direct product, non-abelian, soluble
Aliases: C10×CSU2(𝔽3), C2.5(C10×S4), C10.30(C2×S4), (C2×C10).10S4, C22.4(C5×S4), (C5×Q8).13D6, (Q8×C10).4S3, Q8.1(S3×C10), (C2×SL2(𝔽3)).2C10, (C10×SL2(𝔽3)).5C2, SL2(𝔽3).1(C2×C10), (C5×SL2(𝔽3)).13C22, (C2×Q8).2(C5×S3), SmallGroup(480,1016)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C5×SL2(𝔽3) — C5×CSU2(𝔽3) — C10×CSU2(𝔽3) |
| SL2(𝔽3) — C10×CSU2(𝔽3) |
Subgroups: 226 in 78 conjugacy classes, 24 normal (16 characteristic)
C1, C2, C2 [×2], C3, C4 [×4], C22, C5, C6 [×3], C8 [×2], C2×C4 [×2], Q8, Q8 [×4], C10, C10 [×2], Dic3 [×2], C2×C6, C15, C2×C8, Q16 [×4], C2×Q8, C2×Q8, C20 [×4], C2×C10, SL2(𝔽3), C2×Dic3, C30 [×3], C2×Q16, C40 [×2], C2×C20 [×2], C5×Q8, C5×Q8 [×4], CSU2(𝔽3) [×2], C2×SL2(𝔽3), C5×Dic3 [×2], C2×C30, C2×C40, C5×Q16 [×4], Q8×C10, Q8×C10, C2×CSU2(𝔽3), C5×SL2(𝔽3), C10×Dic3, C10×Q16, C5×CSU2(𝔽3) [×2], C10×SL2(𝔽3), C10×CSU2(𝔽3)
Quotients:
C1, C2 [×3], C22, C5, S3, C10 [×3], D6, C2×C10, S4, C5×S3, CSU2(𝔽3) [×2], C2×S4, S3×C10, C2×CSU2(𝔽3), C5×S4, C5×CSU2(𝔽3) [×2], C10×S4, C10×CSU2(𝔽3)
Generators and relations
G = < a,b,c,d,e | a10=b4=d3=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece-1=b-1, dbd-1=bc, ebe-1=b2c, dcd-1=b, ede-1=d-1 >
►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)
(1 145 117 107)(2 146 118 108)(3 147 119 109)(4 148 120 110)(5 149 111 101)(6 150 112 102)(7 141 113 103)(8 142 114 104)(9 143 115 105)(10 144 116 106)(11 23 57 75)(12 24 58 76)(13 25 59 77)(14 26 60 78)(15 27 51 79)(16 28 52 80)(17 29 53 71)(18 30 54 72)(19 21 55 73)(20 22 56 74)(31 47 157 65)(32 48 158 66)(33 49 159 67)(34 50 160 68)(35 41 151 69)(36 42 152 70)(37 43 153 61)(38 44 154 62)(39 45 155 63)(40 46 156 64)(81 96 124 140)(82 97 125 131)(83 98 126 132)(84 99 127 133)(85 100 128 134)(86 91 129 135)(87 92 130 136)(88 93 121 137)(89 94 122 138)(90 95 123 139)
(1 84 117 127)(2 85 118 128)(3 86 119 129)(4 87 120 130)(5 88 111 121)(6 89 112 122)(7 90 113 123)(8 81 114 124)(9 82 115 125)(10 83 116 126)(11 39 57 155)(12 40 58 156)(13 31 59 157)(14 32 60 158)(15 33 51 159)(16 34 52 160)(17 35 53 151)(18 36 54 152)(19 37 55 153)(20 38 56 154)(21 61 73 43)(22 62 74 44)(23 63 75 45)(24 64 76 46)(25 65 77 47)(26 66 78 48)(27 67 79 49)(28 68 80 50)(29 69 71 41)(30 70 72 42)(91 147 135 109)(92 148 136 110)(93 149 137 101)(94 150 138 102)(95 141 139 103)(96 142 140 104)(97 143 131 105)(98 144 132 106)(99 145 133 107)(100 146 134 108)
(11 155 23)(12 156 24)(13 157 25)(14 158 26)(15 159 27)(16 160 28)(17 151 29)(18 152 30)(19 153 21)(20 154 22)(31 77 59)(32 78 60)(33 79 51)(34 80 52)(35 71 53)(36 72 54)(37 73 55)(38 74 56)(39 75 57)(40 76 58)(81 140 142)(82 131 143)(83 132 144)(84 133 145)(85 134 146)(86 135 147)(87 136 148)(88 137 149)(89 138 150)(90 139 141)(91 109 129)(92 110 130)(93 101 121)(94 102 122)(95 103 123)(96 104 124)(97 105 125)(98 106 126)(99 107 127)(100 108 128)
