,26,83),(4,60,27,84),(5,56,28,85),(6,96,136,125),(7,97,137,121),(8,98,138,122),(9,99,139,123),(10,100,140,124),(11,93,53,63),(12,94,54,64),(13,95,55,65),(14,91,51,61),(15,92,52,62),(16,117,157,146),(17,118,158,147),(18,119,159,148),(19,120,160,149),(20,116,156,150),(21,102,142,132),(22,103,143,133),(23,104,144,134),(24,105,145,135),(25,101,141,131),(31,112,152,126),(32,113,153,127),(33,114,154,128),(34,115,155,129),(35,111,151,130),(36,110,70,76),(37,106,66,77),(38,107,67,78),(39,108,68,79),(40,109,69,80),(41,86,46,72),(42,87,47,73),(43,88,48,74),(44,89,49,75),(45,90,50,71)], [(1,37,29,66),(2,38,30,67),(3,39,26,68),(4,40,27,69),(5,36,28,70),(6,22,136,143),(7,23,137,144),(8,24,138,145),(9,25,139,141),(10,21,140,142),(11,41,53,46),(12,42,54,47),(13,43,55,48),(14,44,51,49),(15,45,52,50),(16,32,157,153),(17,33,158,154),(18,34,159,155),(19,35,160,151),(20,31,156,152),(56,110,85,76),(57,106,81,77),(58,107,82,78),(59,108,83,79),(60,109,84,80),(61,75,91,89),(62,71,92,90),(63,72,93,86),(64,73,94,87),(65,74,95,88),(96,103,125,133),(97,104,121,134),(98,105,122,135),(99,101,123,131),(100,102,124,132),(111,149,130,120),(112,150,126,116),(113,146,127,117),(114,147,128,118),(115,148,129,119)], [(1,47,29,42),(2,48,30,43),(3,49,26,44),(4,50,27,45),(5,46,28,41),(6,152,136,31),(7,153,137,32),(8,154,138,33),(9,155,139,34),(10,151,140,35),(11,36,53,70),(12,37,54,66),(13,38,55,67),(14,39,51,68),(15,40,52,69),(16,23,157,144),(17,24,158,145),(18,25,159,141),(19,21,160,142),(20,22,156,143),(56,72,85,86),(57,73,81,87),(58,74,82,88),(59,75,83,89),(60,71,84,90),(61,79,91,108),(62,80,92,109),(63,76,93,110),(64,77,94,106),(65,78,95,107),(96,126,125,112),(97,127,121,113),(98,128,122,114),(99,129,123,115),(100,130,124,111),(101,148,131,119),(102,149,132,120),(103,150,133,116),(104,146,134,117),(105,147,135,118)], [(11,36,46),(12,37,47),(13,38,48),(14,39,49),(15,40,50),(16,23,153),(17,24,154),(18,25,155),(19,21,151),(20,22,152),(31,156,143),(32,157,144),(33,158,145),(34,159,141),(35,160,142),(41,53,70),(42,54,66),(43,55,67),(44,51,68),(45,52,69),(61,79,89),(62,80,90),(63,76,86),(64,77,87),(65,78,88),(71,92,109),(72,93,110),(73,94,106),(74,95,107),(75,91,108),(101,129,119),(102,130,120),(103,126,116),(104,127,117),(105,128,118),(111,149,132),(112,150,133),(113,146,134),(114,147,135),(115,148,131)], [(1,97,29,121),(2,98,30,122),(3,99,26,123),(4,100,27,124),(5,96,28,125),(6,85,136,56),(7,81,137,57),(8,82,138,58),(9,83,139,59),(10,84,140,60),(11,133,53,103),(12,134,54,104),(13,135,55,105),(14,131,51,101),(15,132,52,102),(16,106,157,77),(17,107,158,78),(18,108,159,79),(19,109,160,80),(20,110,156,76),(21,92,142,62),(22,93,143,63),(23,94,144,64),(24,95,145,65),(25,91,141,61),(31,86,152,72),(32,87,153,73),(33,88,154,74),(34,89,155,75),(35,90,151,71),(36,150,70,116),(37,146,66,117),(38,147,67,118),(39,148,68,119),(40,149,69,120),(41,126,46,112),(42,127,47,113),(43,128,48,114),(44,129,49,115),(45,130,50,111)])

Matrix representation ►G ⊆ GL₇(𝔽₂₄₁)

205	0	0	0	0	0	0
0	205	0	0	0	0	0
0	0	205	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

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	0	142	99	142
0	0	0	99	0	142	142
0	0	0	142	99	0	142
0	0	0	99	99	99	0

0	1	0	0	0	0	0
1	0	0	0	0	0	0
240	240	240	0	0	0	0
0	0	0	0	1	0	0
0	0	0	240	0	0	0
0	0	0	0	0	0	240
0	0	0	0	0	1	0

240	240	240	0	0	0	0
0	0	1	0	0	0	0
0	1	0	0	0	0	0
0	0	0	0	0	0	1
0	0	0	0	0	240	0
0	0	0	0	1	0	0
0	0	0	240	0	0	0

