direct product, metabelian, supersoluble, monomial
Aliases: C5×C32⋊7D4, C62⋊3C10, C30.66D6, (C6×C30)⋊9C2, (C3×C15)⋊26D4, C32⋊7(C5×D4), (C2×C30)⋊10S3, C6.16(S3×C10), C3⋊Dic3⋊3C10, C15⋊14(C3⋊D4), (C3×C30).56C22, (C2×C6)⋊4(C5×S3), C3⋊3(C5×C3⋊D4), (C10×C3⋊S3)⋊8C2, (C2×C3⋊S3)⋊3C10, C2.5(C10×C3⋊S3), C22⋊2(C5×C3⋊S3), (C2×C10)⋊3(C3⋊S3), C10.17(C2×C3⋊S3), (C5×C3⋊Dic3)⋊8C2, (C3×C6).15(C2×C10), SmallGroup(360,109)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C32⋊7D4
G = < a,b,c,d,e | a5=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 280 in 96 conjugacy classes, 42 normal (18 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, D4, C32, C10, C10, Dic3, D6, C2×C6, C15, C3⋊S3, C3×C6, C3×C6, C20, C2×C10, C2×C10, C3⋊D4, C5×S3, C30, C30, C3⋊Dic3, C2×C3⋊S3, C62, C5×D4, C3×C15, C5×Dic3, S3×C10, C2×C30, C32⋊7D4, C5×C3⋊S3, C3×C30, C3×C30, C5×C3⋊D4, C5×C3⋊Dic3, C10×C3⋊S3, C6×C30, C5×C32⋊7D4
Quotients: C1, C2, C22, C5, S3, D4, C10, D6, C3⋊S3, C2×C10, C3⋊D4, C5×S3, C2×C3⋊S3, C5×D4, S3×C10, C32⋊7D4, C5×C3⋊S3, C5×C3⋊D4, C10×C3⋊S3, C5×C32⋊7D4
►On 180 points
Generators in S180
(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)(161 162 163 164 165)(166 167 168 169 170)(171 172 173 174 175)(176 177 178 179 180)
(1 27 77)(2 28 78)(3 29 79)(4 30 80)(5 26 76)(6 112 62)(7 113 63)(8 114 64)(9 115 65)(10 111 61)(11 66 93)(12 67 94)(13 68 95)(14 69 91)(15 70 92)(16 175 145)(17 171 141)(18 172 142)(19 173 143)(20 174 144)(21 88 146)(22 89 147)(23 90 148)(24 86 149)(25 87 150)(31 81 46)(32 82 47)(33 83 48)(34 84 49)(35 85 50)(36 116 166)(37 117 167)(38 118 168)(39 119 169)(40 120 170)(41 98 58)(42 99 59)(43 100 60)(44 96 56)(45 97 57)(51 131 180)(52 132 176)(53 133 177)(54 134 178)(55 135 179)(71 151 121)(72 152 122)(73 153 123)(74 154 124)(75 155 125)(101 156 126)(102 157 127)(103 158 128)(104 159 129)(105 160 130)(106 161 136)(107 162 137)(108 163 138)(109 164 139)(110 165 140)
(1 92 9)(2 93 10)(3 94 6)(4 95 7)(5 91 8)(11 111 28)(12 112 29)(13 113 30)(14 114 26)(15 115 27)(16 169 127)(17 170 128)(18 166 129)(19 167 130)(20 168 126)(21 45 83)(22 41 84)(23 42 85)(24 43 81)(25 44 82)(31 149 60)(32 150 56)(33 146 57)(34 147 58)(35 148 59)(36 104 172)(37 105 173)(38 101 174)(39 102 175)(40 103 171)(46 86 100)(47 87 96)(48 88 97)(49 89 98)(50 90 99)(51 154 107)(52 155 108)(53 151 109)(54 152 110)(55 153 106)(61 78 66)(62 79 67)(63 80 68)(64 76 69)(65 77 70)(71 139 177)(72 140 178)(73 136 179)(74 137 180)(75 138 176)(116 159 142)(117 160 143)(118 156 144)(119 157 145)(120 158 141)(121 164 133)(122 165 134)(123 161 135)(124 162 131)(125 163 132)