(1 48 117 66)(2 49 118 67)(3 50 119 68)(4 41 120 69)(5 42 111 70)(6 43 112 61)(7 44 113 62)(8 45 114 63)(9 46 115 64)(10 47 116 65)(11 140 57 96)(12 131 58 97)(13 132 59 98)(14 133 60 99)(15 134 51 100)(16 135 52 91)(17 136 53 92)(18 137 54 93)(19 138 55 94)(20 139 56 95)(21 150 73 102)(22 141 74 103)(23 142 75 104)(24 143 76 105)(25 144 77 106)(26 145 78 107)(27 146 79 108)(28 147 80 109)(29 148 71 110)(30 149 72 101)(31 126 157 83)(32 127 158 84)(33 128 159 85)(34 129 160 86)(35 130 151 87)(36 121 152 88)(37 122 153 89)(38 123 154 90)(39 124 155 81)(40 125 156 82)
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), (1,145,117,107)(2,146,118,108)(3,147,119,109)(4,148,120,110)(5,149,111,101)(6,150,112,102)(7,141,113,103)(8,142,114,104)(9,143,115,105)(10,144,116,106)(11,23,57,75)(12,24,58,76)(13,25,59,77)(14,26,60,78)(15,27,51,79)(16,28,52,80)(17,29,53,71)(18,30,54,72)(19,21,55,73)(20,22,56,74)(31,47,157,65)(32,48,158,66)(33,49,159,67)(34,50,160,68)(35,41,151,69)(36,42,152,70)(37,43,153,61)(38,44,154,62)(39,45,155,63)(40,46,156,64)(81,96,124,140)(82,97,125,131)(83,98,126,132)(84,99,127,133)(85,100,128,134)(86,91,129,135)(87,92,130,136)(88,93,121,137)(89,94,122,138)(90,95,123,139), (1,84,117,127)(2,85,118,128)(3,86,119,129)(4,87,120,130)(5,88,111,121)(6,89,112,122)(7,90,113,123)(8,81,114,124)(9,82,115,125)(10,83,116,126)(11,39,57,155)(12,40,58,156)(13,31,59,157)(14,32,60,158)(15,33,51,159)(16,34,52,160)(17,35,53,151)(18,36,54,152)(19,37,55,153)(20,38,56,154)(21,61,73,43)(22,62,74,44)(23,63,75,45)(24,64,76,46)(25,65,77,47)(26,66,78,48)(27,67,79,49)(28,68,80,50)(29,69,71,41)(30,70,72,42)(91,147,135,109)(92,148,136,110)(93,149,137,101)(94,150,138,102)(95,141,139,103)(96,142,140,104)(97,143,131,105)(98,144,132,106)(99,145,133,107)(100,146,134,108), (11,155,23)(12,156,24)(13,157,25)(14,158,26)(15,159,27)(16,160,28)(17,151,29)(18,152,30)(19,153,21)(20,154,22)(31,77,59)(32,78,60)(33,79,51)(34,80,52)(35,71,53)(36,72,54)(37,73,55)(38,74,56)(39,75,57)(40,76,58)(81,140,142)(82,131,143)(83,132,144)(84,133,145)(85,134,146)(86,135,147)(87,136,148)(88,137,149)(89,138,150)(90,139,141)(91,109,129)(92,110,130)(93,101,121)(94,102,122)(95,103,123)(96,104,124)(97,105,125)(98,106,126)(99,107,127)(100,108,128), (1,48,117,66)(2,49,118,67)(3,50,119,68)(4,41,120,69)(5,42,111,70)(6,43,112,61)(7,44,113,62)(8,45,114,63)(9,46,115,64)(10,47,116,65)(11,140,57,96)(12,131,58,97)(13,132,59,98)(14,133,60,99)(15,134,51,100)(16,135,52,91)(17,136,53,92)(18,137,54,93)(19,138,55,94)(20,139,56,95)(21,150,73,102)(22,141,74,103)(23,142,75,104)(24,143,76,105)(25,144,77,106)(26,145,78,107)(27,146,79,108)(28,147,80,109)(29,148,71,110)(30,149,72,101)(31,126,157,83)(32,127,158,84)(33,128,159,85)(34,129,160,86)(35,130,151,87)(36,121,152,88)(37,122,153,89)(38,123,154,90)(39,124,155,81)(40,125,156,82)>;
