Aliases: C20.2S4, Q8.3D30, SL2(𝔽3).9D10, C4.2(C5⋊S4), C4.A4.1D5, C10.22(C2×S4), C5⋊2(C4.S4), C4○D4.2D15, Q8.D15⋊2C2, (C5×Q8).10D6, (C5×SL2(𝔽3)).9C22, C2.8(C2×C5⋊S4), (C5×C4.A4).1C2, (C5×C4○D4).2S3, SmallGroup(480,1030)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C5×SL2(𝔽3) — C20.2S4 |
| C5×SL2(𝔽3) — C20.2S4 |
Subgroups: 506 in 72 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C2, C3, C4, C4 [×3], C22, C5, C6, C8 [×2], C2×C4 [×2], D4, Q8, Q8 [×3], C10, C10, Dic3 [×2], C12, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5 [×2], C20, C20, C2×C10, SL2(𝔽3), Dic6, C30, C8.C22, C5⋊2C8 [×2], Dic10 [×3], C2×Dic5, C2×C20, C5×D4, C5×Q8, CSU2(𝔽3) [×2], C4.A4, Dic15 [×2], C60, C4.Dic5, D4.D5 [×2], C5⋊Q16 [×2], C2×Dic10, C5×C4○D4, C4.S4, C5×SL2(𝔽3), Dic30, D4.9D10, Q8.D15 [×2], C5×C4.A4, C20.2S4
Quotients:
C1, C2 [×3], C22, S3, D5, D6, D10, S4, D15, C2×S4, D30, C4.S4, C5⋊S4, C2×C5⋊S4, C20.2S4
Generators and relations
G = < a,b,c,d,e | a20=d3=1, b2=c2=e2=a10, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc-1=a10b, dbd-1=a10bc, ebe-1=bc, dcd-1=b, ece-1=a10c, 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)
(1 157 11 147)(2 158 12 148)(3 159 13 149)(4 160 14 150)(5 141 15 151)(6 142 16 152)(7 143 17 153)(8 144 18 154)(9 145 19 155)(10 146 20 156)(21 97 31 87)(22 98 32 88)(23 99 33 89)(24 100 34 90)(25 81 35 91)(26 82 36 92)(27 83 37 93)(28 84 38 94)(29 85 39 95)(30 86 40 96)(41 127 51 137)(42 128 52 138)(43 129 53 139)(44 130 54 140)(45 131 55 121)(46 132 56 122)(47 133 57 123)(48 134 58 124)(49 135 59 125)(50 136 60 126)(61 108 71 118)(62 109 72 119)(63 110 73 120)(64 111 74 101)(65 112 75 102)(66 113 76 103)(67 114 77 104)(68 115 78 105)(69 116 79 106)(70 117 80 107)
(1 76 11 66)(2 77 12 67)(3 78 13 68)(4 79 14 69)(5 80 15 70)(6 61 16 71)(7 62 17 72)(8 63 18 73)(9 64 19 74)(10 65 20 75)(21 42 31 52)(22 43 32 53)(23 44 33 54)(24 45 34 55)(25 46 35 56)(26 47 36 57)(27 48 37 58)(28 49 38 59)(29 50 39 60)(30 51 40 41)(81 122 91 132)(82 123 92 133)(83 124 93 134)(84 125 94 135)(85 126 95 136)(86 127 96 137)(87 128 97 138)(88 129 98 139)(89 130 99 140)(90 131 100 121)(101 155 111 145)(102 156 112 146)(103 157 113 147)(104 158 114 148)(105 159 115 149)(106 160 116 150)(107 141 117 151)(108 142 118 152)(109 143 119 153)(110 144 120 154)
(41 127 96)(42 128 97)(43 129 98)(44 130 99)(45 131 100)(46 132 81)(47 133 82)(48 134 83)(49 135 84)(50 136 85)(51 137 86)(52 138 87)(53 139 88)(54 140 89)(55 121 90)(56 122 91)(57 123 92)(58 124 93)(59 125 94)(60 126 95)(61 108 142)(62 109 143)(63 110 144)(64 111 145)(65 112 146)(66 113 147)(67 114 148)(68 115 149)(69 116 150)(70 117 151)(71 118 152)(72 119 153)(73 120 154)(74 101 155)(75 102 156)(76 103 157)(77 104 158)(78 105 159)(79 106 160)(80 107 141)
(1 25 11 35)(2 24 12 34)(3 23 13 33)(4 22 14 32)(5 21 15 31)(6 40 16 30)(7 39 17 29)(8 38 18 28)(9 37 19 27)(10 36 20 26)(41 61 51 71)(42 80 52 70)(43 79 53 69)(44 78 54 68)(45 77 55 67)(46 76 56 66)(47 75 57 65)(48 74 58 64)(49 73 59 63)(50 72 60 62)(81 103 91 113)(82 102 92 112)(83 101 93 111)(84 120 94 110)(85 119 95 109)(86 118 96 108)(87 117 97 107)(88 116 98 106)(89 115 99 105)(90 114 100 104)(121 148 131 158)(122 147 132 157)(123 146 133 156)(124 145 134 155)(125 144 135 154)(126 143 136 153)(127 142 137 152)(128 141 138 151)(129 160 139 150)(130 159 140 149)
