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, (C5×Q8).15D6, Q8.3(S3×C10), (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
| C1 — C2 — Q8 — SL2(𝔽3) — C5×SL2(𝔽3) — C5×CSU2(𝔽3) — C5×C4.S4 |
| SL2(𝔽3) — C5×C4.S4 |
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
►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