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), (1,145,117,107)(2,146,118,108)(3,147,119,109)(4,148,120,110)(5,149,111,101)(6,150,112,102)(7,141,113,103)(8,142,114,104)(9,143,115,105)(10,144,116,106)(11,23,57,75)(12,24,58,76)(13,25,59,77)(14,26,60,78)(15,27,51,79)(16,28,52,80)(17,29,53,71)(18,30,54,72)(19,21,55,73)(20,22,56,74)(31,47,157,65)(32,48,158,66)(33,49,159,67)(34,50,160,68)(35,41,151,69)(36,42,152,70)(37,43,153,61)(38,44,154,62)(39,45,155,63)(40,46,156,64)(81,96,124,140)(82,97,125,131)(83,98,126,132)(84,99,127,133)(85,100,128,134)(86,91,129,135)(87,92,130,136)(88,93,121,137)(89,94,122,138)(90,95,123,139), (1,84,117,127)(2,85,118,128)(3,86,119,129)(4,87,120,130)(5,88,111,121)(6,89,112,122)(7,90,113,123)(8,81,114,124)(9,82,115,125)(10,83,116,126)(11,39,57,155)(12,40,58,156)(13,31,59,157)(14,32,60,158)(15,33,51,159)(16,34,52,160)(17,35,53,151)(18,36,54,152)(19,37,55,153)(20,38,56,154)(21,61,73,43)(22,62,74,44)(23,63,75,45)(24,64,76,46)(25,65,77,47)(26,66,78,48)(27,67,79,49)(28,68,80,50)(29,69,71,41)(30,70,72,42)(91,147,135,109)(92,148,136,110)(93,149,137,101)(94,150,138,102)(95,141,139,103)(96,142,140,104)(97,143,131,105)(98,144,132,106)(99,145,133,107)(100,146,134,108), (11,155,23)(12,156,24)(13,157,25)(14,158,26)(15,159,27)(16,160,28)(17,151,29)(18,152,30)(19,153,21)(20,154,22)(31,77,59)(32,78,60)(33,79,51)(34,80,52)(35,71,53)(36,72,54)(37,73,55)(38,74,56)(39,75,57)(40,76,58)(81,140,142)(82,131,143)(83,132,144)(84,133,145)(85,134,146)(86,135,147)(87,136,148)(88,137,149)(89,138,150)(90,139,141)(91,109,129)(92,110,130)(93,101,121)(94,102,122)(95,103,123)(96,104,124)(97,105,125)(98,106,126)(99,107,127)(100,108,128), (1,48,117,66)(2,49,118,67)(3,50,119,68)(4,41,120,69)(5,42,111,70)(6,43,112,61)(7,44,113,62)(8,45,114,63)(9,46,115,64)(10,47,116,65)(11,140,57,96)(12,131,58,97)(13,132,59,98)(14,133,60,99)(15,134,51,100)(16,135,52,91)(17,136,53,92)(18,137,54,93)(19,138,55,94)(20,139,56,95)(21,150,73,102)(22,141,74,103)(23,142,75,104)(24,143,76,105)(25,144,77,106)(26,145,78,107)(27,146,79,108)(28,147,80,109)(29,148,71,110)(30,149,72,101)(31,126,157,83)(32,127,158,84)(33,128,159,85)(34,129,160,86)(35,130,151,87)(36,121,152,88)(37,122,153,89)(38,123,154,90)(39,124,155,81)(40,125,156,82) );
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)], [(1,145,117,107),(2,146,118,108),(3,147,119,109),(4,148,120,110),(5,149,111,101),(6,150,112,102),(7,141,113,103),(8,142,114,104),(9,143,115,105),(10,144,116,106),(11,23,57,75),(12,24,58,76),(13,25,59,77),(14,26,60,78),(15,27,51,79),(16,28,52,80),(17,29,53,71),(18,30,54,72),(19,21,55,73),(20,22,56,74),(31,47,157,65),(32,48,158,66),(33,49,159,67),(34,50,160,68),(35,41,151,69),(36,42,152,70),(37,43,153,61),(38,44,154,62),(39,45,155,63),(40,46,156,64),(81,96,124,140),(82,97,125,131),(83,98,126,132),(84,99,127,133),(85,100,128,134),(86,91,129,135),(87,92,130,136),(88,93,121,137),(89,94,122,138),(90,95,123,139)], [(1,84,117,127),(2,85,118,128),(3,86,119,129),(4,87,120,130),(5,88,111,121),(6,89,112,122),(7,90,113,123),(8,81,114,124),(9,82,115,125),(10,83,116,126),(11,39,57,155),(12,40,58,156),(13,31,59,157),(14,32,60,158),(15,33,51,159),(16,34,52,160),(17,35,53,151),(18,36,54,152),(19,37,55,153),(20,38,56,154),(21,61,73,43),(22,62,74,44),(23,63,75,45),(24,64,76,46),(25,65,77,47),(26,66,78,48),(27,67,79,49),(28,68,80,50),(29,69,71,41),(30,70,72,42),