(1 129 21 134)(2 130 22 135)(3 126 23 131)(4 127 24 132)(5 128 25 133)(6 20 85 124)(7 16 81 125)(8 17 82 121)(9 18 83 122)(10 19 84 123)(11 117 98 106)(12 118 99 107)(13 119 100 108)(14 120 96 109)(15 116 97 110)(26 158 87 53)(27 159 88 54)(28 160 89 55)(29 156 90 51)(30 157 86 52)(31 75 63 175)(32 71 64 171)(33 72 65 172)(34 73 61 173)(35 74 62 174)(36 57 140 70)(37 58 136 66)(38 59 137 67)(39 60 138 68)(40 56 139 69)(41 161 93 167)(42 162 94 168)(43 163 95 169)(44 164 91 170)(45 165 92 166)(46 155 113 145)(47 151 114 141)(48 152 115 142)(49 153 111 143)(50 154 112 144)(76 103 150 177)(77 104 146 178)(78 105 147 179)(79 101 148 180)(80 102 149 176)
(6 94)(7 95)(8 91)(9 92)(10 93)(11 61)(12 62)(13 63)(14 64)(15 65)(16 163)(17 164)(18 165)(19 161)(20 162)(26 76)(27 77)(28 78)(29 79)(30 80)(31 100)(32 96)(33 97)(34 98)(35 99)(36 152)(37 153)(38 154)(39 155)(40 151)(41 84)(42 85)(43 81)(44 82)(45 83)(46 60)(47 56)(48 57)(49 58)(50 59)(51 101)(52 102)(53 103)(54 104)(55 105)(66 111)(67 112)(68 113)(69 114)(70 115)(71 120)(72 116)(73 117)(74 118)(75 119)(86 149)(87 150)(88 146)(89 147)(90 148)(106 173)(107 174)(108 175)(109 171)(110 172)(121 170)(122 166)(123 167)(124 168)(125 169)(126 131)(127 132)(128 133)(129 134)(130 135)(136 143)(137 144)(138 145)(139 141)(140 142)(156 180)(157 176)(158 177)(159 178)(160 179)
G:=sub<Sym(180)| (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)(161,162,163,164,165)(166,167,168,169,170)(171,172,173,174,175)(176,177,178,179,180), (1,27,77)(2,28,78)(3,29,79)(4,30,80)(5,26,76)(6,112,62)(7,113,63)(8,114,64)(9,115,65)(10,111,61)(11,66,93)(12,67,94)(13,68,95)(14,69,91)(15,70,92)(16,175,145)(17,171,141)(18,172,142)(19,173,143)(20,174,144)(21,88,146)(22,89,147)(23,90,148)(24,86,149)(25,87,150)(31,81,46)(32,82,47)(33,83,48)(34,84,49)(35,85,50)(36,116,166)(37,117,167)(38,118,168)(39,119,169)(40,120,170)(41,98,58)(42,99,59)(43,100,60)(44,96,56)(45,97,57)(51,131,180)(52,132,176)(53,133,177)(54,134,178)(55,135,179)(71,151,121)(72,152,122)(73,153,123)(74,154,124)(75,155,125)(101,156,126)(102,157,127)(103,158,128)(104,159,129)(105,160,130)(106,161,136)(107,162,137)(108,163,138)(109,164,139)(110,165,140), (1,92,9)(2,93,10)(3,94,6)(4,95,7)(5,91,8)(11,111,28)(12,112,29)(13,113,30)(14,114,26)(15,115,27)(16,169,127)(17,170,128)(18,166,129)(19,167,130)(20,168,126)(21,45,83)(22,41,84)(23,42,85)(24,43,81)(25,44,82)(31,149,60)(32,150,56)(33,146,57)(34,147,58)(35,148,59)(36,104,172)(37,105,173)(38,101,174)(39,102,175)(40,103,171)(46,86,100)(47,87,96)(48,88,97)(49,89,98)(50,90,99)(51,154,107)(52,155,108)(53,151,109)(54,152,110)(55,153,106)(61,78,66)(62,79,67)(63,80,68)(64,76,69)(65,77,70)(71,139,177)(72,140,178)(73,136,179)(74,137,180)(75,138,176)(116,159,142)(117,160,143)(118,156,144)(119,157,145)(120,158,141)(121,164,133)(122,165,134)(123,161,135)(124,162,131)(125,163,132), (1,129,21,134)(2,130,22,135)(3,126,23,131)(4,127,24,132)(5,128,25,133)(6,20,85,124)(7,16,81,125)(8,17,82,121)(9,18,83,122)(10,19,84,123)(11,117,98,106)(12,118,99,107)(13,119,100,108)(14,120,96,109)(15,116,97,110)(26,158,87,53)(27,159,88,54)(28,160,89,55)(29,156,90,51)(30,157,86,52)(