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 |
Generators and relations for C2×Q8.D15
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 >
Subgroups: 514 in 78 conjugacy classes, 21 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C8, C2×C4, Q8, Q8, C10, C10, Dic3, C2×C6, C15, C2×C8, Q16, C2×Q8, C2×Q8, Dic5, C20, C2×C10, SL2(𝔽3), C2×Dic3, C30, C2×Q16, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C5×Q8, C5×Q8, CSU2(𝔽3), C2×SL2(𝔽3), Dic15, C2×C30, C2×C5⋊2C8, C5⋊Q16, C2×Dic10, Q8×C10, C2×CSU2(𝔽3), C5×SL2(𝔽3), C2×Dic15, C2×C5⋊Q16, Q8.D15, C10×SL2(𝔽3), C2×Q8.D15
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, D15, CSU2(𝔽3), C2×S4, D30, C2×CSU2(𝔽3), C5⋊S4, Q8.D15, C2×C5⋊S4, C2×Q8.D15
►On 160 points
Generators in S160
(1 25)(2 21)(3 22)(4 23)(5 24)(6 20)(7 16)(8 17)(9 18)(10 19)(11 27)(12 28)(13 29)(14 30)(15 26)(31 40)(32 36)(33 37)(34 38)(35 39)(41 158)(42 159)(43 160)(44 146)(45 147)(46 148)(47 149)(48 150)(49 151)(50 152)(51 153)(52 154)(53 155)(54 156)(55 157)(56 131)(57 132)(58 133)(59 134)(60 135)(61 136)(62 137)(63 138)(64 139)(65 140)(66 141)(67 142)(68 143)(69 144)(70 145)(71 118)(72 119)(73 120)(74 121)(75 122)(76 123)(77 124)(78 125)(79 126)(80 127)(81 128)(82 129)(83 130)(84 116)(85 117)(86 113)(87 114)(88 115)(89 101)(90 102)(91 103)(92 104)(93 105)(94 106)(95 107)(96 108)(97 109)(98 110)(99 111)(100 112)
(1 108 15 152)(2 114 11 158)(3 105 12 149)(4 111 13 155)(5 102 14 146)(6 66 39 76)(7 57 40 82)(8 63 36 73)(9 69 37 79)(10 60 38 85)(16 132 31 129)(17 138 32 120)(18 144 33 126)(19 135 34 117)(20 141 35 123)(21 87 27 41)(22 93 28 47)(23 99 29 53)(24 90 30 44)(25 96 26 50)(42 52 88 98)(43 94 89 48)(45 55 91 86)(46 97 92 51)(49 100 95 54)(56 71 81 61)(58 68 83 78)(59 74 84 64)(62 77 72 67)(65 80 75 70)(101 150 160 106)(103 113 147 157)(104 153 148 109)(107 156 151 112)(110 159 154 115)(116 139 134 121)(118 128 136 131)(119 142 137 124)(122 145 140 127)(125 133 143 130)
(1 113 15 157)(2 104 11 148)(3 110 12 154)(4 101 13 160)(5 107 14 151)(6 56 39 81)(7 62 40 72)(8 68 36 78)(9 59 37 84)(10 65 38 75)(16 137 31 119)(17 143 32 125)(18 134 33 116)(19 140 34 122)(20 131 35 128)(21 92 27 46)(22 98 28 52)(23 89 29 43)(24 95 30 49)(25 86 26 55)(41 51 87 97)(42 93 88 47)(44 54 90 100)(45 96 91 50)(48 99 94 53)(57 67 82 77)(58 73 83 63)(60 70 85 80)(61 76 71 66)(64 79 74 69)(102 112 146 156)(103 152 147 108)(105 115 149 159)(106 155 150 111)(109 158 153 114)(117 127 135 145)(118 141 136 123)(120 130 138 133)(121 144 139 126)(124 132 142 129)
