direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20.C23, C20.31C24, D20.28C23, Dic10.27C23, (C2×Q8)⋊27D10, (C22×Q8)⋊4D5, C20.255(C2×D4), (C2×C20).211D4, Q8⋊D5⋊16C22, C4.31(C23×D5), C10⋊4(C8.C22), C5⋊2C8.13C23, (Q8×C10)⋊34C22, C5⋊Q16⋊15C22, Q8.20(C22×D5), (C5×Q8).20C23, (C2×C20).548C23, C4○D20.57C22, (C22×C4).274D10, (C22×C10).210D4, C10.150(C22×D4), C23.93(C5⋊D4), C4.Dic5⋊33C22, (C2×D20).285C22, (C22×C20).280C22, (C2×Dic10).313C22, (Q8×C2×C10)⋊3C2, C5⋊5(C2×C8.C22), (C2×Q8⋊D5)⋊30C2, C4.25(C2×C5⋊D4), (C2×C5⋊Q16)⋊30C2, (C2×C4○D20).24C2, (C2×C10).585(C2×D4), (C2×C4).93(C5⋊D4), (C2×C4.Dic5)⋊27C2, C2.23(C22×C5⋊D4), (C2×C4).240(C22×D5), C22.113(C2×C5⋊D4), (C2×C5⋊2C8).183C22, SmallGroup(320,1480)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 798 in 258 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×6], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×7], Q8 [×4], Q8 [×9], C23, C23, D5 [×2], C10, C10 [×2], C10 [×2], C2×C8 [×2], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8 [×6], C2×Q8 [×4], C4○D4 [×6], Dic5 [×2], C20 [×2], C20 [×2], C20 [×4], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C5⋊2C8 [×4], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20, C2×Dic5, C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×6], C5×Q8 [×4], C5×Q8 [×6], C22×D5, C22×C10, C2×C8.C22, C2×C5⋊2C8 [×2], C4.Dic5 [×4], Q8⋊D5 [×8], C5⋊Q16 [×8], C2×Dic10, C2×C4×D5, C2×D20, C4○D20 [×4], C4○D20 [×2], C2×C5⋊D4, C22×C20, C22×C20, Q8×C10 [×6], Q8×C10 [×3], C2×C4.Dic5, C2×Q8⋊D5 [×2], C20.C23 [×8], C2×C5⋊Q16 [×2], C2×C4○D20, Q8×C2×C10, C2×C20.C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C8.C22 [×2], C22×D4, C5⋊D4 [×4], C22×D5 [×7], C2×C8.C22, C2×C5⋊D4 [×6], C23×D5, C20.C23 [×2], C22×C5⋊D4, C2×C20.C23
Generators and relations
G = < a,b,c,d,e | a2=b20=c2=1, d2=e2=b10, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b11, cd=dc, ece-1=b5c, ede-1=b10d >
►On 160 points
Generators in S160
(1 84)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 91)(9 92)(10 93)(11 94)(12 95)(13 96)(14 97)(15 98)(16 99)(17 100)(18 81)(19 82)(20 83)(21 110)(22 111)(23 112)(24 113)(25 114)(26 115)(27 116)(28 117)(29 118)(30 119)(31 120)(32 101)(33 102)(34 103)(35 104)(36 105)(37 106)(38 107)(39 108)(40 109)(41 133)(42 134)(43 135)(44 136)(45 137)(46 138)(47 139)(48 140)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 152)(62 153)(63 154)(64 155)(65 156)(66 157)(67 158)(68 159)(69 160)(70 141)(71 142)(72 143)(73 144)(74 145)(75 146)(76 147)(77 148)(78 149)(79 150)(80 151)
(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)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 30)(39 40)(41 45)(42 44)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(61 66)(62 65)(63 64)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(81 87)(82 86)(83 85)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 110)(108 109)(117 120)(118 119)(121 129)(122 128)(123 127)(124 126)(130 140)(131 139)(132 138)(133 137)(134 136)(141 148)(142 147)(143 146)(144 145)(149 160)(150 159)(151 158)(152 157)(153 156)(154 155)
