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## G = C5×C4.S4order 480 = 25·3·5

### Direct product of C5 and C4.S4

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

Generators and relations for C5×C4.S4
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 >

Subgroups: 218 in 72 conjugacy classes, 20 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, D4, Q8, Q8, C10, C10, Dic3, C12, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, C20, C20, C2×C10, SL2(𝔽3), Dic6, C30, C8.C22, C40, C2×C20, C5×D4, C5×Q8, C5×Q8, CSU2(𝔽3), C4.A4, C5×Dic3, C60, C5×M4(2), C5×SD16, C5×Q16, Q8×C10, C5×C4○D4, C4.S4, C5×SL2(𝔽3), C5×Dic6, C5×C8.C22, C5×CSU2(𝔽3), C5×C4.A4, C5×C4.S4
Quotients: C1, C2, C22, C5, S3, C10, D6, C2×C10, S4, C5×S3, C2×S4, S3×C10, C4.S4, C5×S4, C10×S4, C5×C4.S4

Smallest permutation representation of C5×C4.S4
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 65 12 81)(2 61 13 82)(3 62 14 83)(4 63 15 84)(5 64 11 85)(6 104 144 125)(7 105 145 121)(8 101 141 122)(9 102 142 123)(10 103 143 124)(16 97 137 135)(17 98 138 131)(18 99 139 132)(19 100 140 133)(20 96 136 134)(21 119 159 148)(22 120 160 149)(23 116 156 150)(24 117 157 146)(25 118 158 147)(26 90 50 71)(27 86 46 72)(28 87 47 73)(29 88 48 74)(30 89 49 75)(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 95 55 57)(42 91 51 58)(43 92 52 59)(44 93 53 60)(45 94 54 56)
(1 37 12 66)(2 38 13 67)(3 39 14 68)(4 40 15 69)(5 36 11 70)(6 19 144 140)(7 20 145 136)(8 16 141 137)(9 17 142 138)(10 18 143 139)(21 34 159 155)(22 35 160 151)(23 31 156 152)(24 32 157 153)(25 33 158 154)(26 52 50 43)(27 53 46 44)(28 54 47 45)(29 55 48 41)(30 51 49 42)(56 73 94 87)(57 74 95 88)(58 75 91 89)(59 71 92 90)(60 72 93 86)(61 107 82 78)(62 108 83 79)(63 109 84 80)(64 110 85 76)(65 106 81 77)(96 121 134 105)(97 122 135 101)(98 123 131 102)(99 124 132 103)(100 125 133 104)(111 149 130 120)(112 150 126 116)(113 146 127 117)(114 147 128 118)(115 148 129 119)
(1 47 12 28)(2 48 13 29)(3 49 14 30)(4 50 15 26)(5 46 11 27)(6 152 144 31)(7 153 145 32)(8 154 141 33)(9 155 142 34)(10 151 143 35)(16 158 137 25)(17 159 138 21)(18 160 139 22)(19 156 140 23)(20 157 136 24)(36 53 70 44)(37 54 66 45)(38 55 67 41)(39 51 68 42)(40 52 69 43)(56 77 94 106)(57 78 95 107)(58 79 91 108)(59 80 92 109)(60 76 93 110)(61 74 82 88)(62 75 83 89)(63 71 84 90)(64 72 85 86)(65 73 81 87)(96 146 134 117)(97 147 135 118)(98 148 131 119)(99 149 132 120)(100 150 133 116)(101 128 122 114)(102 129 123 115)(103 130 124 111)(104 126 125 112)(105 127 121 113)
(16 154 25)(17 155 21)(18 151 22)(19 152 23)(20 153 24)(26 52 69)(27 53 70)(28 54 66)(29 55 67)(30 51 68)(31 156 140)(32 157 136)(33 158 137)(34 159 138)(35 160 139)(36 46 44)(37 47 45)(38 48 41)(39 49 42)(40 50 43)(56 77 87)(57 78 88)(58 79 89)(59 80 90)(60 76 86)(71 92 109)(72 93 110)(73 94 106)(74 95 107)(75 91 108)(96 127 117)(97 128 118)(98 129 119)(99 130 120)(100 126 116)(111 149 132)(112 150 133)(113 146 134)(114 147 135)(115 148 131)
(1 105 12 121)(2 101 13 122)(3 102 14 123)(4 103 15 124)(5 104 11 125)(6 85 144 64)(7 81 145 65)(8 82 141 61)(9 83 142 62)(10 84 143 63)(16 95 137 57)(17 91 138 58)(18 92 139 59)(19 93 140 60)(20 94 136 56)(21 108 159 79)(22 109 160 80)(23 110 156 76)(24 106 157 77)(25 107 158 78)(26 130 50 111)(27 126 46 112)(28 127 47 113)(29 128 48 114)(30 129 49 115)(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)(41 135 55 97)(42 131 51 98)(43 132 52 99)(44 133 53 100)(45 134 54 96)

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

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

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

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

Matrix representation of C5×C4.S4 in 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] >;

C5×C4.S4 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

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