metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×D4).8F5, (C4×D5).37D4, (D4×C10).10C4, C23.14(C2×F5), C10.18(C8○D4), C2.18(D4.F5), C4.14(C22⋊F5), C20.14(C22⋊C4), Dic5.110(C2×D4), (C2×Dic10).13C4, C23.2F5⋊12C2, D10.13(C22⋊C4), C22.94(C22×F5), (C2×Dic5).355C23, (C22×Dic5).188C22, (C2×D5⋊C8)⋊2C2, (C2×C4.F5)⋊2C2, (C2×C5⋊D4).8C4, (C2×C4).82(C2×F5), (C2×C20).56(C2×C4), (C2×C5⋊C8).12C22, C2.21(C2×C22⋊F5), C5⋊2((C22×C8)⋊C2), C10.20(C2×C22⋊C4), (C2×C4×D5).201C22, (C2×D4⋊2D5).16C2, (C2×C10).79(C22×C4), (C22×C10).27(C2×C4), (C2×Dic5).74(C2×C4), (C22×D5).56(C2×C4), SmallGroup(320,1114)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 586 in 158 conjugacy classes, 52 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×10], C5, C8 [×4], C2×C4, C2×C4 [×11], D4 [×6], Q8 [×2], C23 [×2], C23, D5 [×2], C10, C10 [×2], C10 [×2], C2×C8 [×6], M4(2) [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], Dic5 [×2], Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×6], C22⋊C8 [×4], C22×C8, C2×M4(2), C2×C4○D4, C5⋊C8 [×4], Dic10 [×2], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×C10 [×2], (C22×C8)⋊C2, D5⋊C8 [×2], C4.F5 [×2], C2×C5⋊C8 [×4], C2×Dic10, C2×C4×D5, D4⋊2D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, C23.2F5 [×4], C2×D5⋊C8, C2×C4.F5, C2×D4⋊2D5, (C2×D4).8F5
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C8○D4 [×2], C2×F5 [×3], (C22×C8)⋊C2, C22⋊F5 [×2], C22×F5, D4.F5 [×2], C2×C22⋊F5, (C2×D4).8F5
Generators and relations
G = < a,b,c,d,e | a2=b4=c2=d5=1, e4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=ab2c, ede-1=d3 >
►On 160 points
Generators in S160
(1 141)(2 142)(3 143)(4 144)(5 137)(6 138)(7 139)(8 140)(9 130)(10 131)(11 132)(12 133)(13 134)(14 135)(15 136)(16 129)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 147)(26 148)(27 149)(28 150)(29 151)(30 152)(31 145)(32 146)(41 99)(42 100)(43 101)(44 102)(45 103)(46 104)(47 97)(48 98)(49 110)(50 111)(51 112)(52 105)(53 106)(54 107)(55 108)(56 109)(57 121)(58 122)(59 123)(60 124)(61 125)(62 126)(63 127)(64 128)(65 87)(66 88)(67 81)(68 82)(69 83)(70 84)(71 85)(72 86)(73 96)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)(113 155)(114 156)(115 157)(116 158)(117 159)(118 160)(119 153)(120 154)
(1 143 5 139)(2 144 6 140)(3 137 7 141)(4 138 8 142)(9 63 13 59)(10 64 14 60)(11 57 15 61)(12 58 16 62)(17 37 21 33)(18 38 22 34)(19 39 23 35)(20 40 24 36)(25 42 29 46)(26 43 30 47)(27 44 31 48)(28 45 32 41)(49 84 53 88)(50 85 54 81)(51 86 55 82)(52 87 56 83)(65 109 69 105)(66 110 70 106)(67 111 71 107)(68 112 72 108)(73 117 77 113)(74 118 78 114)(75 119 79 115)(76 120 80 116)(89 160 93 156)(90 153 94 157)(91 154 95 158)(92 155 96 159)(97 148 101 152)(98 149 102 145)(99 150 103 146)(100 151 104 147)(121 136 125 132)(122 129 126 133)(123 130 127 134)(124 131 128 135)
(1 34)(2 17)(3 36)(4 19)(5 38)(6 21)(7 40)(8 23)(9 147)(10 30)(11 149)(12 32)(13 151)(14 26)(15 145)(16 28)(18 139)(20 141)(22 143)(24 137)(25 130)(27 132)(29 134)(31 136)(33 144)(35 138)(37 140)(39 142)(41 62)(42 123)(43 64)(44 125)(45 58)(46 127)(47 60)(48 121)(49 78)(50 90)(51 80)(52 92)(53 74)(54 94)(55 76)(56 96)(57 98)(59 100)(61 102)(63 104)(65 117)(66 156)(67 119)(68 158)(69 113)(70 160)(71 115)(72 154)(73 109)(75 111)(77 105)(79 107)(81 153)(82 116)(83 155)(84 118)(85 157)(86 120)(87 159)(88 114)(89 106)(91 108)(93 110)(95 112)(97 124)(99 126)(101 128)(103 122)(129 150)(131 152)(133 146)(135 148)