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), (1,157,11,147)(2,158,12,148)(3,159,13,149)(4,160,14,150)(5,141,15,151)(6,142,16,152)(7,143,17,153)(8,144,18,154)(9,145,19,155)(10,146,20,156)(21,97,31,87)(22,98,32,88)(23,99,33,89)(24,100,34,90)(25,81,35,91)(26,82,36,92)(27,83,37,93)(28,84,38,94)(29,85,39,95)(30,86,40,96)(41,127,51,137)(42,128,52,138)(43,129,53,139)(44,130,54,140)(45,131,55,121)(46,132,56,122)(47,133,57,123)(48,134,58,124)(49,135,59,125)(50,136,60,126)(61,108,71,118)(62,109,72,119)(63,110,73,120)(64,111,74,101)(65,112,75,102)(66,113,76,103)(67,114,77,104)(68,115,78,105)(69,116,79,106)(70,117,80,107), (1,76,11,66)(2,77,12,67)(3,78,13,68)(4,79,14,69)(5,80,15,70)(6,61,16,71)(7,62,17,72)(8,63,18,73)(9,64,19,74)(10,65,20,75)(21,42,31,52)(22,43,32,53)(23,44,33,54)(24,45,34,55)(25,46,35,56)(26,47,36,57)(27,48,37,58)(28,49,38,59)(29,50,39,60)(30,51,40,41)(81,122,91,132)(82,123,92,133)(83,124,93,134)(84,125,94,135)(85,126,95,136)(86,127,96,137)(87,128,97,138)(88,129,98,139)(89,130,99,140)(90,131,100,121)(101,155,111,145)(102,156,112,146)(103,157,113,147)(104,158,114,148)(105,159,115,149)(106,160,116,150)(107,141,117,151)(108,142,118,152)(109,143,119,153)(110,144,120,154), (41,127,96)(42,128,97)(43,129,98)(44,130,99)(45,131,100)(46,132,81)(47,133,82)(48,134,83)(49,135,84)(50,136,85)(51,137,86)(52,138,87)(53,139,88)(54,140,89)(55,121,90)(56,122,91)(57,123,92)(58,124,93)(59,125,94)(60,126,95)(61,108,142)(62,109,143)(63,110,144)(64,111,145)(65,112,146)(66,113,147)(67,114,148)(68,115,149)(69,116,150)(70,117,151)(71,118,152)(72,119,153)(73,120,154)(74,101,155)(75,102,156)(76,103,157)(77,104,158)(78,105,159)(79,106,160)(80,107,141), (1,25,11,35)(2,24,12,34)(3,23,13,33)(4,22,14,32)(5,21,15,31)(6,40,16,30)(7,39,17,29)(8,38,18,28)(9,37,19,27)(10,36,20,26)(41,61,51,71)(42,80,52,70)(43,79,53,69)(44,78,54,68)(45,77,55,67)(46,76,56,66)(47,75,57,65)(48,74,58,64)(49,73,59,63)(50,72,60,62)(81,103,91,113)(82,102,92,112)(83,101,93,111)(84,120,94,110)(85,119,95,109)(86,118,96,108)(87,117,97,107)(88,116,98,106)(89,115,99,105)(90,114,100,104)(121,148,131,158)(122,147,132,157)(123,146,133,156)(124,145,134,155)(125,144,135,154)(126,143,136,153)(127,142,137,152)(128,141,138,151)(129,160,139,150)(130,159,140,149)>;
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), (1,157,11,147)(2,158,12,148)(3,159,13,149)(4,160,14,150)(5,141,15,151)(6,142,16,152)(7,143,17,153)(8,144,18,154)(9,145,19,155)(10,146,20,156)(21,97,31,87)(22,98,32,88)(23,99,33,89)(24,100,34,90)(25,81,35,91)(26,82,36,92)(27,83,37,93)(28,84,38,94)(29,85,39,95)(30,86,40,96)(41,127,51,137)(42,128,52,138)(43,129,53,139)(44,130,54,140)(45,131,55,121)(46,132,56,122)(47,133,57,123)(48,134,58,124)(49,135,59,125)(50,136,60,126)(61,108,71,118)(62,109,72,119)(63,110,73,120)(64,111,74,101)(65,112,75,102)(66,113,76,103)(67,114,77,104)(68,115,78,105)(69,116,