direct product, non-abelian, soluble
Aliases: C2×Q8.D15, Q8.1D30, C10⋊CSU2(𝔽3), SL2(𝔽3).7D10, (C2×C10).5S4, (C5×Q8).8D6, C10.19(C2×S4), C22.4(C5⋊S4), (C2×Q8).2D15, (Q8×C10).2S3, C5⋊2(C2×CSU2(𝔽3)), (C2×SL2(𝔽3)).2D5, (C10×SL2(𝔽3)).2C2, (C5×SL2(𝔽3)).7C22, C2.5(C2×C5⋊S4), SmallGroup(480,1027)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C5×SL2(𝔽3) — C2×Q8.D15 |
| C5×SL2(𝔽3) — C2×Q8.D15 |
Subgroups: 514 in 78 conjugacy classes, 21 normal (15 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, Dic5 [×2], C20 [×2], C2×C10, SL2(𝔽3), C2×Dic3, C30 [×3], C2×Q16, C5⋊2C8 [×2], Dic10 [×3], C2×Dic5, C2×C20, C5×Q8, C5×Q8, CSU2(𝔽3) [×2], C2×SL2(𝔽3), Dic15 [×2], C2×C30, C2×C5⋊2C8, C5⋊Q16 [×4], C2×Dic10, Q8×C10, C2×CSU2(𝔽3), C5×SL2(𝔽3), C2×Dic15, C2×C5⋊Q16, Q8.D15 [×2], C10×SL2(𝔽3), C2×Q8.D15
Quotients:
C1, C2 [×3], C22, S3, D5, D6, D10, S4, D15, CSU2(𝔽3) [×2], C2×S4, D30, C2×CSU2(𝔽3), C5⋊S4, Q8.D15 [×2], C2×C5⋊S4, C2×Q8.D15
Generators and relations
G = < a,b,c,d,e | a2=b4=d15=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, dbd-1=c, ebe-1=b-1c, dcd-1=bc, ece-1=b2c, ede-1=d-1 >
►On 160 points
Generators in S160
(1 22)(2 23)(3 24)(4 25)(5 21)(6 30)(7 26)(8 27)(9 28)(10 29)(11 39)(12 40)(13 36)(14 37)(15 38)(16 31)(17 32)(18 33)(19 34)(20 35)(41 70)(42 56)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(71 105)(72 106)(73 107)(74 108)(75 109)(76 110)(77 111)(78 112)(79 113)(80 114)(81 115)(82 101)(83 102)(84 103)(85 104)(86 136)(87 137)(88 138)(89 139)(90 140)(91 141)(92 142)(93 143)(94 144)(95 145)(96 131)(97 132)(98 133)(99 134)(100 135)(116 148)(117 149)(118 150)(119 151)(120 152)(121 153)(122 154)(123 155)(124 156)(125 157)(126 158)(127 159)(128 160)(129 146)(130 147)
(1 65 9 150)(2 56 10 156)(3 62 6 147)(4 68 7 153)(5 59 8 159)(11 139 31 71)(12 145 32 77)(13 136 33 83)(14 142 34 74)(15 133 35 80)(16 105 39 89)(17 111 40 95)(18 102 36 86)(19 108 37 92)(20 114 38 98)(21 45 27 127)(22 51 28 118)(23 42 29 124)(24 48 30 130)(25 54 26 121)(41 128 123 46)(43 53 125 120)(44 116 126 49)(47 119 129 52)(50 122 117 55)(57 67 157 152)(58 148 158 63)(60 70 160 155)(61 151 146 66)(64 154 149 69)(72 82 140 135)(73 131 141 78)(75 85 143 138)(76 134 144 81)(79 137 132 84)(87 97 103 113)(88 109 104 93)(90 100 106 101)(91 112 107 96)(94 115 110 99)
(1 70 9 155)(2 61 10 146)(3 67 6 152)(4 58 7 158)(5 64 8 149)(11 144 31 76)(12 135 32 82)(13 141 33 73)(14 132 34 79)(15 138 35 85)(16 110 39 94)(17 101 40 100)(18 107 36 91)(19 113 37 97)(20 104 38 88)(21 50 27 117)(22 41 28 123)(23 47 29 129)(24 53 30 120)(25 44 26 126)(42 52 124 119)(43 130 125 48)(45 55 127 122)(46 118 128 51)(49 121 116 54)(56 66 156 151)(57 147 157 62)(59 69 159 154)(60 150 160 65)(63 153 148 68)(71 81 139 134)(72 145 140 77)(74 84 142 137)(75 133 143 80)(78 136 131 83)(86 96 102 112)(87 108 103 92)(89 99 105 115)(90 111 106 95)(93 114 109 98)