(1 94 132 63 113)(2 64 95 114 133)(3 115 57 134 96)(4 135 116 89 58)(5 90 136 59 117)(6 60 91 118 129)(7 119 61 130 92)(8 131 120 93 62)(9 77 139 153 125)(10 154 78 126 140)(11 127 155 141 79)(12 142 128 80 156)(13 73 143 157 121)(14 158 74 122 144)(15 123 159 137 75)(16 138 124 76 160)(17 43 112 88 146)(18 81 44 147 105)(19 148 82 106 45)(20 107 149 46 83)(21 47 108 84 150)(22 85 48 151 109)(23 152 86 110 41)(24 111 145 42 87)(25 52 40 67 102)(26 68 53 103 33)(27 104 69 34 54)(28 35 97 55 70)(29 56 36 71 98)(30 72 49 99 37)(31 100 65 38 50)(32 39 101 51 66)
(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)
G:=sub<Sym(160)| (1,141)(2,142)(3,143)(4,144)(5,137)(6,138)(7,139)(8,140)(9,130)(10,131)(11,132)(12,133)(13,134)(14,135)(15,136)(16,129)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,147)(26,148)(27,149)(28,150)(29,151)(30,152)(31,145)(32,146)(41,99)(42,100)(43,101)(44,102)(45,103)(46,104)(47,97)(48,98)(49,110)(50,111)(51,112)(52,105)(53,106)(54,107)(55,108)(56,109)(57,121)(58,122)(59,123)(60,124)(61,125)(62,126)(63,127)(64,128)(65,87)(66,88)(67,81)(68,82)(69,83)(70,84)(71,85)(72,86)(73,96)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(113,155)(114,156)(115,157)(116,158)(117,159)(118,160)(119,153)(120,154), (1,143,5,139)(2,144,6,140)(3,137,7,141)(4,138,8,142)(9,63,13,59)(10,64,14,60)(11,57,15,61)(12,58,16,62)(17,37,21,33)(18,38,22,34)(19,39,23,35)(20,40,24,36)(25,42,29,46)(26,43,30,47)(27,44,31,48)(28,45,32,41)(49,84,53,88)(50,85,54,81)(51,86,55,82)(52,87,56,83)(65,109,69,105)(66,110,70,106)(67,111,71,107)(68,112,72,108)(73,117,77,113)(74,118,78,114)(75,119,79,115)(76,120,80,116)(89,160,93,156)(90,153,94,157)(91,154,95,158)(92,155,96,159)(97,148,101,152)(98,149,102,145)(99,150,103,146)(100,151,104,147)(121,136,125,132)(122,129,126,133)(123,130,127,134)(124,131,128,135), (1,34)(2,17)(3,36)(4,19)(5,38)(6,21)(7,40)(8,23)(9,147)(10,30)(11,149)(12,32)(13,151)(14,26)(15,145)(16,28)(18,139)(20,141)(22,143)(24,137)(25,130)(27,132)(29,134)(31,136)(33,144)(35,138)(37,140)(39,142)(41,62)(42,123)(43,64)(44,125)(45,58)(46,127)(47,60)(48,121)(49,78)(50,90)(51,80)(52,92)(53,74)(54,94)(55,76)(56,96)(57,98)(59,100)(61,102)(63,104)(65,117)(66,156)(67,119)(68,158)(69,113)(70,160)(71,115)(72,154)(73,109)(75,111)(77,105)(79,107)(81,153)(82,116)(83,155)(84,118)(85,157)(86,120)(87,159)(88,114)(89,106)(91,108)(93,110)(95,112)(97,124)(99,126)(101,128)(103,122)(129,150)(131,152)(133,146)(135,148), (1,94,132,63,113)(2,64,95,114,133)(3,115,57,134,96)(4,135,116,89,58)(5,90,136,59,117)(6,60,91,118,129)(7,119,61,130,92)(8,131,120,93,62)(9,77,139,153,125)(10,154,78,126,140)(11,127,155,141,79)(12,142,128,80,156)(13,73,143,157,121)(14,158,74,122,144)(15,123,159,137,75)(16,138,124,76,160)(17,43,112,88,146)(18,81,44,147,105)(19,148,82,106,45)(20,107,149,46,83)(21,47,108,84,150)(22,85,48,151,109)(23,152,86,110,41)(24,111,145,42,87)(25,52,40,67,102)(26,68,53,103,33)(27,104,69,34,54)(28,35,97,55,70)(29,56,36,71,98)(30,72,49,99,37)(31,100,65,38,50)(32,39,101,51,66), (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)>;