(1 135 11 125)(2 136 12 126)(3 137 13 127)(4 138 14 128)(5 139 15 129)(6 140 16 130)(7 121 17 131)(8 122 18 132)(9 123 19 133)(10 124 20 134)(21 156 31 146)(22 157 32 147)(23 158 33 148)(24 159 34 149)(25 160 35 150)(26 141 36 151)(27 142 37 152)(28 143 38 153)(29 144 39 154)(30 145 40 155)(41 92 51 82)(42 93 52 83)(43 94 53 84)(44 95 54 85)(45 96 55 86)(46 97 56 87)(47 98 57 88)(48 99 58 89)(49 100 59 90)(50 81 60 91)(61 116 71 106)(62 117 72 107)(63 118 73 108)(64 119 74 109)(65 120 75 110)(66 101 76 111)(67 102 77 112)(68 103 78 113)(69 104 79 114)(70 105 80 115)
(1 37 11 27)(2 28 12 38)(3 39 13 29)(4 30 14 40)(5 21 15 31)(6 32 16 22)(7 23 17 33)(8 34 18 24)(9 25 19 35)(10 36 20 26)(41 69 51 79)(42 80 52 70)(43 71 53 61)(44 62 54 72)(45 73 55 63)(46 64 56 74)(47 75 57 65)(48 66 58 76)(49 77 59 67)(50 68 60 78)(81 113 91 103)(82 104 92 114)(83 115 93 105)(84 106 94 116)(85 117 95 107)(86 108 96 118)(87 119 97 109)(88 110 98 120)(89 101 99 111)(90 112 100 102)(121 148 131 158)(122 159 132 149)(123 150 133 160)(124 141 134 151)(125 152 135 142)(126 143 136 153)(127 154 137 144)(128 145 138 155)(129 156 139 146)(130 147 140 157)
G:=sub<Sym(160)| (1,84)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,91)(9,92)(10,93)(11,94)(12,95)(13,96)(14,97)(15,98)(16,99)(17,100)(18,81)(19,82)(20,83)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,133)(42,134)(43,135)(44,136)(45,137)(46,138)(47,139)(48,140)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,152)(62,153)(63,154)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151), (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), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(39,40)(41,45)(42,44)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,66)(62,65)(63,64)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,87)(82,86)(83,85)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)(117,120)(118,119)(121,129)(122,128)(123,127)(124,126)(130,140)(131,139)(132,138)(133,137)(134,136)(141,148)(142,147)(143,146)(144,145)(149,160)(150,159)(151,158)(152,157)(153,156)(154,155), (1,135,11,125)(2,136,12,126)(3,137,13,127)(4,138,14,128)(5,139,15,129)(6,140,16,130)(7,121,17,131)(8,122,18,132)(9,123,19,133)(10,124,20,134)(21,156,31,146)(22,157,32,147)(23,158,33,148)(24,159,34,149)(25,160,35,150)(26,141,36,151)(27,142,37,152)(28,143,38,153)(29,144,39,154)(30,145,40,155)(41,92,51,82)(42,93,52,83)(43,94,53,84)(44,95,54,85)(45,96,55,86)(46,97,56,87)(47,98,57,88)(48,99,58,89)(49,100,59,90)(50,81,60,91)(61,116,71,106)(62,117,72,107)(63,118,73,108)(64,119,74,109)(65,120,75,110)(66,101,76,111)(67,102,77,112)(68,103,78,113)(69,104,79,114)(70,105,80,115), (1,37,11,27)(2,28,12,38)(3,39,13,29)(4,30,14,40)(5,21,15,31)(6,32,16,22)(7,23,17,33)(8,34,18,24)(9,25,19,35)(10,36,20,26)(41,69,51,79)(42,80,52,70)(43,71,53,61)(44,62,54,72)(45,73,55,63)(46,64,56,74)(47,75,57,65)(48,66,58,76)(49,77,59,67)(50,68,60,78)(81,113,91,103)(82,104,92,114)(83,115,93,105)(84,106,94,116)(85,117,95,107)(86,108,96,118)(87,119,97,109)(88,110,98,120)(89,101,99,111)(90,112,100,102)(121,148,131,158)(122,159,132,149)(123,150,133,160)(124,141,134,151)(125,152,135,142)(126,143,136,153)(127,154,137,144)(128,145,138,155)(129,156,139,146)(130,147,140,157)>;