79,106)(70,117,80,107), (1,76,11,66)(2,77,12,67)(3,78,13,68)(4,79,14,69)(5,80,15,70)(6,61,16,71)(7,62,17,72)(8,63,18,73)(9,64,19,74)(10,65,20,75)(21,42,31,52)(22,43,32,53)(23,44,33,54)(24,45,34,55)(25,46,35,56)(26,47,36,57)(27,48,37,58)(28,49,38,59)(29,50,39,60)(30,51,40,41)(81,122,91,132)(82,123,92,133)(83,124,93,134)(84,125,94,135)(85,126,95,136)(86,127,96,137)(87,128,97,138)(88,129,98,139)(89,130,99,140)(90,131,100,121)(101,155,111,145)(102,156,112,146)(103,157,113,147)(104,158,114,148)(105,159,115,149)(106,160,116,150)(107,141,117,151)(108,142,118,152)(109,143,119,153)(110,144,120,154), (41,127,96)(42,128,97)(43,129,98)(44,130,99)(45,131,100)(46,132,81)(47,133,82)(48,134,83)(49,135,84)(50,136,85)(51,137,86)(52,138,87)(53,139,88)(54,140,89)(55,121,90)(56,122,91)(57,123,92)(58,124,93)(59,125,94)(60,126,95)(61,108,142)(62,109,143)(63,110,144)(64,111,145)(65,112,146)(66,113,147)(67,114,148)(68,115,149)(69,116,150)(70,117,151)(71,118,152)(72,119,153)(73,120,154)(74,101,155)(75,102,156)(76,103,157)(77,104,158)(78,105,159)(79,106,160)(80,107,141), (1,25,11,35)(2,24,12,34)(3,23,13,33)(4,22,14,32)(5,21,15,31)(6,40,16,30)(7,39,17,29)(8,38,18,28)(9,37,19,27)(10,36,20,26)(41,61,51,71)(42,80,52,70)(43,79,53,69)(44,78,54,68)(45,77,55,67)(46,76,56,66)(47,75,57,65)(48,74,58,64)(49,73,59,63)(50,72,60,62)(81,103,91,113)(82,102,92,112)(83,101,93,111)(84,120,94,110)(85,119,95,109)(86,118,96,108)(87,117,97,107)(88,116,98,106)(89,115,99,105)(90,114,100,104)(121,148,131,158)(122,147,132,157)(123,146,133,156)(124,145,134,155)(125,144,135,154)(126,143,136,153)(127,142,137,152)(128,141,138,151)(129,160,139,150)(130,159,140,149) );
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)], [(1,157,11,147),(2,158,12,148),(3,159,13,149),(4,160,14,150),(5,141,15,151),(6,142,16,152),(7,143,17,153),(8,144,18,154),(9,145,19,155),(10,146,20,156),(21,97,31,87),(22,98,32,88),(23,99,33,89),(24,100,34,90),(25,81,35,91),(26,82,36,92),(27,83,37,93),(28,84,38,94),(29,85,39,95),(30,86,40,96),(41,127,51,137),(42,128,52,138),(43,129,53,139),(44,130,54,140),(45,131,55,121),(46,132,56,122),(47,133,57,123),(48,134,58,124),(49,135,59,125),(50,136,60,126),(61,108,71,118),(62,109,72,119),(63,110,73,120),(64,111,74,101),(65,112,75,102),(66,113,76,103),(67,114,77,104),(68,115,78,105),(69,116,79,106),(70,117,80,107)], [(1,76,11,66),(2,77,12,67),(3,78,13,68),(4,79,14,69),(5,80,15,70),(6,61,16,71),(7,62,17,72),(8,63,18,73),(9,64,19,74),(10,65,20,75),(21,42,31,52),(22,43,32,53),(23,44,33,54),(24,45,34,55),(25,46,35,56),(26,47,36,57),(27,48,37,58),(28,49,38,59),(29,50,39,60),(30,51,40,41),(81,122,91,132),(82,123,92,133),(83,124,93,134),(84,125,94,135),(85,126,95,136),(86,127,96,137),(87,128,97,138),(88,129,98,139),(89,130,99,140),(90,131,100,121),(101,155,111,145),(102,156,112,146),(103,157,113,147),(104,158,114,148),(105,159,115,149),(106,160,116,150),(107,141,117,151),(108,142,118,152),(109,143,119,153),(110,144,120,154)], [(41,127,96),(42,128,97),(43,129,98),(44,130,99),(45,131,100),(46,132,81),(47,133,82),(48,134,83),(49,135,84),(50,136,85),(51,137,86),(52,138,87),(53,139,88),(54