(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 35 15 20)(2 34 11 19)(3 33 12 18)(4 32 13 17)(5 31 14 16)(6 25 39 26)(7 24 40 30)(8 23 36 29)(9 22 37 28)(10 21 38 27)(41 80 87 70)(42 79 88 69)(43 78 89 68)(44 77 90 67)(45 76 91 66)(46 75 92 65)(47 74 93 64)(48 73 94 63)(49 72 95 62)(50 71 96 61)(51 85 97 60)(52 84 98 59)(53 83 99 58)(54 82 100 57)(55 81 86 56)(101 143 160 125)(102 142 146 124)(103 141 147 123)(104 140 148 122)(105 139 149 121)(106 138 150 120)(107 137 151 119)(108 136 152 118)(109 135 153 117)(110 134 154 116)(111 133 155 130)(112 132 156 129)(113 131 157 128)(114 145 158 127)(115 144 159 126)
G:=sub<Sym(160)| (1,25)(2,21)(3,22)(4,23)(5,24)(6,20)(7,16)(8,17)(9,18)(10,19)(11,27)(12,28)(13,29)(14,30)(15,26)(31,40)(32,36)(33,37)(34,38)(35,39)(41,158)(42,159)(43,160)(44,146)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,154)(53,155)(54,156)(55,157)(56,131)(57,132)(58,133)(59,134)(60,135)(61,136)(62,137)(63,138)(64,139)(65,140)(66,141)(67,142)(68,143)(69,144)(70,145)(71,118)(72,119)(73,120)(74,121)(75,122)(76,123)(77,124)(78,125)(79,126)(80,127)(81,128)(82,129)(83,130)(84,116)(85,117)(86,113)(87,114)(88,115)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110)(99,111)(100,112), (1,108,15,152)(2,114,11,158)(3,105,12,149)(4,111,13,155)(5,102,14,146)(6,66,39,76)(7,57,40,82)(8,63,36,73)(9,69,37,79)(10,60,38,85)(16,132,31,129)(17,138,32,120)(18,144,33,126)(19,135,34,117)(20,141,35,123)(21,87,27,41)(22,93,28,47)(23,99,29,53)(24,90,30,44)(25,96,26,50)(42,52,88,98)(43,94,89,48)(45,55,91,86)(46,97,92,51)(49,100,95,54)(56,71,81,61)(58,68,83,78)(59,74,84,64)(62,77,72,67)(65,80,75,70)(101,150,160,106)(103,113,147,157)(104,153,148,109)(107,156,151,112)(110,159,154,115)(116,139,134,121)(118,128,136,131)(119,142,137,124)(122,145,140,127)(125,133,143,130), (1,113,15,157)(2,104,11,148)(3,110,12,154)(4,101,13,160)(5,107,14,151)(6,56,39,81)(7,62,40,72)(8,68,36,78)(9,59,37,84)(10,65,38,75)(16,137,31,119)(17,143,32,125)(18,134,33,116)(19,140,34,122)(20,131,35,128)(21,92,27,46)(22,98,28,52)(23,89,29,43)(24,95,30,49)(25,86,26,55)(41,51,87,97)(42,93,88,47)(44,54,90,100)(45,96,91,50)(48,99,94,53)(57,67,82,77)(58,73,83,63)(60,70,85,80)(61,76,71,66)(64,79,74,69)(102,112,146,156)(103,152,147,108)(105,115,149,159)(106,155,150,111)(109,158,153,114)(117,127,135,145)(118,141,136,123)(120,130,138,133)(121,144,139,126)(124,132,142,129), (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), (1,35,15,20)(2,34,11,19)(3,33,12,18)(4,32,13,17)(5,31,14,16)(6,25,39,26)(7,24,40,30)(8,23,36,29)(9,22,37,28)(10,21,38,27)(41,80,87,70)(42,79,88,69)(43,78,89,68)(44,77,90,67)(45,76,91,66)(46,75,92,65)(47,74,93,64)(48,73,94,63)(49,72,95,62)(50,71,96,61)(51,85,97,60)(52,84,98,59)(53,83,99,58)(54,82,100,57)(55,81,86,56)(101,143,160,125)(102,142,146,124)(103,141,147,123)(104,140,148,122)(105,139,149,121)(106,138,150,120)(107,137,151,119)(108,136,152,118)(109,135,153,117)(110,134,154,116)(111,133,155,130)(112,132,156,129)(113,131,157,128)(114,145,158,127)(115,144,159,126)>;