G:=Group( (1,84)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,91)(9,92)(10,93)(11,94)(12,95)(13,96)(14,97)(15,98)(16,99)(17,100)(18,81)(19,82)(20,83)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,133)(42,134)(43,135)(44,136)(45,137)(46,138)(47,139)(48,140)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,152)(62,153)(63,154)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151), (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), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(39,40)(41,45)(42,44)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(61,66)(62,65)(63,64)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,87)(82,86)(83,85)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)(117,120)(118,119)(121,129)(122,128)(123,127)(124,126)(130,140)(131,139)(132,138)(133,137)(134,136)(141,148)(142,147)(143,146)(144,145)(149,160)(150,159)(151,158)(152,157)(153,156)(154,155), (1,135,11,125)(2,136,12,126)(3,137,13,127)(4,138,14,128)(5,139,15,129)(6,140,16,130)(7,121,17,131)(8,122,18,132)(9,123,19,133)(10,124,20,134)(21,156,31,146)(22,157,32,147)(23,158,33,148)(24,159,34,149)(25,160,35,150)(26,141,36,151)(27,142,37,152)(28,143,38,153)(29,144,39,154)(30,145,40,155)(41,92,51,82)(42,93,52,83)(43,94,53,84)(44,95,54,85)(45,96,55,86)(46,97,56,87)(47,98,57,88)(48,99,58,89)(49,100,59,90)(50,81,60,91)(61,116,71,106)(62,117,72,107)(63,118,73,108)(64,119,74,109)(65,120,75,110)(66,101,76,111)(67,102,77,112)(68,103,78,113)(69,104,79,114)(70,105,80,115), (1,37,11,27)(2,28,12,38)(3,39,13,29)(4,30,14,40)(5,21,15,31)(6,32,16,22)(7,23,17,33)(8,34,18,24)(9,25,19,35)(10,36,20,26)(41,69,51,79)(42,80,52,70)(43,71,53,61)(44,62,54,72)(45,73,55,63)(46,64,56,74)(47,75,57,65)(48,66,58,76)(49,77,59,67)(50,68,60,78)(81,113,91,103)(82,104,92,114)(83,115,93,105)(84,106,94,116)(85,117,95,107)(86,108,96,118)(87,119,97,109)(88,110,98,120)(89,101,99,111)(90,112,100,102)(121,148,131,158)(122,159,132,149)(123,150,133,160)(124,141,134,151)(125,152,135,142)(126,143,136,153)(127,154,137,144)(128,145,138,155)(129,156,139,146)(130,147,140,157) );
G=PermutationGroup([(1,84),(2,85),(3,86),(4,87),(5,88),(6,89),(7,90),(8,91),(9,92),(10,93),(11,94),(12,95),(13,96),(14,97),(15,98),(16,99),(17,100),(18,81),(19,82),(20,83),(21,110),(22,111),(23,112),(24,113),(25,114),(26,115),(27,116),(28,117),(29,118),(30,119),(31,120),(32,101),(33,102),(34,103),(35,104),(36,105),(37,106),(38,107),(39,108),(40,109),(41,133),(42,134),(43,135),(44,136),(45,137),(46,138),(47,139),(48,140),(49,121),(50,122),(51,123),(52,124),(53,125),(54,126),(55,127),(56,128),(57,129),(58,130),(59,131),(60,132),(61,152),(62,153),(63,154),(64,155),(65,156),(66,157),(67,158),(68,159),(69,160),(70,141),(71,142),(72,143),(73,144),(74,145),(75,146),(76,147),(77,148),(78,149),(79,150),(80,151)], [(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)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,38),(22,37),(23,36),(24,35),(25,34),(26,33),(27,32),(28,31),(29,30),(39,40),(41,45),(42,44),(46,60),(47,59),(48,58),(49,57),(50,56),(51,55),(52,54),(61,66),(62,65),(63,64),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(81,87),(82,86),(83,85),(88,100),(89,99),(90,98),(91,97),(92,96),(93,95),(101,116),(102,115),(103,114),(104,113),(105,112),(106,111),(107,110),(108,109),