31,75,63,175)(32,71,64,171)(33,72,65,172)(34,73,61,173)(35,74,62,174)(36,57,140,70)(37,58,136,66)(38,59,137,67)(39,60,138,68)(40,56,139,69)(41,161,93,167)(42,162,94,168)(43,163,95,169)(44,164,91,170)(45,165,92,166)(46,155,113,145)(47,151,114,141)(48,152,115,142)(49,153,111,143)(50,154,112,144)(76,103,150,177)(77,104,146,178)(78,105,147,179)(79,101,148,180)(80,102,149,176), (6,94)(7,95)(8,91)(9,92)(10,93)(11,61)(12,62)(13,63)(14,64)(15,65)(16,163)(17,164)(18,165)(19,161)(20,162)(26,76)(27,77)(28,78)(29,79)(30,80)(31,100)(32,96)(33,97)(34,98)(35,99)(36,152)(37,153)(38,154)(39,155)(40,151)(41,84)(42,85)(43,81)(44,82)(45,83)(46,60)(47,56)(48,57)(49,58)(50,59)(51,101)(52,102)(53,103)(54,104)(55,105)(66,111)(67,112)(68,113)(69,114)(70,115)(71,120)(72,116)(73,117)(74,118)(75,119)(86,149)(87,150)(88,146)(89,147)(90,148)(106,173)(107,174)(108,175)(109,171)(110,172)(121,170)(122,166)(123,167)(124,168)(125,169)(126,131)(127,132)(128,133)(129,134)(130,135)(136,143)(137,144)(138,145)(139,141)(140,142)(156,180)(157,176)(158,177)(159,178)(160,179)>;
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)(161,162,163,164,165)(166,167,168,169,170)(171,172,173,174,175)(176,177,178,179,180), (1,27,77)(2,28,78)(3,29,79)(4,30,80)(5,26,76)(6,112,62)(7,113,63)(8,114,64)(9,115,65)(10,111,61)(11,66,93)(12,67,94)(13,68,95)(14,69,91)(15,70,92)(16,175,145)(17,171,141)(18,172,142)(19,173,143)(20,174,144)(21,88,146)(22,89,147)(23,90,148)(24,86,149)(25,87,150)(31,81,46)(32,82,47)(33,83,48)(34,84,49)(35,85,50)(36,116,166)(37,117,167)(38,118,168)(39,119,169)(40,120,170)(41,98,58)(42,99,59)(43,100,60)(44,96,56)(45,97,57)(51,131,180)(52,132,176)(53,133,177)(54,134,178)(55,135,179)(71,151,121)(72,152,122)(73,153,123)(74,154,124)(75,155,125)(101,156,126)(102,157,127)(103,158,128)(104,159,129)(105,160,130)(106,161,136)(107,162,137)(108,163,138)(109,164,139)(110,165,140), (1,92,9)(2,93,10)(3,94,6)(4,95,7)(5,91,8)(11,111,28)(12,112,29)(13,113,30)(14,114,26)(15,115,27)(16,169,127)(17,170,128)(18,166,129)(19,167,130)(20,168,126)(21,45,83)(22,41,84)(23,42,85)(24,43,81)(25,44,82)(31,149,60)(32,150,56)(33,146,57)(34,147,58)(35,148,59)(36,104,172)(37,105,173)(38,101,174)(39,102,175)(40,103,171)(46,86,100)(47,87,96)(48,88,97)(49,89,98)(50,90,99)(51,154,107)(52,155,108)(53,151,109)(54,152,110)(55,153,106)(61,78,66)(62,79,67)(63,80,68)(64,76,69)(65,77,70)(71,139,177)(72,140,178)(73,136,179)(74,137,180)(75,138,176)(116,159,142)(117,160,143)(118,156,144)(119,157,145)(120,158,141)(121,164,133)(122,165,134)(123,161,135)(124,162,131)(125,163,132), (1,129,21,134)(2,130,22,135)(3,126,23,131)(4,127,24,132)(5,128,25,133)(6,20,85,124)(7,16,81,125)(8,17,82,121)(9,18,83,122)(10,19,84,123)(11,117,98,106)(12,118,99,107)(13,119,100,108)(14,120,96,109)(15,116,97,110)(26,158,87,53)(27,159,88,54)(28,160,89,55)(29,156,90,51)(30,157,86,52)(31,75,63,175)(32,71,64,