direct product, non-abelian, soluble
Aliases: C5×Q8⋊Dic3, SL2(𝔽3)⋊1C20, C10.2GL2(𝔽3), C10.2CSU2(𝔽3), Q8⋊(C5×Dic3), (C2×C10).9S4, (C5×Q8)⋊3Dic3, C22.3(C5×S4), (Q8×C10).3S3, C10.8(A4⋊C4), C2.(C5×GL2(𝔽3)), C2.(C5×CSU2(𝔽3)), (C5×SL2(𝔽3))⋊7C4, (C2×SL2(𝔽3)).1C10, (C10×SL2(𝔽3)).4C2, C2.2(C5×A4⋊C4), (C2×Q8).1(C5×S3), SmallGroup(480,256)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C10×SL2(𝔽3) — C5×Q8⋊Dic3 |
| SL2(𝔽3) — C5×Q8⋊Dic3 |
Generators and relations for C5×Q8⋊Dic3
G = < a,b,c,d,e | a5=b4=d6=1, c2=b2, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece-1=b-1, dbd-1=c, ebe-1=b2c, dcd-1=bc, ede-1=d-1 >
Smallest permutation representation of C5×Q8⋊Dic3
►On 160 points
Generators in S160
(1 26 3 20 24)(2 25 4 19 23)(5 35 14 12 15)(6 36 13 11 16)(7 22 28 37 40)(8 21 27 38 39)(9 34 18 32 29)(10 33 17 31 30)(41 85 55 61 124)(42 86 56 62 119)(43 87 57 63 120)(44 88 58 64 121)(45 83 53 59 122)(46 84 54 60 123)(47 103 130 115 147)(48 104 125 116 148)(49 105 126 117 143)(50 106 127 118 144)(51 101 128 113 145)(52 102 129 114 146)(65 153 97 73 81)(66 154 98 74 82)(67 149 99 75 77)(68 150 100 76 78)(69 151 95 71 79)(70 152 96 72 80)(89 157 137 108 135)(90 158 138 109 136)(91 159 139 110 131)(92 160 140 111 132)(93 155 141 112 133)(94 156 142 107 134)
(1 59 9 154)(2 62 10 151)(3 45 18 74)(4 42 17 71)(5 93 28 114)(6 90 27 117)(7 102 12 112)(8 105 11 109)(13 138 39 49)(14 141 40 52)(15 133 22 129)(16 136 21 126)(19 86 31 79)(20 83 32 82)(23 56 30 69)(24 53 29 66)(25 119 33 95)(26 122 34 98)(35 155 37 146)(36 158 38 143)(41 72 76 43)(44 75 73 46)(47 51 142 140)(48 139 137 50)(54 58 67 65)(55 70 68 57)(60 64 149 153)(61 152 150 63)(77 81 84 88)(78 87 85 80)(89 118 116 91)(92 115 113 94)(96 100 120 124)(97 123 121 99)(101 107 111 103)(104 110 108 106)(125 131 135 127)(128 134 132 130)(144 148 159 157)(145 156 160 147)
(1 61 9 150)(2 64 10 153)(3 41 18 76)(4 44 17 73)(5 89 28 116)(6 92 27 113)(7 104 12 108)(8 101 11 111)(13 140 39 51)(14 137 40 48)(15 135 22 125)(16 132 21 128)(19 88 31 81)(20 85 32 78)(23 58 30 65)(24 55 29 68)(25 121 33 97)(26 124 34 100)(35 157 37 148)(36 160 38 145)(42 46 71 75)(43 74 72 45)(47 138 142 49)(50 141 139 52)(53 57 66 70)(54 69 67 56)(59 63 154 152)(60 151 149 62)(77 86 84 79)(80 83 87 82)(90 94 117 115)(91 114 118 93)(95 99 119 123)(96 122 120 98)(102 106 112 110)(103 109 107 105)(126 130 136 134)(127 133 131 129)(143 147 158 156)(144 155 159 146)
(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 11 2 12)(3 6 4 5)(7 9 8 10)(13 23 14 24)(15 26 16 25)(17 28 18 27)(19 35 20 36)(21 33 22 34)(29 39 30 40)(31 37 32 38)(41 117 44 114)(42 116 45 113)(43 115 46 118)(47 54 50 57)(48 53 51 56)(49 58 52 55)(59 101 62 104)(60 106 63 103)(61 105 64 102)(65 141 68 138)(66 140 69 137)(67 139 70 142)(71 89 74 92)(72 94 75 91)(73 93 76 90)(77 159 80 156)(78 158 81 155)(79 157 82 160)(83 145 86 148)(84 144 87 147)(85 143 88 146)(95 135 98 132)(96 134 99 131)(97 133 100 136)(107 149 110 152)(108 154 111 151)(109 153 112 150)(119 125 122 128)(120 130 123 127)(121 129 124 126)