(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 32 9 12)(2 31 10 11)(3 35 6 15)(4 34 7 14)(5 33 8 13)(16 29 39 23)(17 28 40 22)(18 27 36 21)(19 26 37 25)(20 30 38 24)(41 100 123 101)(42 99 124 115)(43 98 125 114)(44 97 126 113)(45 96 127 112)(46 95 128 111)(47 94 129 110)(48 93 130 109)(49 92 116 108)(50 91 117 107)(51 90 118 106)(52 89 119 105)(53 88 120 104)(54 87 121 103)(55 86 122 102)(56 134 156 81)(57 133 157 80)(58 132 158 79)(59 131 159 78)(60 145 160 77)(61 144 146 76)(62 143 147 75)(63 142 148 74)(64 141 149 73)(65 140 150 72)(66 139 151 71)(67 138 152 85)(68 137 153 84)(69 136 154 83)(70 135 155 82)
G:=sub<Sym(160)| (1,22)(2,23)(3,24)(4,25)(5,21)(6,30)(7,26)(8,27)(9,28)(10,29)(11,39)(12,40)(13,36)(14,37)(15,38)(16,31)(17,32)(18,33)(19,34)(20,35)(41,70)(42,56)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(71,105)(72,106)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,113)(80,114)(81,115)(82,101)(83,102)(84,103)(85,104)(86,136)(87,137)(88,138)(89,139)(90,140)(91,141)(92,142)(93,143)(94,144)(95,145)(96,131)(97,132)(98,133)(99,134)(100,135)(116,148)(117,149)(118,150)(119,151)(120,152)(121,153)(122,154)(123,155)(124,156)(125,157)(126,158)(127,159)(128,160)(129,146)(130,147), (1,65,9,150)(2,56,10,156)(3,62,6,147)(4,68,7,153)(5,59,8,159)(11,139,31,71)(12,145,32,77)(13,136,33,83)(14,142,34,74)(15,133,35,80)(16,105,39,89)(17,111,40,95)(18,102,36,86)(19,108,37,92)(20,114,38,98)(21,45,27,127)(22,51,28,118)(23,42,29,124)(24,48,30,130)(25,54,26,121)(41,128,123,46)(43,53,125,120)(44,116,126,49)(47,119,129,52)(50,122,117,55)(57,67,157,152)(58,148,158,63)(60,70,160,155)(61,151,146,66)(64,154,149,69)(72,82,140,135)(73,131,141,78)(75,85,143,138)(76,134,144,81)(79,137,132,84)(87,97,103,113)(88,109,104,93)(90,100,106,101)(91,112,107,96)(94,115,110,99), (1,70,9,155)(2,61,10,146)(3,67,6,152)(4,58,7,158)(5,64,8,149)(11,144,31,76)(12,135,32,82)(13,141,33,73)(14,132,34,79)(15,138,35,85)(16,110,39,94)(17,101,40,100)(18,107,36,91)(19,113,37,97)(20,104,38,88)(21,50,27,117)(22,41,28,123)(23,47,29,129)(24,53,30,120)(25,44,26,126)(42,52,124,119)(43,130,125,48)(45,55,127,122)(46,118,128,51)(49,121,116,54)(56,66,156,151)(57,147,157,62)(59,69,159,154)(60,150,160,65)(63,153,148,68)(71,81,139,134)(72,145,140,77)(74,84,142,137)(75,133,143,80)(78,136,131,83)(86,96,102,112)(87,108,103,92)(89,99,105,115)(90,111,106,95)(93,114,109,98), (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), (1,32,9,12)(2,31,10,11)(3,35,6,15)(4,34,7,14)(5,33,8,13)(16,29,39,23)(17,28,40,22)(18,27,36,21)(19,26,37,25)(20,30,38,24)(41,100,123,101)(42,99,124,115)(43,98,125,114)(44,97,126,113)(45,96,127,112)(46,95,128,111)(47,94,129,110)(48,93,130,109)(49,92,116,108)(50,91,117,107)(51,90,118,106)(52,89,119,105)(53,88,120,104)(54,87,121,103)(55,86,122,102)(56,134,156,81)(57,133,157,80)(58,132,158,79)(59,131,159,78)(60,145,160,77)(61,144,146,76)(62,143,147,75)(63,142,148,74)(64,141,149,73)(65,140,150,72)(66,139,151,71)(67,138,152,85)(68,137,153,84)(69,136,154,83)(70,135,155,82)>;