171)(33,72,65,172)(34,73,61,173)(35,74,62,174)(36,57,140,70)(37,58,136,66)(38,59,137,67)(39,60,138,68)(40,56,139,69)(41,161,93,167)(42,162,94,168)(43,163,95,169)(44,164,91,170)(45,165,92,166)(46,155,113,145)(47,151,114,141)(48,152,115,142)(49,153,111,143)(50,154,112,144)(76,103,150,177)(77,104,146,178)(78,105,147,179)(79,101,148,180)(80,102,149,176), (6,94)(7,95)(8,91)(9,92)(10,93)(11,61)(12,62)(13,63)(14,64)(15,65)(16,163)(17,164)(18,165)(19,161)(20,162)(26,76)(27,77)(28,78)(29,79)(30,80)(31,100)(32,96)(33,97)(34,98)(35,99)(36,152)(37,153)(38,154)(39,155)(40,151)(41,84)(42,85)(43,81)(44,82)(45,83)(46,60)(47,56)(48,57)(49,58)(50,59)(51,101)(52,102)(53,103)(54,104)(55,105)(66,111)(67,112)(68,113)(69,114)(70,115)(71,120)(72,116)(73,117)(74,118)(75,119)(86,149)(87,150)(88,146)(89,147)(90,148)(106,173)(107,174)(108,175)(109,171)(110,172)(121,170)(122,166)(123,167)(124,168)(125,169)(126,131)(127,132)(128,133)(129,134)(130,135)(136,143)(137,144)(138,145)(139,141)(140,142)(156,180)(157,176)(158,177)(159,178)(160,179) );
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),(161,162,163,164,165),(166,167,168,169,170),(171,172,173,174,175),(176,177,178,179,180)], [(1,27,77),(2,28,78),(3,29,79),(4,30,80),(5,26,76),(6,112,62),(7,113,63),(8,114,64),(9,115,65),(10,111,61),(11,66,93),(12,67,94),(13,68,95),(14,69,91),(15,70,92),(16,175,145),(17,171,141),(18,172,142),(19,173,143),(20,174,144),(21,88,146),(22,89,147),(23,90,148),(24,86,149),(25,87,150),(31,81,46),(32,82,47),(33,83,48),(34,84,49),(35,85,50),(36,116,166),(37,117,167),(38,118,168),(39,119,169),(40,120,170),(41,98,58),(42,99,59),(43,100,60),(44,96,56),(45,97,57),(51,131,180),(52,132,176),(53,133,177),(54,134,178),(55,135,179),(71,151,121),(72,152,122),(73,153,123),(74,154,124),(75,155,125),(101,156,126),(102,157,127),(103,158,128),(104,159,129),(105,160,130),(106,161,136),(107,162,137),(108,163,138),(109,164,139),(110,165,140)], [(1,92,9),(2,93,10),(3,94,6),(4,95,7),(5,91,8),(11,111,28),(12,112,29),(13,113,30),(14,114,26),(15,115,27),(16,169,127),(17,170,128),(18,166,129),(19,167,130),(20,168,126),(21,45,83),(22,41,84),(23,42,85),(24,43,81),(25,44,82),(31,149,60),(32,150,56),(33,146,57),(34,147,58),(35,148,59),(36,104,172),(37,105,173),(38,101,174),(39,102,175),(40,103,171),(46,86,100),(47,87,96),(48,88,97),(49,89,98),(50,90,99),(51,154,107),(52,155,108),(53,151,109),(54,152,110),(55,153,106),(61,78,66),(62,79,67),(63,80,68),(64,76,69),(65,77,70),(71,139,177),(72,140,178),(73,136,179),(74,137,180),(75,138,176),(116,159,142),(117,160,143),(118,156,144),(119,157,145),(120,158,141),(121,164,133),(122,165,134),(123,161,135),(124,162,131),(125,163,132)], [(1,129,21,134),(2,130,22,135),(3,126,23,131),(4,127,24,132),(5,128,25,133),(6,20,85,124),(7,16,81,125),(8,17,82,121),(9,18,83,122),(10,19,84,123),(11,117,98,106),(12,118,99,107),(13,119,100,108),(14,