direct product, non-abelian, soluble
Aliases: C2×C10×SL2(𝔽3), Q8⋊(C2×C30), (C2×Q8)⋊2C30, (Q8×C10)⋊4C6, C23.6(C5×A4), (C22×Q8)⋊1C15, (C22×C10).6A4, C22.8(C10×A4), C10.14(C22×A4), (Q8×C2×C10)⋊1C3, C2.3(A4×C2×C10), (C5×Q8)⋊4(C2×C6), (C2×C10).17(C2×A4), SmallGroup(480,1128)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — C10×SL2(𝔽3) — C2×C10×SL2(𝔽3) |
| Q8 — C2×C10×SL2(𝔽3) |
Generators and relations for C2×C10×SL2(𝔽3)
G = < a,b,c,d,e | a2=b10=c4=e3=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >
Subgroups: 294 in 122 conjugacy classes, 52 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C2×C4, Q8, Q8, C23, C10, C10, C2×C6, C15, C22×C4, C2×Q8, C2×Q8, C20, C2×C10, SL2(𝔽3), C22×C6, C30, C22×Q8, C2×C20, C5×Q8, C5×Q8, C22×C10, C2×SL2(𝔽3), C2×C30, C22×C20, Q8×C10, Q8×C10, C22×SL2(𝔽3), C5×SL2(𝔽3), C22×C30, Q8×C2×C10, C10×SL2(𝔽3), C2×C10×SL2(𝔽3)
Quotients: C1, C2, C3, C22, C5, C6, C10, A4, C2×C6, C15, C2×C10, SL2(𝔽3), C2×A4, C30, C2×SL2(𝔽3), C22×A4, C5×A4, C2×C30, C22×SL2(𝔽3), C5×SL2(𝔽3), C10×A4, C10×SL2(𝔽3), A4×C2×C10, C2×C10×SL2(𝔽3)
►On 160 points
Generators in S160
(1 152)(2 153)(3 154)(4 155)(5 156)(6 157)(7 158)(8 159)(9 160)(10 151)(11 143)(12 144)(13 145)(14 146)(15 147)(16 148)(17 149)(18 150)(19 141)(20 142)(21 137)(22 138)(23 139)(24 140)(25 131)(26 132)(27 133)(28 134)(29 135)(30 136)(31 106)(32 107)(33 108)(34 109)(35 110)(36 101)(37 102)(38 103)(39 104)(40 105)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 97)(48 98)(49 99)(50 100)(51 88)(52 89)(53 90)(54 81)(55 82)(56 83)(57 84)(58 85)(59 86)(60 87)(61 111)(62 112)(63 113)(64 114)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 125)(72 126)(73 127)(74 128)(75 129)(76 130)(77 121)(78 122)(79 123)(80 124)
(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 98 125 88)(2 99 126 89)(3 100 127 90)(4 91 128 81)(5 92 129 82)(6 93 130 83)(7 94 121 84)(8 95 122 85)(9 96 123 86)(10 97 124 87)(11 35 24 63)(12 36 25 64)(13 37 26 65)(14 38 27 66)(15 39 28 67)(16 40 29 68)(17 31 30 69)(18 32 21 70)(19 33 22 61)(20 34 23 62)(41 74 54 155)(42 75 55 156)(43 76 56 157)(44 77 57 158)(45 78 58 159)(46 79 59 160)(47 80 60 151)(48 71 51 152)(49 72 52 153)(50 73 53 154)(101 131 114 144)(102 132 115 145)(103 133 116 146)(104 134 117 147)(105 135 118 148)(106 136 119 149)(107 137 120 150)(108 138 111 141)(109 139 112 142)(110 140 113 143)
(1 120 125 107)(2 111 126 108)(3 112 127 109)(4 113 128 110)(5 114 129 101)(6 115 130 102)(7 116 121 103)(8 117 122 104)(9 118 123 105)(10 119 124 106)(11 41 24 54)(12 42 25 55)(13 43 26 56)(14 44 27 57)(15 45 28 58)(16 46 29 59)(17 47 30 60)(18 48 21 51)(19 49 22 52)(20 50 23 53)(31 151 69 80)(32 152 70 71)(33 153 61 72)(34 154 62 73)(35 155 63 74)(36 156 64 75)(37 157 65 76)(38 158 66 77)(39 159 67 78)(40 160 68 79)(81 143 91 140)(82 144 92 131)(83 145 93 132)(84 146 94 133)(85 147 95 134)(86 148 96 135)(87 149 97 136)(88 150 98 137)(89 141 99 138)(90 142 100 139)