G:=sub<Sym(160)| (1,26,3,20,24)(2,25,4,19,23)(5,35,14,12,15)(6,36,13,11,16)(7,22,28,37,40)(8,21,27,38,39)(9,34,18,32,29)(10,33,17,31,30)(41,85,55,61,124)(42,86,56,62,119)(43,87,57,63,120)(44,88,58,64,121)(45,83,53,59,122)(46,84,54,60,123)(47,103,130,115,147)(48,104,125,116,148)(49,105,126,117,143)(50,106,127,118,144)(51,101,128,113,145)(52,102,129,114,146)(65,153,97,73,81)(66,154,98,74,82)(67,149,99,75,77)(68,150,100,76,78)(69,151,95,71,79)(70,152,96,72,80)(89,157,137,108,135)(90,158,138,109,136)(91,159,139,110,131)(92,160,140,111,132)(93,155,141,112,133)(94,156,142,107,134), (1,59,9,154)(2,62,10,151)(3,45,18,74)(4,42,17,71)(5,93,28,114)(6,90,27,117)(7,102,12,112)(8,105,11,109)(13,138,39,49)(14,141,40,52)(15,133,22,129)(16,136,21,126)(19,86,31,79)(20,83,32,82)(23,56,30,69)(24,53,29,66)(25,119,33,95)(26,122,34,98)(35,155,37,146)(36,158,38,143)(41,72,76,43)(44,75,73,46)(47,51,142,140)(48,139,137,50)(54,58,67,65)(55,70,68,57)(60,64,149,153)(61,152,150,63)(77,81,84,88)(78,87,85,80)(89,118,116,91)(92,115,113,94)(96,100,120,124)(97,123,121,99)(101,107,111,103)(104,110,108,106)(125,131,135,127)(128,134,132,130)(144,148,159,157)(145,156,160,147), (1,61,9,150)(2,64,10,153)(3,41,18,76)(4,44,17,73)(5,89,28,116)(6,92,27,113)(7,104,12,108)(8,101,11,111)(13,140,39,51)(14,137,40,48)(15,135,22,125)(16,132,21,128)(19,88,31,81)(20,85,32,78)(23,58,30,65)(24,55,29,68)(25,121,33,97)(26,124,34,100)(35,157,37,148)(36,160,38,145)(42,46,71,75)(43,74,72,45)(47,138,142,49)(50,141,139,52)(53,57,66,70)(54,69,67,56)(59,63,154,152)(60,151,149,62)(77,86,84,79)(80,83,87,82)(90,94,117,115)(91,114,118,93)(95,99,119,123)(96,122,120,98)(102,106,112,110)(103,109,107,105)(126,130,136,134)(127,133,131,129)(143,147,158,156)(144,155,159,146), (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), (1,11,2,12)(3,6,4,5)(7,9,8,10)(13,23,14,24)(15,26,16,25)(17,28,18,27)(19,35,20,36)(21,33,22,34)(29,39,30,40)(31,37,32,38)(41,117,44,114)(42,116,45,113)(43,115,46,118)(47,54,50,57)(48,53,51,56)(49,58,52,55)(59,101,62,104)(60,106,63,103)(61,105,64,102)(65,141,68,138)(66,140,69,137)(67,139,70,142)(71,89,74,92)(72,94,75,91)(73,93,76,90)(77,159,80,156)(78,158,81,155)(79,157,82,160)(83,145,86,148)(84,144,87,147)(85,143,88,146)(95,135,98,132)(96,134,99,131)(97,133,100,136)(107,149,110,152)(108,154,111,151)(109,153,112,150)(119,125,122,128)(120,130,123,127)(121,129,124,126)>;
G:=Group( (1,26,3,20,24)(2,25,4,19,23)(5,35,14,12,15)(6,36,13,11,16)(7,22,28,37,40)(8,21,27,38,39)(9,34,18,32,29)(10,33,17,31,30)(41,85,55,61,124)(42,86,56,62,119)(43,87,57,63,120)(44,88,58,64,121)(45,83,53,59,122)(46,84,54,60,123)(47,103,130,115,147)(48,104,125,116,148)(49,105,126,117,143)(50,106,127,118,144)(51,101,128,113,145)(52,102,129,114,146)(65,153,97,73,81)(66,154,98,74,82)(67,149,99,75,77)(68,150,100,76,78)(69,151,95,71,79)(70,152,96,72,80)(89,157,137,108,135)(90,158,138,109,136)(91,159,139,110,131)(92,160,140,111,132)(93,155,141,112,133)(94,156,142,107,134), (1,59,9,154)(2,62,10,151)(3,45,18,74)(4,42,17,71)(5,93,28,114)(6,90,27,117)(7,102,12,112)(8,105,11,109)(13,138,39,49)(14,141,40,52)(15,133,22,