G:=Group( (1,141)(2,142)(3,143)(4,144)(5,137)(6,138)(7,139)(8,140)(9,130)(10,131)(11,132)(12,133)(13,134)(14,135)(15,136)(16,129)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,147)(26,148)(27,149)(28,150)(29,151)(30,152)(31,145)(32,146)(41,99)(42,100)(43,101)(44,102)(45,103)(46,104)(47,97)(48,98)(49,110)(50,111)(51,112)(52,105)(53,106)(54,107)(55,108)(56,109)(57,121)(58,122)(59,123)(60,124)(61,125)(62,126)(63,127)(64,128)(65,87)(66,88)(67,81)(68,82)(69,83)(70,84)(71,85)(72,86)(73,96)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(113,155)(114,156)(115,157)(116,158)(117,159)(118,160)(119,153)(120,154), (1,143,5,139)(2,144,6,140)(3,137,7,141)(4,138,8,142)(9,63,13,59)(10,64,14,60)(11,57,15,61)(12,58,16,62)(17,37,21,33)(18,38,22,34)(19,39,23,35)(20,40,24,36)(25,42,29,46)(26,43,30,47)(27,44,31,48)(28,45,32,41)(49,84,53,88)(50,85,54,81)(51,86,55,82)(52,87,56,83)(65,109,69,105)(66,110,70,106)(67,111,71,107)(68,112,72,108)(73,117,77,113)(74,118,78,114)(75,119,79,115)(76,120,80,116)(89,160,93,156)(90,153,94,157)(91,154,95,158)(92,155,96,159)(97,148,101,152)(98,149,102,145)(99,150,103,146)(100,151,104,147)(121,136,125,132)(122,129,126,133)(123,130,127,134)(124,131,128,135), (1,34)(2,17)(3,36)(4,19)(5,38)(6,21)(7,40)(8,23)(9,147)(10,30)(11,149)(12,32)(13,151)(14,26)(15,145)(16,28)(18,139)(20,141)(22,143)(24,137)(25,130)(27,132)(29,134)(31,136)(33,144)(35,138)(37,140)(39,142)(41,62)(42,123)(43,64)(44,125)(45,58)(46,127)(47,60)(48,121)(49,78)(50,90)(51,80)(52,92)(53,74)(54,94)(55,76)(56,96)(57,98)(59,100)(61,102)(63,104)(65,117)(66,156)(67,119)(68,158)(69,113)(70,160)(71,115)(72,154)(73,109)(75,111)(77,105)(79,107)(81,153)(82,116)(83,155)(84,118)(85,157)(86,120)(87,159)(88,114)(89,106)(91,108)(93,110)(95,112)(97,124)(99,126)(101,128)(103,122)(129,150)(131,152)(133,146)(135,148), (1,94,132,63,113)(2,64,95,114,133)(3,115,57,134,96)(4,135,116,89,58)(5,90,136,59,117)(6,60,91,118,129)(7,119,61,130,92)(8,131,120,93,62)(9,77,139,153,125)(10,154,78,126,140)(11,127,155,141,79)(12,142,128,80,156)(13,73,143,157,121)(14,158,74,122,144)(15,123,159,137,75)(16,138,124,76,160)(17,43,112,88,146)(18,81,44,147,105)(19,148,82,106,45)(20,107,149,46,83)(21,47,108,84,150)(22,85,48,151,109)(23,152,86,110,41)(24,111,145,42,87)(25,52,40,67,102)(26,68,53,103,33)(27,104,69,34,54)(28,35,97,55,70)(29,56,36,71,98)(30,72,49,99,37)(31,100,65,38,50)(32,39,101,51,66), (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) );
G=PermutationGroup([(1,141),(2,142),(3,143),(4,144),(5,137),(6,138),(7,139),(8,140),(9,130),(10,131),(11,132),(12,133),(13,134),(14,135),(15,136),(16,129),(17,39),(18,40),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,147),(26,148),(27,149),(28,150),(29,151),(30,152),(31,145),(32,146),(41,99),(42,100),(43,101),(44,102),(45,103),(46,104),(47,97),(48,98),(49,110),(50,111),(51,112),(52,105),(53,106),(54,107),(55,108),(56,109),(57,121),(58,122),(59,123),(60,124),(61,125),(62,126),(63,127),(64,128),(65,87),(66,88),(67,81),(68,82),(69,83),(70,84),(71,85),(72,86),(73,96),(74,89),(75,90),(76,91),(77,92),(78,93),(79,94),(80,95),(113,155),(114,156),(115,157),(116,158),(117,159),(118,160),(119,153),(120,154)], [(1,143,5,139),(2,144,6,140),(3,137,7,141),(4,138,8,142),(9,63,13,59),(10,64,14,60),(11,57,15,61),(12,58,16,62),(17,37,21,33),(18,38,22,34),(19,39,23,35),(20,40,24,36),(25,42,29,46),(26,43,30,47),(27,44,31,48),(28,45,32,41),(49,84,53,88),(50,85,54,81),(51,86,55,82),(52,