(11 35 54)(12 36 55)(13 37 56)(14 38 57)(15 39 58)(16 40 59)(17 31 60)(18 32 51)(19 33 52)(20 34 53)(21 70 48)(22 61 49)(23 62 50)(24 63 41)(25 64 42)(26 65 43)(27 66 44)(28 67 45)(29 68 46)(30 69 47)(81 143 110)(82 144 101)(83 145 102)(84 146 103)(85 147 104)(86 148 105)(87 149 106)(88 150 107)(89 141 108)(90 142 109)(91 140 113)(92 131 114)(93 132 115)(94 133 116)(95 134 117)(96 135 118)(97 136 119)(98 137 120)(99 138 111)(100 139 112)
G:=sub<Sym(160)| (1,152)(2,153)(3,154)(4,155)(5,156)(6,157)(7,158)(8,159)(9,160)(10,151)(11,143)(12,144)(13,145)(14,146)(15,147)(16,148)(17,149)(18,150)(19,141)(20,142)(21,137)(22,138)(23,139)(24,140)(25,131)(26,132)(27,133)(28,134)(29,135)(30,136)(31,106)(32,107)(33,108)(34,109)(35,110)(36,101)(37,102)(38,103)(39,104)(40,105)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,99)(50,100)(51,88)(52,89)(53,90)(54,81)(55,82)(56,83)(57,84)(58,85)(59,86)(60,87)(61,111)(62,112)(63,113)(64,114)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,125)(72,126)(73,127)(74,128)(75,129)(76,130)(77,121)(78,122)(79,123)(80,124), (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), (1,98,125,88)(2,99,126,89)(3,100,127,90)(4,91,128,81)(5,92,129,82)(6,93,130,83)(7,94,121,84)(8,95,122,85)(9,96,123,86)(10,97,124,87)(11,35,24,63)(12,36,25,64)(13,37,26,65)(14,38,27,66)(15,39,28,67)(16,40,29,68)(17,31,30,69)(18,32,21,70)(19,33,22,61)(20,34,23,62)(41,74,54,155)(42,75,55,156)(43,76,56,157)(44,77,57,158)(45,78,58,159)(46,79,59,160)(47,80,60,151)(48,71,51,152)(49,72,52,153)(50,73,53,154)(101,131,114,144)(102,132,115,145)(103,133,116,146)(104,134,117,147)(105,135,118,148)(106,136,119,149)(107,137,120,150)(108,138,111,141)(109,139,112,142)(110,140,113,143), (1,120,125,107)(2,111,126,108)(3,112,127,109)(4,113,128,110)(5,114,129,101)(6,115,130,102)(7,116,121,103)(8,117,122,104)(9,118,123,105)(10,119,124,106)(11,41,24,54)(12,42,25,55)(13,43,26,56)(14,44,27,57)(15,45,28,58)(16,46,29,59)(17,47,30,60)(18,48,21,51)(19,49,22,52)(20,50,23,53)(31,151,69,80)(32,152,70,71)(33,153,61,72)(34,154,62,73)(35,155,63,74)(36,156,64,75)(37,157,65,76)(38,158,66,77)(39,159,67,78)(40,160,68,79)(81,143,91,140)(82,144,92,131)(83,145,93,132)(84,146,94,133)(85,147,95,134)(86,148,96,135)(87,149,97,136)(88,150,98,137)(89,141,99,138)(90,142,100,139), (11,35,54)(12,36,55)(13,37,56)(14,38,57)(15,39,58)(16,40,59)(17,31,60)(18,32,51)(19,33,52)(20,34,53)(21,70,48)(22,61,49)(23,62,50)(24,63,41)(25,64,42)(26,65,43)(27,66,44)(28,67,45)(29,68,46)(30,69,47)(81,143,110)(82,144,101)(83,145,102)(84,146,103)(85,147,104)(86,148,105)(87,149,106)(88,150,107)(89,141,108)(90,142,109)(91,140,113)(92,131,114)(93,132,115)(94,133,116)(95,134,117)(96,135,118)(97,136,119)(98,137,120)(99,138,111)(100,139,112)>;