129)(16,136,21,126)(19,86,31,79)(20,83,32,82)(23,56,30,69)(24,53,29,66)(25,119,33,95)(26,122,34,98)(35,155,37,146)(36,158,38,143)(41,72,76,43)(44,75,73,46)(47,51,142,140)(48,139,137,50)(54,58,67,65)(55,70,68,57)(60,64,149,153)(61,152,150,63)(77,81,84,88)(78,87,85,80)(89,118,116,91)(92,115,113,94)(96,100,120,124)(97,123,121,99)(101,107,111,103)(104,110,108,106)(125,131,135,127)(128,134,132,130)(144,148,159,157)(145,156,160,147), (1,61,9,150)(2,64,10,153)(3,41,18,76)(4,44,17,73)(5,89,28,116)(6,92,27,113)(7,104,12,108)(8,101,11,111)(13,140,39,51)(14,137,40,48)(15,135,22,125)(16,132,21,128)(19,88,31,81)(20,85,32,78)(23,58,30,65)(24,55,29,68)(25,121,33,97)(26,124,34,100)(35,157,37,148)(36,160,38,145)(42,46,71,75)(43,74,72,45)(47,138,142,49)(50,141,139,52)(53,57,66,70)(54,69,67,56)(59,63,154,152)(60,151,149,62)(77,86,84,79)(80,83,87,82)(90,94,117,115)(91,114,118,93)(95,99,119,123)(96,122,120,98)(102,106,112,110)(103,109,107,105)(126,130,136,134)(127,133,131,129)(143,147,158,156)(144,155,159,146), (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), (1,11,2,12)(3,6,4,5)(7,9,8,10)(13,23,14,24)(15,26,16,25)(17,28,18,27)(19,35,20,36)(21,33,22,34)(29,39,30,40)(31,37,32,38)(41,117,44,114)(42,116,45,113)(43,115,46,118)(47,54,50,57)(48,53,51,56)(49,58,52,55)(59,101,62,104)(60,106,63,103)(61,105,64,102)(65,141,68,138)(66,140,69,137)(67,139,70,142)(71,89,74,92)(72,94,75,91)(73,93,76,90)(77,159,80,156)(78,158,81,155)(79,157,82,160)(83,145,86,148)(84,144,87,147)(85,143,88,146)(95,135,98,132)(96,134,99,131)(97,133,100,136)(107,149,110,152)(108,154,111,151)(109,153,112,150)(119,125,122,128)(120,130,123,127)(121,129,124,126) );
G=PermutationGroup([[(1,26,3,20,24),(2,25,4,19,23),(5,35,14,12,15),(6,36,13,11,16),(7,22,28,37,40),(8,21,27,38,39),(9,34,18,32,29),(10,33,17,31,30),(41,85,55,61,124),(42,86,56,62,119),(43,87,57,63,120),(44,88,58,64,121),(45,83,53,59,122),(46,84,54,60,123),(47,103,130,115,147),(48,104,125,116,148),(49,105,126,117,143),(50,106,127,118,144),(51,101,128,113,145),(52,102,129,114,146),(65,153,97,73,81),(66,154,98,74,82),(67,149,99,75,77),(68,150,100,76,78),(69,151,95,71,79),(70,152,96,72,80),(89,157,137,108,135),(90,158,138,109,136),(91,159,139,110,131),(92,160,140,111,132),(93,155,141,112,133),(94,156,142,107,134)], [(1,59,9,154),(2,62,10,151),(3,45,18,74),(4,42,17,71),(5,93,28,114),(6,90,27,117),(7,102,12,112),(8,105,11,109),(13,138,39,49),(14,141,40,52),(15,133,22,129),(16,136,21,126),(19,86,31,79),(20,83,32,82),(23,56,30,69),(24,53,29,66),(25,119,33,95),(26,122,34,98),(35,155,37,146),(36,158,38,143),(41,72,76,43),(44,75,73,46),(47,51,142,140),(48,139,137,50),(54,58,67,65),(55,70,68,57),(60,64,149,153),(61,152,150,63),(77,81,84,88),(78,87,85,80),(89,118,116,91),(92,115,113,94),(96,100,120,124),(97,123,121,99),(101,107,111,103),(104,110,108,106),(125,131,135,127),(128,134,132,130),(144,148,159,157),(145,156,160,147)], [(1,61,9,150),(2,64,10,153),(3,41,18,76),(4,44,17,73),(5,89,28,116),(6,92,27,113),(7,104,12,108),(8,101,11,111),(13,140,39,51),(14,137,40,48),(15,135,22,125),(16,132,21,128),(19,88,31,81),