1	0	0	0	0	0	0
0	0	1	0	0	0	0
240	240	240	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	0	240	0
0	0	0	0	0	0	240
0	0	0	0	1	0	0

1	0	0	0	0	0	0
0	0	1	0	0	0	0
0	1	0	0	0	0	0
0	0	0	182	98	143	98
0	0	0	98	143	182	98
0	0	0	143	182	98	98
0	0	0	98	98	98	59

G:=sub<GL(7,GF(241))| [205,0,0,0,0,0,0,0,205,0,0,0,0,0,0,0,205,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,99,142,99,0,0,0,142,0,99,99,0,0,0,99,142,0,99,0,0,0,142,142,142,0],[0,1,240,0,0,0,0,1,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,240,0],[240,0,0,0,0,0,0,240,0,1,0,0,0,0,240,1,0,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,1,0,0,0,0,0,240,0,0,0,0,0,1,0,0,0],[1,0,240,0,0,0,0,0,0,240,0,0,0,0,0,1,240,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,240,0,0,0,0,0,0,0,240,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,182,98,143,98,0,0,0,98,143,182,98,0,0,0,143,182,98,98,0,0,0,98,98,98,59] >;

65 conjugacy classes

class	1	2A	2B	3	4A	4B	4C	4D	5A	5B	5C	5D	6	8A	8B	10A	10B	10C	10D	10E	10F	10G	10H	12A	12B	15A	15B	15C	15D	20A	20B	20C	20D	20E	20F	20G	20H	20I	···	20P	30A	30B	30C	30D	40A	···	40H	60A	···	60H
order	1	2	2	3	4	4	4	4	5	5	5	5	6	8	8	10	10	10	10	10	10	10	10	12	12	15	15	15	15	20	20	20	20	20	20	20	20	20	···	20	30	30	30	30	40	···	40	60	···	60
size	1	1	6	8	2	6	12	12	1	1	1	1	8	12	12	1	1	1	1	6	6	6	6	8	8	8	8	8	8	2	2	2	2	6	6	6	6	12	···	12	8	8	8	8	12	···	12	8	···	8

65 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	3	3	3	3	4	4
type	+	+	+				+	+			+	+			-
image	C₁	C₂	C₂	C₅	C₁₀	C₁₀	S₃	D₆	C₅×S₃	S₃×C₁₀	S₄	C₂×S₄	C₅×S₄	C₁₀×S₄	C₄.S₄	C₅×C₄.S₄
kernel	C₅×C₄.S₄	C₅×CSU₂(𝔽₃)	C₅×C₄.A₄	C₄.S₄	CSU₂(𝔽₃)	C₄.A₄	C₅×C₄○D₄	C₅×Q₈	C₄○D₄	Q₈	C₂₀	C₁₀	C₄	C₂	C₅	C₁
# reps	1	2	1	4	8	4	1	1	4	4	2	2	8	8	3	12

In GAP, Magma, Sage, TeX

C_5\times C_4.S_4

% in TeX

G:=Group("C5xC4.S4");

// GroupNames label

G:=SmallGroup(480,1019);

// by ID

G=gap.SmallGroup(480,1019);

# by ID

G:=PCGroup([7,-2,-2,-5,-3,-2,2,-2,1680,3389,1688,1123,4204,655,172,2525,404,285,124]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^4=e^3=1,c^2=d^2=f^2=b^2,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^2*c,e*c*e^-1=b^2*c*d,f*c*f^-1=c*d,e*d*e^-1=c,f*d*f^-1=b^2*d,f*e*f^-1=e^-1>;

// generators/relations

G = C₅×C₄.S₄ order 480 = 2⁵·3·5

Direct product of C₅ and C₄.S₄

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	0	142	99	142
0	0	0	99	0	142	142
0	0	0	142	99	0	142
0	0	0	99	99	99	0

1	0	0	0	0	0	0
0	0	1	0	0	0	0
0	1	0	0	0	0	0
0	0	0	182	98	143	98
0	0	0	98	143	182	98
0	0	0	143	182	98	98
0	0	0	98	98	98	59

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	0	142	99	142
0	0	0	99	0	142	142
0	0	0	142	99	0	142
0	0	0	99	99	99	0

1	0	0	0	0	0	0
0	0	1	0	0	0	0
0	1	0	0	0	0	0
0	0	0	182	98	143	98
0	0	0	98	143	182	98
0	0	0	143	182	98	98
0	0	0	98	98	98	59

G = C5×C4.S4 order 480 = 25·3·5

Direct product of C5 and C4.S4

G = C₅×C₄.S₄ order 480 = 2⁵·3·5

Direct product of C₅ and C₄.S₄

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	0	142	99	142
0	0	0	99	0	142	142
0	0	0	142	99	0	142
0	0	0	99	99	99	0

1	0	0	0	0	0	0
0	0	1	0	0	0	0
0	1	0	0	0	0	0
0	0	0	182	98	143	98
0	0	0	98	143	182	98
0	0	0	143	182	98	98
0	0	0	98	98	98	59