G:=Group( (1,22)(2,23)(3,24)(4,25)(5,21)(6,30)(7,26)(8,27)(9,28)(10,29)(11,39)(12,40)(13,36)(14,37)(15,38)(16,31)(17,32)(18,33)(19,34)(20,35)(41,70)(42,56)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(71,105)(72,106)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,113)(80,114)(81,115)(82,101)(83,102)(84,103)(85,104)(86,136)(87,137)(88,138)(89,139)(90,140)(91,141)(92,142)(93,143)(94,144)(95,145)(96,131)(97,132)(98,133)(99,134)(100,135)(116,148)(117,149)(118,150)(119,151)(120,152)(121,153)(122,154)(123,155)(124,156)(125,157)(126,158)(127,159)(128,160)(129,146)(130,147), (1,65,9,150)(2,56,10,156)(3,62,6,147)(4,68,7,153)(5,59,8,159)(11,139,31,71)(12,145,32,77)(13,136,33,83)(14,142,34,74)(15,133,35,80)(16,105,39,89)(17,111,40,95)(18,102,36,86)(19,108,37,92)(20,114,38,98)(21,45,27,127)(22,51,28,118)(23,42,29,124)(24,48,30,130)(25,54,26,121)(41,128,123,46)(43,53,125,120)(44,116,126,49)(47,119,129,52)(50,122,117,55)(57,67,157,152)(58,148,158,63)(60,70,160,155)(61,151,146,66)(64,154,149,69)(72,82,140,135)(73,131,141,78)(75,85,143,138)(76,134,144,81)(79,137,132,84)(87,97,103,113)(88,109,104,93)(90,100,106,101)(91,112,107,96)(94,115,110,99), (1,70,9,155)(2,61,10,146)(3,67,6,152)(4,58,7,158)(5,64,8,149)(11,144,31,76)(12,135,32,82)(13,141,33,73)(14,132,34,79)(15,138,35,85)(16,110,39,94)(17,101,40,100)(18,107,36,91)(19,113,37,97)(20,104,38,88)(21,50,27,117)(22,41,28,123)(23,47,29,129)(24,53,30,120)(25,44,26,126)(42,52,124,119)(43,130,125,48)(45,55,127,122)(46,118,128,51)(49,121,116,54)(56,66,156,151)(57,147,157,62)(59,69,159,154)(60,150,160,65)(63,153,148,68)(71,81,139,134)(72,145,140,77)(74,84,142,137)(75,133,143,80)(78,136,131,83)(86,96,102,112)(87,108,103,92)(89,99,105,115)(90,111,106,95)(93,114,109,98), (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), (1,32,9,12)(2,31,10,11)(3,35,6,15)(4,34,7,14)(5,33,8,13)(16,29,39,23)(17,28,40,22)(18,27,36,21)(19,26,37,25)(20,30,38,24)(41,100,123,101)(42,99,124,115)(43,98,125,114)(44,97,126,113)(45,96,127,112)(46,95,128,111)(47,94,129,110)(48,93,130,109)(49,92,116,108)(50,91,117,107)(51,90,118,106)(52,89,119,105)(53,88,120,104)(54,87,121,103)(55,86,122,102)(56,134,156,81)(57,133,157,80)(58,132,158,79)(59,131,159,78)(60,145,160,77)(61,144,146,76)(62,143,147,75)(63,142,148,74)(64,141,149,73)(65,140,150,72)(66,139,151,71)(67,138,152,85)(68,137,153,84)(69,136,154,83)(70,135,155,82) );
G=PermutationGroup([(1,22),(2,23),(3,24),(4,25),(5,21),(6,30),(7,26),(8,27),(9,28),(10,29),(11,39),(12,40),(13,36),(14,37),(15,38),(16,31),(17,32),(18,33),(19,34),(20,35),(41,70),(42,56),(43,57),(44,58),(45,59),(46,60),(47,61),(48,62),(49,63),(50,64),(51,65),(52,66),(53,67),(54,68),(55,69),(71,105),(72,106),(73,107),(74,108),(75,109),(76,110),(77,111),(78,112),(79,113),(80,114),(81,115),(82,101),(83,102),(84,103),(85,104),(86,136),(87,137),(88,138),(89,139),(90,140),(91,141),(92,142),(93,143),(94,144),(95,145),(96,131),(97,132),(98,133),(99,134),(100,135),(116,148),(117,149),(118,150),(119,151),(120,152),(121,153),(122,154),(123,155),(124,156),(125,157),(126,158),(127,159),(128,160),(129,146),(130,147)], [(1,65,9,150),(2,56,10,156),(3,62,6,147),(4,68,7,153),(5,59,8,159),(11,139,31,71),(12,145,32,77),(13,136,33,83),(14,142,34,74),(15,133,35,80),(16,105,39,89),(17,111,40,95),(18,102,36,86),(19,108,37,92),(20,114,38,98),(21,45,27,127),(22,51,28,118),(23,42,29,124),(24,48,30,130),(25,54,26,121),(41,128,123,46),(43,53,125,120),(44,116,126,49),(47,119,129,52),(50,122,117,55),(57,67,157,152),(58,148,158,63),(60,70,160,155),(61,151,146,66),(64,154,149,69