(20,85,32,78),(23,58,30,65),(24,55,29,68),(25,121,33,97),(26,124,34,100),(35,157,37,148),(36,160,38,145),(42,46,71,75),(43,74,72,45),(47,138,142,49),(50,141,139,52),(53,57,66,70),(54,69,67,56),(59,63,154,152),(60,151,149,62),(77,86,84,79),(80,83,87,82),(90,94,117,115),(91,114,118,93),(95,99,119,123),(96,122,120,98),(102,106,112,110),(103,109,107,105),(126,130,136,134),(127,133,131,129),(143,147,158,156),(144,155,159,146)], [(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)], [(1,11,2,12),(3,6,4,5),(7,9,8,10),(13,23,14,24),(15,26,16,25),(17,28,18,27),(19,35,20,36),(21,33,22,34),(29,39,30,40),(31,37,32,38),(41,117,44,114),(42,116,45,113),(43,115,46,118),(47,54,50,57),(48,53,51,56),(49,58,52,55),(59,101,62,104),(60,106,63,103),(61,105,64,102),(65,141,68,138),(66,140,69,137),(67,139,70,142),(71,89,74,92),(72,94,75,91),(73,93,76,90),(77,159,80,156),(78,158,81,155),(79,157,82,160),(83,145,86,148),(84,144,87,147),(85,143,88,146),(95,135,98,132),(96,134,99,131),(97,133,100,136),(107,149,110,152),(108,154,111,151),(109,153,112,150),(119,125,122,128),(120,130,123,127),(121,129,124,126)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 6 | 6 | 12 | 12 | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 6 | ··· | 6 | 12 | ··· | 12 | 8 | ··· | 8 | 6 | ··· | 6 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | - | + | - | + | ||||||||||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | S3 | Dic3 | C5×S3 | CSU2(𝔽3) | GL2(𝔽3) | C5×Dic3 | C5×CSU2(𝔽3) | C5×GL2(𝔽3) | S4 | A4⋊C4 | C5×S4 | C5×A4⋊C4 | CSU2(𝔽3) | GL2(𝔽3) | C5×CSU2(𝔽3) | C5×GL2(𝔽3) |
| kernel | C5×Q8⋊Dic3 | C10×SL2(𝔽3) | C5×SL2(𝔽3) | Q8⋊Dic3 | C2×SL2(𝔽3) | SL2(𝔽3) | Q8×C10 | C5×Q8 | C2×Q8 | C10 | C10 | Q8 | C2 | C2 | C2×C10 | C10 | C22 | C2 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 1 | 4 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 8 | 8 | 1 | 1 | 4 | 4 |
Matrix representation of C5×Q8⋊Dic3 ►in GL4(𝔽241) generated by
| 91 | 0 | 0 | 0 |
| 0 | 91 | 0 | 0 |
| 0 | 0 | 87 | 0 |
| 0 | 0 | 0 | 87 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 115 | 221 |
| 0 | 0 | 107 | 126 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 106 | 125 |
| 0 | 0 | 20 | 135 |
| 0 | 240 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 240 |
| 64 | 0 | 0 | 0 |
| 177 | 177 | 0 | 0 |
| 0 | 0 | 171 | 137 |
| 0 | 0 | 207 | 70 |
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,87,0,0,0,0,87],[1,0,0,0,0,1,0,0,0,0,115,107,0,0,221,126],[1,0,0,0,0,1,0,0,0,0,106,20,0,0,125,135],[0,1,0,0,240,1,0,0,0,0,0,240,0,0,1,240],[64,177,0,0,0,177,0,0,0,0,171,207,0,0,137,70] >;
C5×Q8⋊Dic3 in GAP, Magma, Sage, TeX
C_5\times Q_8\rtimes {\rm Dic}_3 % in TeX
G:=Group("C5xQ8:Dic3"); // GroupNames label
G:=SmallGroup(480,256);
// by ID
G=gap.SmallGroup(480,256);
# by ID
G:=PCGroup([7,-2,-5,-2,-3,-2,2,-2,70,1123,4204,655,172,2525,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=d^6=1,c^2=b^2,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*c*e^-1=b^-1,d*b*d^-1=c,e*b*e^-1=b^2*c,d*c*d^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations
Export