(117,120),(118,119),(121,129),(122,128),(123,127),(124,126),(130,140),(131,139),(132,138),(133,137),(134,136),(141,148),(142,147),(143,146),(144,145),(149,160),(150,159),(151,158),(152,157),(153,156),(154,155)], [(1,135,11,125),(2,136,12,126),(3,137,13,127),(4,138,14,128),(5,139,15,129),(6,140,16,130),(7,121,17,131),(8,122,18,132),(9,123,19,133),(10,124,20,134),(21,156,31,146),(22,157,32,147),(23,158,33,148),(24,159,34,149),(25,160,35,150),(26,141,36,151),(27,142,37,152),(28,143,38,153),(29,144,39,154),(30,145,40,155),(41,92,51,82),(42,93,52,83),(43,94,53,84),(44,95,54,85),(45,96,55,86),(46,97,56,87),(47,98,57,88),(48,99,58,89),(49,100,59,90),(50,81,60,91),(61,116,71,106),(62,117,72,107),(63,118,73,108),(64,119,74,109),(65,120,75,110),(66,101,76,111),(67,102,77,112),(68,103,78,113),(69,104,79,114),(70,105,80,115)], [(1,37,11,27),(2,28,12,38),(3,39,13,29),(4,30,14,40),(5,21,15,31),(6,32,16,22),(7,23,17,33),(8,34,18,24),(9,25,19,35),(10,36,20,26),(41,69,51,79),(42,80,52,70),(43,71,53,61),(44,62,54,72),(45,73,55,63),(46,64,56,74),(47,75,57,65),(48,66,58,76),(49,77,59,67),(50,68,60,78),(81,113,91,103),(82,104,92,114),(83,115,93,105),(84,106,94,116),(85,117,95,107),(86,108,96,118),(87,119,97,109),(88,110,98,120),(89,101,99,111),(90,112,100,102),(121,148,131,158),(122,159,132,149),(123,150,133,160),(124,141,134,151),(125,152,135,142),(126,143,136,153),(127,154,137,144),(128,145,138,155),(129,156,139,146),(130,147,140,157)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 35 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 40 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 6 |
| 0 | 0 | 0 | 0 | 1 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 6 |
| 0 | 0 | 0 | 0 | 35 | 23 |
| 0 | 0 | 23 | 35 | 0 | 0 |
| 0 | 0 | 6 | 18 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 36 | 20 | 5 |
| 0 | 0 | 5 | 20 | 36 | 31 |
| 0 | 0 | 20 | 5 | 10 | 5 |
| 0 | 0 | 36 | 31 | 36 | 21 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,35,1,0,0,0,0,40,0,0,0,6,40,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,6,40,0,0,0,0,35,35,0,0,0,0,0,0,35,1,0,0,0,0,6,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,23,6,0,0,0,0,35,18,0,0,18,35,0,0,0,0,6,23,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,31,5,20,36,0,0,36,20,5,31,0,0,20,36,10,36,0,0,5,31,5,21] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | C5⋊D4 | C5⋊D4 | C8.C22 | C20.C23 |
| kernel | C2×C20.C23 | C2×C4.Dic5 | C2×Q8⋊D5 | C20.C23 | C2×C5⋊Q16 | C2×C4○D20 | Q8×C2×C10 | C2×C20 | C22×C10 | C22×Q8 | C22×C4 | C2×Q8 | C2×C4 | C23 | C10 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 1 | 1 | 3 | 1 | 2 | 2 | 12 | 12 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_{20}.C_2^3 % in TeX
G:=Group("C2xC20.C2^3"); // GroupNames label
G:=SmallGroup(320,1480);
// by ID
G=gap.SmallGroup(320,1480);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,136,1684,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^20=c^2=1,d^2=e^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=b^11,c*d=d*c,e*c*e^-1=b^5*c,e*d*e^-1=b^10*d>;
// generators/relations
×
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ℤ
𝔽
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