G:=Group( (1,25)(2,21)(3,22)(4,23)(5,24)(6,20)(7,16)(8,17)(9,18)(10,19)(11,27)(12,28)(13,29)(14,30)(15,26)(31,40)(32,36)(33,37)(34,38)(35,39)(41,158)(42,159)(43,160)(44,146)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,154)(53,155)(54,156)(55,157)(56,131)(57,132)(58,133)(59,134)(60,135)(61,136)(62,137)(63,138)(64,139)(65,140)(66,141)(67,142)(68,143)(69,144)(70,145)(71,118)(72,119)(73,120)(74,121)(75,122)(76,123)(77,124)(78,125)(79,126)(80,127)(81,128)(82,129)(83,130)(84,116)(85,117)(86,113)(87,114)(88,115)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110)(99,111)(100,112), (1,108,15,152)(2,114,11,158)(3,105,12,149)(4,111,13,155)(5,102,14,146)(6,66,39,76)(7,57,40,82)(8,63,36,73)(9,69,37,79)(10,60,38,85)(16,132,31,129)(17,138,32,120)(18,144,33,126)(19,135,34,117)(20,141,35,123)(21,87,27,41)(22,93,28,47)(23,99,29,53)(24,90,30,44)(25,96,26,50)(42,52,88,98)(43,94,89,48)(45,55,91,86)(46,97,92,51)(49,100,95,54)(56,71,81,61)(58,68,83,78)(59,74,84,64)(62,77,72,67)(65,80,75,70)(101,150,160,106)(103,113,147,157)(104,153,148,109)(107,156,151,112)(110,159,154,115)(116,139,134,121)(118,128,136,131)(119,142,137,124)(122,145,140,127)(125,133,143,130), (1,113,15,157)(2,104,11,148)(3,110,12,154)(4,101,13,160)(5,107,14,151)(6,56,39,81)(7,62,40,72)(8,68,36,78)(9,59,37,84)(10,65,38,75)(16,137,31,119)(17,143,32,125)(18,134,33,116)(19,140,34,122)(20,131,35,128)(21,92,27,46)(22,98,28,52)(23,89,29,43)(24,95,30,49)(25,86,26,55)(41,51,87,97)(42,93,88,47)(44,54,90,100)(45,96,91,50)(48,99,94,53)(57,67,82,77)(58,73,83,63)(60,70,85,80)(61,76,71,66)(64,79,74,69)(102,112,146,156)(103,152,147,108)(105,115,149,159)(106,155,150,111)(109,158,153,114)(117,127,135,145)(118,141,136,123)(120,130,138,133)(121,144,139,126)(124,132,142,129), (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), (1,35,15,20)(2,34,11,19)(3,33,12,18)(4,32,13,17)(5,31,14,16)(6,25,39,26)(7,24,40,30)(8,23,36,29)(9,22,37,28)(10,21,38,27)(41,80,87,70)(42,79,88,69)(43,78,89,68)(44,77,90,67)(45,76,91,66)(46,75,92,65)(47,74,93,64)(48,73,94,63)(49,72,95,62)(50,71,96,61)(51,85,97,60)(52,84,98,59)(53,83,99,58)(54,82,100,57)(55,81,86,56)(101,143,160,125)(102,142,146,124)(103,141,147,123)(104,140,148,122)(105,139,149,121)(106,138,150,120)(107,137,151,119)(108,136,152,118)(109,135,153,117)(110,134,154,116)(111,133,155,130)(112,132,156,129)(113,131,157,128)(114,145,158,127)(115,144,159,126) );