,140,89),(55,121,90),(56,122,91),(57,123,92),(58,124,93),(59,125,94),(60,126,95),(61,108,142),(62,109,143),(63,110,144),(64,111,145),(65,112,146),(66,113,147),(67,114,148),(68,115,149),(69,116,150),(70,117,151),(71,118,152),(72,119,153),(73,120,154),(74,101,155),(75,102,156),(76,103,157),(77,104,158),(78,105,159),(79,106,160),(80,107,141)], [(1,25,11,35),(2,24,12,34),(3,23,13,33),(4,22,14,32),(5,21,15,31),(6,40,16,30),(7,39,17,29),(8,38,18,28),(9,37,19,27),(10,36,20,26),(41,61,51,71),(42,80,52,70),(43,79,53,69),(44,78,54,68),(45,77,55,67),(46,76,56,66),(47,75,57,65),(48,74,58,64),(49,73,59,63),(50,72,60,62),(81,103,91,113),(82,102,92,112),(83,101,93,111),(84,120,94,110),(85,119,95,109),(86,118,96,108),(87,117,97,107),(88,116,98,106),(89,115,99,105),(90,114,100,104),(121,148,131,158),(122,147,132,157),(123,146,133,156),(124,145,134,155),(125,144,135,154),(126,143,136,153),(127,142,137,152),(128,141,138,151),(129,160,139,150),(130,159,140,149)])
Matrix representation ►G ⊆ GL6(𝔽241)
| 22 | 8 | 0 | 0 | 0 | 0 |
| 233 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 142 | 99 | 142 |
| 0 | 0 | 99 | 0 | 142 | 142 |
| 0 | 0 | 142 | 99 | 0 | 142 |
| 0 | 0 | 99 | 99 | 99 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 1 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 152 | 170 | 0 | 0 | 0 | 0 |
| 81 | 89 | 0 | 0 | 0 | 0 |
| 0 | 0 | 239 | 221 | 20 | 221 |
| 0 | 0 | 221 | 20 | 239 | 221 |
| 0 | 0 | 20 | 239 | 221 | 221 |
| 0 | 0 | 221 | 221 | 221 | 2 |
G:=sub<GL(6,GF(241))| [22,233,0,0,0,0,8,30,0,0,0,0,0,0,0,99,142,99,0,0,142,0,99,99,0,0,99,142,0,99,0,0,142,142,142,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,240,0,0,0,0,1,0,0,0,0,240,0,0,0],[0,1,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,1,0,0],[152,81,0,0,0,0,170,89,0,0,0,0,0,0,239,221,20,221,0,0,221,20,239,221,0,0,20,239,221,221,0,0,221,221,221,2] >;
41 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 30A | 30B | 30C | 30D | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 6 | 8 | 2 | 6 | 60 | 60 | 2 | 2 | 8 | 60 | 60 | 2 | 2 | 12 | 12 | 8 | 8 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 12 | 12 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
41 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | - | + | + |
| image | C1 | C2 | C2 | S3 | D5 | D6 | D10 | D15 | D30 | S4 | C2×S4 | C4.S4 | C20.2S4 | C5⋊S4 | C2×C5⋊S4 |
| kernel | C20.2S4 | Q8.D15 | C5×C4.A4 | C5×C4○D4 | C4.A4 | C5×Q8 | SL2(𝔽3) | C4○D4 | Q8 | C20 | C10 | C5 | C1 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 2 | 3 | 12 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_{20}._2S_4 % in TeX
G:=Group("C20.2S4"); // GroupNames label
G:=SmallGroup(480,1030);
// by ID
G=gap.SmallGroup(480,1030);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,1680,3389,1688,170,1347,4204,3168,172,2525,1909,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^20=d^3=1,b^2=c^2=e^2=a^10,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,c*b*c^-1=a^10*b,d*b*d^-1=a^10*b*c,e*b*e^-1=b*c,d*c*d^-1=b,e*c*e^-1=a^10*c,e*d*e^-1=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