),(72,82,140,135),(73,131,141,78),(75,85,143,138),(76,134,144,81),(79,137,132,84),(87,97,103,113),(88,109,104,93),(90,100,106,101),(91,112,107,96),(94,115,110,99)], [(1,70,9,155),(2,61,10,146),(3,67,6,152),(4,58,7,158),(5,64,8,149),(11,144,31,76),(12,135,32,82),(13,141,33,73),(14,132,34,79),(15,138,35,85),(16,110,39,94),(17,101,40,100),(18,107,36,91),(19,113,37,97),(20,104,38,88),(21,50,27,117),(22,41,28,123),(23,47,29,129),(24,53,30,120),(25,44,26,126),(42,52,124,119),(43,130,125,48),(45,55,127,122),(46,118,128,51),(49,121,116,54),(56,66,156,151),(57,147,157,62),(59,69,159,154),(60,150,160,65),(63,153,148,68),(71,81,139,134),(72,145,140,77),(74,84,142,137),(75,133,143,80),(78,136,131,83),(86,96,102,112),(87,108,103,92),(89,99,105,115),(90,111,106,95),(93,114,109,98)], [(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)], [(1,32,9,12),(2,31,10,11),(3,35,6,15),(4,34,7,14),(5,33,8,13),(16,29,39,23),(17,28,40,22),(18,27,36,21),(19,26,37,25),(20,30,38,24),(41,100,123,101),(42,99,124,115),(43,98,125,114),(44,97,126,113),(45,96,127,112),(46,95,128,111),(47,94,129,110),(48,93,130,109),(49,92,116,108),(50,91,117,107),(51,90,118,106),(52,89,119,105),(53,88,120,104),(54,87,121,103),(55,86,122,102),(56,134,156,81),(57,133,157,80),(58,132,158,79),(59,131,159,78),(60,145,160,77),(61,144,146,76),(62,143,147,75),(63,142,148,74),(64,141,149,73),(65,140,150,72),(66,139,151,71),(67,138,152,85),(68,137,153,84),(69,136,154,83),(70,135,155,82)])
Matrix representation ►G ⊆ GL4(𝔽241) generated by
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 205 |
| 0 | 0 | 208 | 237 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 32 | 5 |
| 0 | 0 | 36 | 209 |
| 66 | 31 | 0 | 0 |
| 117 | 128 | 0 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 0 | 1 | 240 |
| 56 | 188 | 0 | 0 |
| 141 | 185 | 0 | 0 |
| 0 | 0 | 31 | 36 |
| 0 | 0 | 67 | 210 |
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240],[1,0,0,0,0,1,0,0,0,0,4,208,0,0,205,237],[1,0,0,0,0,1,0,0,0,0,32,36,0,0,5,209],[66,117,0,0,31,128,0,0,0,0,0,1,0,0,240,240],[56,141,0,0,188,185,0,0,0,0,31,67,0,0,36,210] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | ··· | 30L |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 8 | 6 | 6 | 60 | 60 | 2 | 2 | 8 | 8 | 8 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | - | + | + |
| image | C1 | C2 | C2 | S3 | D5 | D6 | D10 | D15 | CSU2(𝔽3) | D30 | S4 | C2×S4 | CSU2(𝔽3) | Q8.D15 | C5⋊S4 | C2×C5⋊S4 |
| kernel | C2×Q8.D15 | Q8.D15 | C10×SL2(𝔽3) | Q8×C10 | C2×SL2(𝔽3) | C5×Q8 | SL2(𝔽3) | C2×Q8 | C10 | Q8 | C2×C10 | C10 | C10 | C2 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 12 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_2\times Q_8.D_{15} % in TeX
G:=Group("C2xQ8.D15"); // GroupNames label
G:=SmallGroup(480,1027);
// by ID
G=gap.SmallGroup(480,1027);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,1680,170,1347,4204,3168,172,2525,1909,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^15=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=b^-1,d*b*d^-1=c,e*b*e^-1=b^-1*c,d*c*d^-1=b*c,e*c*e^-1=b^2*c,e*d*e^-1=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