87,56,83),(65,109,69,105),(66,110,70,106),(67,111,71,107),(68,112,72,108),(73,117,77,113),(74,118,78,114),(75,119,79,115),(76,120,80,116),(89,160,93,156),(90,153,94,157),(91,154,95,158),(92,155,96,159),(97,148,101,152),(98,149,102,145),(99,150,103,146),(100,151,104,147),(121,136,125,132),(122,129,126,133),(123,130,127,134),(124,131,128,135)], [(1,34),(2,17),(3,36),(4,19),(5,38),(6,21),(7,40),(8,23),(9,147),(10,30),(11,149),(12,32),(13,151),(14,26),(15,145),(16,28),(18,139),(20,141),(22,143),(24,137),(25,130),(27,132),(29,134),(31,136),(33,144),(35,138),(37,140),(39,142),(41,62),(42,123),(43,64),(44,125),(45,58),(46,127),(47,60),(48,121),(49,78),(50,90),(51,80),(52,92),(53,74),(54,94),(55,76),(56,96),(57,98),(59,100),(61,102),(63,104),(65,117),(66,156),(67,119),(68,158),(69,113),(70,160),(71,115),(72,154),(73,109),(75,111),(77,105),(79,107),(81,153),(82,116),(83,155),(84,118),(85,157),(86,120),(87,159),(88,114),(89,106),(91,108),(93,110),(95,112),(97,124),(99,126),(101,128),(103,122),(129,150),(131,152),(133,146),(135,148)], [(1,94,132,63,113),(2,64,95,114,133),(3,115,57,134,96),(4,135,116,89,58),(5,90,136,59,117),(6,60,91,118,129),(7,119,61,130,92),(8,131,120,93,62),(9,77,139,153,125),(10,154,78,126,140),(11,127,155,141,79),(12,142,128,80,156),(13,73,143,157,121),(14,158,74,122,144),(15,123,159,137,75),(16,138,124,76,160),(17,43,112,88,146),(18,81,44,147,105),(19,148,82,106,45),(20,107,149,46,83),(21,47,108,84,150),(22,85,48,151,109),(23,152,86,110,41),(24,111,145,42,87),(25,52,40,67,102),(26,68,53,103,33),(27,104,69,34,54),(28,35,97,55,70),(29,56,36,71,98),(30,72,49,99,37),(31,100,65,38,50),(32,39,101,51,66)], [(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)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 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 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 38 | 0 | 3 |
| 0 | 0 | 0 | 19 | 38 | 3 |
| 0 | 0 | 3 | 38 | 19 | 0 |
| 0 | 0 | 3 | 0 | 38 | 22 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 38 | 0 | 0 | 0 | 0 | 0 |
| 0 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 28 | 20 | 0 |
| 0 | 0 | 33 | 28 | 0 | 13 |
| 0 | 0 | 13 | 0 | 28 | 33 |
| 0 | 0 | 0 | 20 | 28 | 13 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,22,0,3,3,0,0,38,19,38,0,0,0,0,38,19,38,0,0,3,3,0,22],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[38,0,0,0,0,0,0,38,0,0,0,0,0,0,13,33,13,0,0,0,28,28,0,20,0,0,20,0,28,28,0,0,0,13,33,13] >;
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 20A | 20B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 2 | 2 | 5 | 5 | 5 | 5 | 20 | 20 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | C8○D4 | F5 | C2×F5 | C2×F5 | C22⋊F5 | D4.F5 |
| kernel | (C2×D4).8F5 | C23.2F5 | C2×D5⋊C8 | C2×C4.F5 | C2×D4⋊2D5 | C2×Dic10 | C2×C5⋊D4 | D4×C10 | C4×D5 | C10 | C2×D4 | C2×C4 | C23 | C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 4 | 2 |
In GAP, Magma, Sage, TeX
(C_2\times D_4)._8F_5
% in TeX
G:=Group("(C2xD4).8F5"); // GroupNames label
G:=SmallGroup(320,1114);
// by ID
G=gap.SmallGroup(320,1114);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^5=1,e^4=b^2,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,b*e=e*b,c*d=d*c,e*c*e^-1=a*b^2*c,e*d*e^-1=d^3>;
// generators/relations
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