G:=Group( (1,152)(2,153)(3,154)(4,155)(5,156)(6,157)(7,158)(8,159)(9,160)(10,151)(11,143)(12,144)(13,145)(14,146)(15,147)(16,148)(17,149)(18,150)(19,141)(20,142)(21,137)(22,138)(23,139)(24,140)(25,131)(26,132)(27,133)(28,134)(29,135)(30,136)(31,106)(32,107)(33,108)(34,109)(35,110)(36,101)(37,102)(38,103)(39,104)(40,105)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,99)(50,100)(51,88)(52,89)(53,90)(54,81)(55,82)(56,83)(57,84)(58,85)(59,86)(60,87)(61,111)(62,112)(63,113)(64,114)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,125)(72,126)(73,127)(74,128)(75,129)(76,130)(77,121)(78,122)(79,123)(80,124), (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), (1,98,125,88)(2,99,126,89)(3,100,127,90)(4,91,128,81)(5,92,129,82)(6,93,130,83)(7,94,121,84)(8,95,122,85)(9,96,123,86)(10,97,124,87)(11,35,24,63)(12,36,25,64)(13,37,26,65)(14,38,27,66)(15,39,28,67)(16,40,29,68)(17,31,30,69)(18,32,21,70)(19,33,22,61)(20,34,23,62)(41,74,54,155)(42,75,55,156)(43,76,56,157)(44,77,57,158)(45,78,58,159)(46,79,59,160)(47,80,60,151)(48,71,51,152)(49,72,52,153)(50,73,53,154)(101,131,114,144)(102,132,115,145)(103,133,116,146)(104,134,117,147)(105,135,118,148)(106,136,119,149)(107,137,120,150)(108,138,111,141)(109,139,112,142)(110,140,113,143), (1,120,125,107)(2,111,126,108)(3,112,127,109)(4,113,128,110)(5,114,129,101)(6,115,130,102)(7,116,121,103)(8,117,122,104)(9,118,123,105)(10,119,124,106)(11,41,24,54)(12,42,25,55)(13,43,26,56)(14,44,27,57)(15,45,28,58)(16,46,29,59)(17,47,30,60)(18,48,21,51)(19,49,22,52)(20,50,23,53)(31,151,69,80)(32,152,70,71)(33,153,61,72)(34,154,62,73)(35,155,63,74)(36,156,64,75)(37,157,65,76)(38,158,66,77)(39,159,67,78)(40,160,68,79)(81,143,91,140)(82,144,92,131)(83,145,93,132)(84,146,94,133)(85,147,95,134)(86,148,96,135)(87,149,97,136)(88,150,98,137)(89,141,99,138)(90,142,100,139), (11,35,54)(12,36,55)(13,37,56)(14,38,57)(15,39,58)(16,40,59)(17,31,60)(18,32,51)(19,33,52)(20,34,53)(21,70,48)(22,61,49)(23,62,50)(24,63,41)(25,64,42)(26,65,43)(27,66,44)(28,67,45)(29,68,46)(30,69,47)(81,143,110)(82,144,101)(83,145,102)(84,146,103)(85,147,104)(86,148,105)(87,149,106)(88,150,107)(89,141,108)(90,142,109)(91,140,113)(92,131,114)(93,132,115)(94,133,116)(95,134,117)(96,135,118)(97,136,119)(98,137,120)(99,138,111)(100,139,112) );
G=PermutationGroup([[(1,152),(2,153),(3,154),(4,155),(5,156),(6,157),(7,158),(8,159),(9,160),(10,151),(11,143),(12,144),(13,145),(14,146),(15,147),(16,148),(17,149),(18,150),(19,141),(20,142),(21,137),(22,138),(23,139),(24,140),(25,131),(26,132),(27,133),(28,134),(29,135),(30,136),(31,106),(32,107),(33,108),(34,109),(35,110),(36,101),(37,102),(38,103),(39,104),(40,105),(41,91),(42,92),(43,93),(44,94),(45,95),(46,96),(47,97),(48,98),(49,99),(50,100),(51,88),(52,89),(53,90),(54,81),(55,82),(56,83),(57,84),(58,85),(59,86),(60,87),(61,111),(62,112),(63,113),(64,114),(65,115),(66,116),(67,117),(68,118),(69,119),(70,120),(71,125),(72,126),(73,127),(74,128),(75,129),(76,130),(77,121),(78,122),(79,123),(80,124)], [(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)], [(1,98,125,88),(2,99,126,89),(3,100,127,90),(4,91,128,81),(5,92,129,82),(6,93,130,83),(7,94,121,84),(8,95,122,85),(9,96,123,86),(10,97,124,87),(11,35,24,63),(12,36,25,64),(13,37,26,65),(14,38,27,66),(15,39