(91,147,135,109),(92,148,136,110),(93,149,137,101),(94,150,138,102),(95,141,139,103),(96,142,140,104),(97,143,131,105),(98,144,132,106),(99,145,133,107),(100,146,134,108)], [(11,155,23),(12,156,24),(13,157,25),(14,158,26),(15,159,27),(16,160,28),(17,151,29),(18,152,30),(19,153,21),(20,154,22),(31,77,59),(32,78,60),(33,79,51),(34,80,52),(35,71,53),(36,72,54),(37,73,55),(38,74,56),(39,75,57),(40,76,58),(81,140,142),(82,131,143),(83,132,144),(84,133,145),(85,134,146),(86,135,147),(87,136,148),(88,137,149),(89,138,150),(90,139,141),(91,109,129),(92,110,130),(93,101,121),(94,102,122),(95,103,123),(96,104,124),(97,105,125),(98,106,126),(99,107,127),(100,108,128)], [(1,48,117,66),(2,49,118,67),(3,50,119,68),(4,41,120,69),(5,42,111,70),(6,43,112,61),(7,44,113,62),(8,45,114,63),(9,46,115,64),(10,47,116,65),(11,140,57,96),(12,131,58,97),(13,132,59,98),(14,133,60,99),(15,134,51,100),(16,135,52,91),(17,136,53,92),(18,137,54,93),(19,138,55,94),(20,139,56,95),(21,150,73,102),(22,141,74,103),(23,142,75,104),(24,143,76,105),(25,144,77,106),(26,145,78,107),(27,146,79,108),(28,147,80,109),(29,148,71,110),(30,149,72,101),(31,126,157,83),(32,127,158,84),(33,128,159,85),(34,129,160,86),(35,130,151,87),(36,121,152,88),(37,122,153,89),(38,123,154,90),(39,124,155,81),(40,125,156,82)])
Matrix representation ►G ⊆ GL4(𝔽241) generated by
| 143 | 0 | 0 | 0 |
| 0 | 143 | 0 | 0 |
| 0 | 0 | 150 | 0 |
| 0 | 0 | 0 | 150 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 12 |
| 0 | 0 | 26 | 228 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 215 |
| 0 | 0 | 229 | 228 |
| 0 | 240 | 0 | 0 |
| 1 | 240 | 0 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 0 | 1 | 240 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 177 |
| 0 | 0 | 177 | 0 |
G:=sub<GL(4,GF(241))| [143,0,0,0,0,143,0,0,0,0,150,0,0,0,0,150],[1,0,0,0,0,1,0,0,0,0,13,26,0,0,12,228],[1,0,0,0,0,1,0,0,0,0,13,229,0,0,215,228],[0,1,0,0,240,240,0,0,0,0,0,1,0,0,240,240],[0,1,0,0,1,0,0,0,0,0,0,177,0,0,177,0] >;
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 | 3 | 3 | 3 | 3 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | |||||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | S3 | D6 | C5×S3 | CSU2(𝔽3) | S3×C10 | C5×CSU2(𝔽3) | S4 | C2×S4 | C5×S4 | C10×S4 | CSU2(𝔽3) | C5×CSU2(𝔽3) |
| kernel | C10×CSU2(𝔽3) | C5×CSU2(𝔽3) | C10×SL2(𝔽3) | C2×CSU2(𝔽3) | CSU2(𝔽3) | C2×SL2(𝔽3) | Q8×C10 | C5×Q8 | C2×Q8 | C10 | Q8 | C2 | C2×C10 | C10 | C22 | C2 | C10 | C2 |
| # reps | 1 | 2 | 1 | 4 | 8 | 4 | 1 | 1 | 4 | 4 | 4 | 16 | 2 | 2 | 8 | 8 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_{10}\times CSU_2({\mathbb F}_3) % in TeX
G:=Group("C10xCSU(2,3)"); // GroupNames label
G:=SmallGroup(480,1016);
// by ID
G=gap.SmallGroup(480,1016);
# by ID
G:=PCGroup([7,-2,-2,-5,-3,-2,2,-2,1680,1123,4204,655,172,2525,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=d^3=1,c^2=e^2=b^2,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=b*c,e*b*e^-1=b^2*c,d*c*d^-1=b,e*d*e^-1=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