G=PermutationGroup([[(1,25),(2,21),(3,22),(4,23),(5,24),(6,20),(7,16),(8,17),(9,18),(10,19),(11,27),(12,28),(13,29),(14,30),(15,26),(31,40),(32,36),(33,37),(34,38),(35,39),(41,158),(42,159),(43,160),(44,146),(45,147),(46,148),(47,149),(48,150),(49,151),(50,152),(51,153),(52,154),(53,155),(54,156),(55,157),(56,131),(57,132),(58,133),(59,134),(60,135),(61,136),(62,137),(63,138),(64,139),(65,140),(66,141),(67,142),(68,143),(69,144),(70,145),(71,118),(72,119),(73,120),(74,121),(75,122),(76,123),(77,124),(78,125),(79,126),(80,127),(81,128),(82,129),(83,130),(84,116),(85,117),(86,113),(87,114),(88,115),(89,101),(90,102),(91,103),(92,104),(93,105),(94,106),(95,107),(96,108),(97,109),(98,110),(99,111),(100,112)], [(1,108,15,152),(2,114,11,158),(3,105,12,149),(4,111,13,155),(5,102,14,146),(6,66,39,76),(7,57,40,82),(8,63,36,73),(9,69,37,79),(10,60,38,85),(16,132,31,129),(17,138,32,120),(18,144,33,126),(19,135,34,117),(20,141,35,123),(21,87,27,41),(22,93,28,47),(23,99,29,53),(24,90,30,44),(25,96,26,50),(42,52,88,98),(43,94,89,48),(45,55,91,86),(46,97,92,51),(49,100,95,54),(56,71,81,61),(58,68,83,78),(59,74,84,64),(62,77,72,67),(65,80,75,70),(101,150,160,106),(103,113,147,157),(104,153,148,109),(107,156,151,112),(110,159,154,115),(116,139,134,121),(118,128,136,131),(119,142,137,124),(122,145,140,127),(125,133,143,130)], [(1,113,15,157),(2,104,11,148),(3,110,12,154),(4,101,13,160),(5,107,14,151),(6,56,39,81),(7,62,40,72),(8,68,36,78),(9,59,37,84),(10,65,38,75),(16,137,31,119),(17,143,32,125),(18,134,33,116),(19,140,34,122),(20,131,35,128),(21,92,27,46),(22,98,28,52),(23,89,29,43),(24,95,30,49),(25,86,26,55),(41,51,87,97),(42,93,88,47),(44,54,90,100),(45,96,91,50),(48,99,94,53),(57,67,82,77),(58,73,83,63),(60,70,85,80),(61,76,71,66),(64,79,74,69),(102,112,146,156),(103,152,147,108),(105,115,149,159),(106,155,150,111),(109,158,153,114),(117,127,135,145),(118,141,136,123),(120,130,138,133),(121,144,139,126),(124,132,142,129)], [(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)], [(1,35,15,20),(2,34,11,19),(3,33,12,18),(4,32,13,17),(5,31,14,16),(6,25,39,26),(7,24,40,30),(8,23,36,29),(9,22,37,28),(10,21,38,27),(41,80,87,70),(42,79,88,69),(43,78,89,68),(44,77,90,67),(45,76,91,66),(46,75,92,65),(47,74,93,64),(48,73,94,63),(49,72,95,62),(50,71,96,61),(51,85,97,60),(52,84,98,59),(53,83,99,58),(54,82,100,57),(55,81,86,56),(101,143,160,125),(102,142,146,124),(103,141,147,123),(104,140,148,122),(105,139,149,121),(106,138,150,120),(107,137,151,119),(108,136,152,118),(109,135,153,117),(110,134,154,116),(111,133,155,130),(112,132,156,129),(113,131,157,128),(114,145,158,127),(115,144,159,126)]])
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 |
Matrix representation of C2×Q8.D15 ►in 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] >;
C2×Q8.D15 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