120,96,109),(15,116,97,110),(26,158,87,53),(27,159,88,54),(28,160,89,55),(29,156,90,51),(30,157,86,52),(31,75,63,175),(32,71,64,171),(33,72,65,172),(34,73,61,173),(35,74,62,174),(36,57,140,70),(37,58,136,66),(38,59,137,67),(39,60,138,68),(40,56,139,69),(41,161,93,167),(42,162,94,168),(43,163,95,169),(44,164,91,170),(45,165,92,166),(46,155,113,145),(47,151,114,141),(48,152,115,142),(49,153,111,143),(50,154,112,144),(76,103,150,177),(77,104,146,178),(78,105,147,179),(79,101,148,180),(80,102,149,176)], [(6,94),(7,95),(8,91),(9,92),(10,93),(11,61),(12,62),(13,63),(14,64),(15,65),(16,163),(17,164),(18,165),(19,161),(20,162),(26,76),(27,77),(28,78),(29,79),(30,80),(31,100),(32,96),(33,97),(34,98),(35,99),(36,152),(37,153),(38,154),(39,155),(40,151),(41,84),(42,85),(43,81),(44,82),(45,83),(46,60),(47,56),(48,57),(49,58),(50,59),(51,101),(52,102),(53,103),(54,104),(55,105),(66,111),(67,112),(68,113),(69,114),(70,115),(71,120),(72,116),(73,117),(74,118),(75,119),(86,149),(87,150),(88,146),(89,147),(90,148),(106,173),(107,174),(108,175),(109,171),(110,172),(121,170),(122,166),(123,167),(124,168),(125,169),(126,131),(127,132),(128,133),(129,134),(130,135),(136,143),(137,144),(138,145),(139,141),(140,142),(156,180),(157,176),(158,177),(159,178),(160,179)]])
105 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4 | 5A | 5B | 5C | 5D | 6A | ··· | 6L | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 15A | ··· | 15P | 20A | 20B | 20C | 20D | 30A | ··· | 30AV |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 15 | ··· | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 18 | 2 | 2 | 2 | 2 | 18 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | S3 | D4 | D6 | C3⋊D4 | C5×S3 | C5×D4 | S3×C10 | C5×C3⋊D4 |
| kernel | C5×C32⋊7D4 | C5×C3⋊Dic3 | C10×C3⋊S3 | C6×C30 | C32⋊7D4 | C3⋊Dic3 | C2×C3⋊S3 | C62 | C2×C30 | C3×C15 | C30 | C15 | C2×C6 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 1 | 4 | 8 | 16 | 4 | 16 | 32 |
Matrix representation of C5×C32⋊7D4 ►in GL4(𝔽61) generated by
| 58 | 0 | 0 | 0 |
| 0 | 58 | 0 | 0 |
| 0 | 0 | 58 | 0 |
| 0 | 0 | 0 | 58 |
| 60 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 60 |
| 60 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 52 | 43 |
| 0 | 0 | 52 | 9 |
| 1 | 60 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 60 | 60 |
G:=sub<GL(4,GF(61))| [58,0,0,0,0,58,0,0,0,0,58,0,0,0,0,58],[60,60,0,0,1,0,0,0,0,0,0,60,0,0,1,60],[60,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,1,1,0,0,0,0,52,52,0,0,43,9],[1,0,0,0,60,60,0,0,0,0,1,60,0,0,0,60] >;
C5×C32⋊7D4 in GAP, Magma, Sage, TeX
C_5\times C_3^2\rtimes_7D_4
% in TeX
G:=Group("C5xC3^2:7D4"); // GroupNames label
G:=SmallGroup(360,109);
// by ID
G=gap.SmallGroup(360,109);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-3,-3,265,2404,8645]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations