direct product, non-abelian, soluble
Aliases: C2×C10×SL2(𝔽3), Q8⋊(C2×C30), (Q8×C10)⋊4C6, (C2×Q8)⋊2C30, C23.6(C5×A4), (C22×Q8)⋊1C15, C22.8(C10×A4), (C22×C10).6A4, 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) |
Subgroups: 294 in 122 conjugacy classes, 52 normal (14 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C22 [×7], C5, C6 [×7], C2×C4 [×6], Q8, Q8 [×5], C23, C10, C10 [×6], C2×C6 [×7], C15, C22×C4, C2×Q8 [×3], C2×Q8 [×3], C20 [×4], C2×C10 [×7], SL2(𝔽3), C22×C6, C30 [×7], C22×Q8, C2×C20 [×6], C5×Q8, C5×Q8 [×5], C22×C10, C2×SL2(𝔽3) [×3], C2×C30 [×7], C22×C20, Q8×C10 [×3], Q8×C10 [×3], C22×SL2(𝔽3), C5×SL2(𝔽3), C22×C30, Q8×C2×C10, C10×SL2(𝔽3) [×3], C2×C10×SL2(𝔽3)
Quotients:
C1, C2 [×3], C3, C22, C5, C6 [×3], C10 [×3], A4, C2×C6, C15, C2×C10, SL2(𝔽3) [×4], C2×A4 [×3], C30 [×3], C2×SL2(𝔽3) [×6], C22×A4, C5×A4, C2×C30, C22×SL2(𝔽3), C5×SL2(𝔽3) [×4], C10×A4 [×3], C10×SL2(𝔽3) [×6], A4×C2×C10, C2×C10×SL2(𝔽3)
Generators and relations
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 >
►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)(11 102)(12 103)(13 104)(14 105)(15 106)(16 107)(17 108)(18 109)(19 110)(20 101)(21 137)(22 138)(23 139)(24 140)(25 131)(26 132)(27 133)(28 134)(29 135)(30 136)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 129)(42 130)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 127)(50 128)(51 142)(52 143)(53 144)(54 145)(55 146)(56 147)(57 148)(58 149)(59 150)(60 141)(61 87)(62 88)(63 89)(64 90)(65 81)(66 82)(67 83)(68 84)(69 85)(70 86)(71 97)(72 98)(73 99)(74 100)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)
(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 98 125 142)(2 99 126 143)(3 100 127 144)(4 91 128 145)(5 92 129 146)(6 93 130 147)(7 94 121 148)(8 95 122 149)(9 96 123 150)(10 97 124 141)(11 26 65 39)(12 27 66 40)(13 28 67 31)(14 29 68 32)(15 30 69 33)(16 21 70 34)(17 22 61 35)(18 23 62 36)(19 24 63 37)(20 25 64 38)(41 55 156 76)(42 56 157 77)(43 57 158 78)(44 58 159 79)(45 59 160 80)(46 60 151 71)(47 51 152 72)(48 52 153 73)(49 53 154 74)(50 54 155 75)(81 119 102 132)(82 120 103 133)(83 111 104 134)(84 112 105 135)(85 113 106 136)(86 114 107 137)(87 115 108 138)(88 116 109 139)(89 117 110 140)(90 118 101 131)
(1 86 125 107)(2 87 126 108)(3 88 127 109)(4 89 128 110)(5 90 129 101)(6 81 130 102)(7 82 121 103)(8 83 122 104)(9 84 123 105)(10 85 124 106)(11 157 65 42)(12 158 66 43)(13 159 67 44)(14 160 68 45)(15 151 69 46)(16 152 70 47)(17 153 61 48)(18 154 62 49)(19 155 63 50)(20 156 64 41)(21 51 34 72)(22 52 35 73)(23 53 36 74)(24 54 37 75)(25 55 38 76)(26 56 39 77)(27 57 40 78)(28 58 31 79)(29 59 32 80)(30 60 33 71)(91 140 145 117)(92 131 146 118)(93 132 147 119)(94 133 148 120)(95 134 149 111)(96 135 150 112)(97 136 141 113)(98 137 142 114)(99 138 143 115)(100 139 144 116)
(11 56 39)(12 57 40)(13 58 31)(14 59 32)(15 60 33)(16 51 34)(17 52 35)(18 53 36)(19 54 37)(20 55 38)(21 70 72)(22 61 73)(23 62 74)(24 63 75)(25 64 76)(26 65 77)(27 66 78)(28 67 79)(29 68 80)(30 69 71)(81 93 132)(82 94 133)(83 95 134)(84 96 135)(85 97 136)(86 98 137)(87 99 138)(88 100 139)(89 91 140)(90 92 131)(101 146 118)(102 147 119)(103 148 120)(104 149 111)(105 150 112)(106 141 113)(107 142 114)(108 143 115)(109 144 116)(110 145 117)
G:=sub<Sym(160)| (1,152)(2,153)(3,154)(4,155)(5,156)(6,157)(7,158)(8,159)(9,160)(10,151)(11,102)(12,103)(13,104)(14,105)(15,106)(16,107)(17,108)(18,109)(19,110)(20,101)(21,137)(22,138)(23,139)(24,140)(25,131)(26,132)(27,133)(28,134)(29,135)(30,136)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,129)(42,130)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,127)(50,128)(51,142)(52,143)(53,144)(54,145)(55,146)(56,147)(57,148)(58,149)(59,150)(60,141)(61,87)(62,88)(63,89)(64,90)(65,81)(66,82)(67,83)(68,84)(69,85)(70,86)(71,97)(72,98)(73,99)(74,100)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96), (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), (1,98,125,142)(2,99,126,143)(3,100,127,144)(4,91,128,145)(5,92,129,146)(6,93,130,147)(7,94,121,148)(8,95,122,149)(9,96,123,150)(10,97,124,141)(11,26,65,39)(12,27,66,40)(13,28,67,31)(14,29,68,32)(15,30,69,33)(16,21,70,34)(17,22,61,35)(18,23,62,36)(19,24,63,37)(20,25,64,38)(41,55,156,76)(42,56,157,77)(43,57,158,78)(44,58,159,79)(45,59,160,80)(46,60,151,71)(47,51,152,72)(48,52,153,73)(49,53,154,74)(50,54,155,75)(81,119,102,132)(82,120,103,133)(83,111,104,134)(84,112,105,135)(85,113,106,136)(86,114,107,137)(87,115,108,138)(88,116,109,139)(89,117,110,140)(90,118,101,131), (1,86,125,107)(2,87,126,108)(3,88,127,109)(4,89,128,110)(5,90,129,101)(6,81,130,102)(7,82,121,103)(8,83,122,104)(9,84,123,105)(10,85,124,106)(11,157,65,42)(12,158,66,43)(13,159,67,44)(14,160,68,45)(15,151,69,46)(16,152,70,47)(17,153,61,48)(18,154,62,49)(19,155,63,50)(20,156,64,41)(21,51,34,72)(22,52,35,73)(23,53,36,74)(24,54,37,75)(25,55,38,76)(26,56,39,77)(27,57,40,78)(28,58,31,79)(29,59,32,80)(30,60,33,71)(91,140,145,117)(92,131,146,118)(93,132,147,119)(94,133,148,120)(95,134,149,111)(96,135,150,112)(97,136,141,113)(98,137,142,114)(99,138,143,115)(100,139,144,116), (11,56,39)(12,57,40)(13,58,31)(14,59,32)(15,60,33)(16,51,34)(17,52,35)(18,53,36)(19,54,37)(20,55,38)(21,70,72)(22,61,73)(23,62,74)(24,63,75)(25,64,76)(26,65,77)(27,66,78)(28,67,79)(29,68,80)(30,69,71)(81,93,132)(82,94,133)(83,95,134)(84,96,135)(85,97,136)(86,98,137)(87,99,138)(88,100,139)(89,91,140)(90,92,131)(101,146,118)(102,147,119)(103,148,120)(104,149,111)(105,150,112)(106,141,113)(107,142,114)(108,143,115)(109,144,116)(110,145,117)>;
G:=Group( (1,152)(2,153)(3,154)(4,155)(5,156)(6,157)(7,158)(8,159)(9,160)(10,151)(11,102)(12,103)(13,104)(14,105)(15,106)(16,107)(17,108)(18,109)(19,110)(20,101)(21,137)(22,138)(23,139)(24,140)(25,131)(26,132)(27,133)(28,134)(29,135)(30,136)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,129)(42,130)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,127)(50,128)(51,142)(52,143)(53,144)(54,145)(55,146)(56,147)(57,148)(58,149)(59,150)(60,141)(61,87)(62,88)(63,89)(64,90)(65,81)(66,82)(67,83)(68,84)(69,85)(70,86)(71,97)(72,98)(73,99)(74,100)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96), (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), (1,98,125,142)(2,99,126,143)(3,100,127,144)(4,91,128,145)(5,92,129,146)(6,93,130,147)(7,94,121,148)(8,95,122,149)(9,96,123,150)(10,97,124,141)(11,26,65,39)(12,27,66,40)(13,28,67,31)(14,29,68,32)(15,30,69,33)(16,21,70,34)(17,22,61,35)(18,23,62,36)(19,24,63,37)(20,25,64,38)(41,55,156,76)(42,56,157,77)(43,57,158,78)(44,58,159,79)(45,59,160,80)(46,60,151,71)(47,51,152,72)(48,52,153,73)(49,53,154,74)(50,54,155,75)(81,119,102,132)(82,120,103,133)(83,111,104,134)(84,112,105,135)(85,113,106,136)(86,114,107,137)(87,115,108,138)(88,116,109,139)(89,117,110,140)(90,118,101,131), (1,86,125,107)(2,87,126,108)(3,88,127,109)(4,89,128,110)(5,90,129,101)(6,81,130,102)(7,82,121,103)(8,83,122,104)(9,84,123,105)(10,85,124,106)(11,157,65,42)(12,158,66,43)(13,159,67,44)(14,160,68,45)(15,151,69,46)(16,152,70,47)(17,153,61,48)(18,154,62,49)(19,155,63,50)(20,156,64,41)(21,51,34,72)(22,52,35,73)(23,53,36,74)(24,54,37,75)(25,55,38,76)(26,56,39,77)(27,57,40,78)(28,58,31,79)(29,59,32,80)(30,60,33,71)(91,140,145,117)(92,131,146,118)(93,132,147,119)(94,133,148,120)(95,134,149,111)(96,135,150,112)(97,136,141,113)(98,137,142,114)(99,138,143,115)(100,139,144,116), (11,56,39)(12,57,40)(13,58,31)(14,59,32)(15,60,33)(16,51,34)(17,52,35)(18,53,36)(19,54,37)(20,55,38)(21,70,72)(22,61,73)(23,62,74)(24,63,75)(25,64,76)(26,65,77)(27,66,78)(28,67,79)(29,68,80)(30,69,71)(81,93,132)(82,94,133)(83,95,134)(84,96,135)(85,97,136)(86,98,137)(87,99,138)(88,100,139)(89,91,140)(90,92,131)(101,146,118)(102,147,119)(103,148,120)(104,149,111)(105,150,112)(106,141,113)(107,142,114)(108,143,115)(109,144,116)(110,145,117) );
G=PermutationGroup([(1,152),(2,153),(3,154),(4,155),(5,156),(6,157),(7,158),(8,159),(9,160),(10,151),(11,102),(12,103),(13,104),(14,105),(15,106),(16,107),(17,108),(18,109),(19,110),(20,101),(21,137),(22,138),(23,139),(24,140),(25,131),(26,132),(27,133),(28,134),(29,135),(30,136),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,129),(42,130),(43,121),(44,122),(45,123),(46,124),(47,125),(48,126),(49,127),(50,128),(51,142),(52,143),(53,144),(54,145),(55,146),(56,147),(57,148),(58,149),(59,150),(60,141),(61,87),(62,88),(63,89),(64,90),(65,81),(66,82),(67,83),(68,84),(69,85),(70,86),(71,97),(72,98),(73,99),(74,100),(75,91),(76,92),(77,93),(78,94),(79,95),(80,96)], [(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)], [(1,98,125,142),(2,99,126,143),(3,100,127,144),(4,91,128,145),(5,92,129,146),(6,93,130,147),(7,94,121,148),(8,95,122,149),(9,96,123,150),(10,97,124,141),(11,26,65,39),(12,27,66,40),(13,28,67,31),(14,29,68,32),(15,30,69,33),(16,21,70,34),(17,22,61,35),(18,23,62,36),(19,24,63,37),(20,25,64,38),(41,55,156,76),(42,56,157,77),(43,57,158,78),(44,58,159,79),(45,59,160,80),(46,60,151,71),(47,51,152,72),(48,52,153,73),(49,53,154,74),(50,54,155,75),(81,119,102,132),(82,120,103,133),(83,111,104,134),(84,112,105,135),(85,113,106,136),(86,114,107,137),(87,115,108,138),(88,116,109,139),(89,117,110,140),(90,118,101,131)], [(1,86,125,107),(2,87,126,108),(3,88,127,109),(4,89,128,110),(5,90,129,101),(6,81,130,102),(7,82,121,103),(8,83,122,104),(9,84,123,105),(10,85,124,106),(11,157,65,42),(12,158,66,43),(13,159,67,44),(14,160,68,45),(15,151,69,46),(16,152,70,47),(17,153,61,48),(18,154,62,49),(19,155,63,50),(20,156,64,41),(21,51,34,72),(22,52,35,73),(23,53,36,74),(24,54,37,75),(25,55,38,76),(26,56,39,77),(27,57,40,78),(28,58,31,79),(29,59,32,80),(30,60,33,71),(91,140,145,117),(92,131,146,118),(93,132,147,119),(94,133,148,120),(95,134,149,111),(96,135,150,112),(97,136,141,113),(98,137,142,114),(99,138,143,115),(100,139,144,116)], [(11,56,39),(12,57,40),(13,58,31),(14,59,32),(15,60,33),(16,51,34),(17,52,35),(18,53,36),(19,54,37),(20,55,38),(21,70,72),(22,61,73),(23,62,74),(24,63,75),(25,64,76),(26,65,77),(27,66,78),(28,67,79),(29,68,80),(30,69,71),(81,93,132),(82,94,133),(83,95,134),(84,96,135),(85,97,136),(86,98,137),(87,99,138),(88,100,139),(89,91,140),(90,92,131),(101,146,118),(102,147,119),(103,148,120),(104,149,111),(105,150,112),(106,141,113),(107,142,114),(108,143,115),(109,144,116),(110,145,117)])
Matrix representation ►G ⊆ 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] >;
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 |
In GAP, Magma, Sage, TeX
C_2\times C_{10}\times 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
×
⋊
ℤ
𝔽
○
≀