,28,67),(16,40,29,68),(17,31,30,69),(18,32,21,70),(19,33,22,61),(20,34,23,62),(41,74,54,155),(42,75,55,156),(43,76,56,157),(44,77,57,158),(45,78,58,159),(46,79,59,160),(47,80,60,151),(48,71,51,152),(49,72,52,153),(50,73,53,154),(101,131,114,144),(102,132,115,145),(103,133,116,146),(104,134,117,147),(105,135,118,148),(106,136,119,149),(107,137,120,150),(108,138,111,141),(109,139,112,142),(110,140,113,143)], [(1,120,125,107),(2,111,126,108),(3,112,127,109),(4,113,128,110),(5,114,129,101),(6,115,130,102),(7,116,121,103),(8,117,122,104),(9,118,123,105),(10,119,124,106),(11,41,24,54),(12,42,25,55),(13,43,26,56),(14,44,27,57),(15,45,28,58),(16,46,29,59),(17,47,30,60),(18,48,21,51),(19,49,22,52),(20,50,23,53),(31,151,69,80),(32,152,70,71),(33,153,61,72),(34,154,62,73),(35,155,63,74),(36,156,64,75),(37,157,65,76),(38,158,66,77),(39,159,67,78),(40,160,68,79),(81,143,91,140),(82,144,92,131),(83,145,93,132),(84,146,94,133),(85,147,95,134),(86,148,96,135),(87,149,97,136),(88,150,98,137),(89,141,99,138),(90,142,100,139)], [(11,35,54),(12,36,55),(13,37,56),(14,38,57),(15,39,58),(16,40,59),(17,31,60),(18,32,51),(19,33,52),(20,34,53),(21,70,48),(22,61,49),(23,62,50),(24,63,41),(25,64,42),(26,65,43),(27,66,44),(28,67,45),(29,68,46),(30,69,47),(81,143,110),(82,144,101),(83,145,102),(84,146,103),(85,147,104),(86,148,105),(87,149,106),(88,150,107),(89,141,108),(90,142,109),(91,140,113),(92,131,114),(93,132,115),(94,133,116),(95,134,117),(96,135,118),(97,136,119),(98,137,120),(99,138,111),(100,139,112)]])
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | ··· | 6N | 10A | ··· | 10AB | 15A | ··· | 15H | 20A | ··· | 20P | 30A | ··· | 30BD |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 |
| type | + | + | - | + | + | ||||||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 | SL2(𝔽3) | SL2(𝔽3) | C5×SL2(𝔽3) | A4 | C2×A4 | C5×A4 | C10×A4 |
| kernel | C2×C10×SL2(𝔽3) | C10×SL2(𝔽3) | Q8×C2×C10 | C22×SL2(𝔽3) | Q8×C10 | C2×SL2(𝔽3) | C22×Q8 | C2×Q8 | C2×C10 | C2×C10 | C22 | C22×C10 | C2×C10 | C23 | C22 |
| # reps | 1 | 3 | 2 | 4 | 6 | 12 | 8 | 24 | 4 | 8 | 48 | 1 | 3 | 4 | 12 |
Matrix representation of C2×C10×SL2(𝔽3) ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 14 | 48 |
| 0 | 0 | 48 | 47 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 0 |
| 47 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 48 | 47 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,14,48,0,0,48,47],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,60,0],[47,0,0,0,0,13,0,0,0,0,1,48,0,0,0,47] >;
C2×C10×SL2(𝔽3) in GAP, Magma, Sage, TeX
C_2\times C_{10}\times {\rm SL}_2({\mathbb F}_3) % in TeX
G:=Group("C2xC10xSL(2,3)"); // GroupNames label
G:=SmallGroup(480,1128);
// by ID
G=gap.SmallGroup(480,1128);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,1068,172,1909,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^10=c^4=e^3=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,e*c*e^-1=d,e*d*e^-1=c*d>;
// generators/relations