direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4.F5, Dic5.17C24, C5⋊C8.1C23, C10⋊1(C8○D4), D5⋊C8⋊6C22, (C2×D4).13F5, D4.11(C2×F5), C4.F5⋊7C22, C2.8(C23×F5), D4⋊2D5.2C4, (D4×C10).12C4, C10.7(C23×C4), C23.30(C2×F5), C4.25(C22×F5), C20.25(C22×C4), D10.1(C22×C4), (C4×D5).47C23, C22.F5⋊3C22, (C2×Dic10).15C4, Dic10.11(C2×C4), C22.1(C22×F5), Dic5.1(C22×C4), D4⋊2D5.15C22, (C2×Dic5).177C23, (C22×Dic5).191C22, C5⋊1(C2×C8○D4), (C2×D5⋊C8)⋊5C2, (C22×C5⋊C8)⋊9C2, (C2×C4.F5)⋊6C2, (C2×C5⋊C8)⋊11C22, C5⋊D4.1(C2×C4), (C2×C4).91(C2×F5), (C2×C20).70(C2×C4), (C5×D4).11(C2×C4), (C4×D5).31(C2×C4), (C2×C5⋊D4).10C4, (C2×C10).1(C22×C4), (C2×C4×D5).214C22, (C2×C22.F5)⋊10C2, (C2×D4⋊2D5).17C2, (C22×C10).31(C2×C4), (C2×Dic5).80(C2×C4), (C22×D5).60(C2×C4), SmallGroup(320,1593)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 730 in 266 conjugacy classes, 140 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×4], C22 [×8], C5, C8 [×8], C2×C4, C2×C4 [×15], D4 [×4], D4 [×8], Q8 [×4], C23 [×2], C23, D5 [×2], C10, C10 [×2], C10 [×4], C2×C8 [×16], M4(2) [×12], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×8], Dic5 [×2], Dic5 [×4], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×4], C2×C10 [×4], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, C5⋊C8 [×8], Dic10 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×10], C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5, C22×C10 [×2], C2×C8○D4, D5⋊C8 [×4], C4.F5 [×4], C2×C5⋊C8 [×2], C2×C5⋊C8 [×10], C22.F5 [×8], C2×Dic10, C2×C4×D5, D4⋊2D5 [×8], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, C2×D5⋊C8, C2×C4.F5, D4.F5 [×8], C22×C5⋊C8 [×2], C2×C22.F5 [×2], C2×D4⋊2D5, C2×D4.F5
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, F5, C8○D4 [×2], C23×C4, C2×F5 [×7], C2×C8○D4, C22×F5 [×7], D4.F5 [×2], C23×F5, C2×D4.F5
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, ce=ec, ede-1=d3 >
►On 160 points
Generators in S160
(1 159)(2 160)(3 153)(4 154)(5 155)(6 156)(7 157)(8 158)(9 71)(10 72)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 121)(26 122)(27 123)(28 124)(29 125)(30 126)(31 127)(32 128)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 61)(50 62)(51 63)(52 64)(53 57)(54 58)(55 59)(56 60)(73 129)(74 130)(75 131)(76 132)(77 133)(78 134)(79 135)(80 136)(81 137)(82 138)(83 139)(84 140)(85 141)(86 142)(87 143)(88 144)(89 151)(90 152)(91 145)(92 146)(93 147)(94 148)(95 149)(96 150)(105 119)(106 120)(107 113)(108 114)(109 115)(110 116)(111 117)(112 118)
(1 7 5 3)(2 8 6 4)(9 95 13 91)(10 96 14 92)(11 89 15 93)(12 90 16 94)(17 130 21 134)(18 131 22 135)(19 132 23 136)(20 133 24 129)(25 139 29 143)(26 140 30 144)(27 141 31 137)(28 142 32 138)(33 74 37 78)(34 75 38 79)(35 76 39 80)(36 77 40 73)(41 115 45 119)(42 116 46 120)(43 117 47 113)(44 118 48 114)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 151 69 147)(66 152 70 148)(67 145 71 149)(68 146 72 150)(81 123 85 127)(82 124 86 128)(83 125 87 121)(84 126 88 122)(97 109 101 105)(98 110 102 106)(99 111 103 107)(100 112 104 108)(153 159 157 155)(154 160 158 156)
(1 61)(2 62)(3 63)(4 64)(5 57)(6 58)(7 59)(8 60)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 28)(18 29)(19 30)(20 31)(21 32)(22 25)(23 26)(24 27)(33 124)(34 125)(35 126)(36 127)(37 128)(38 121)(39 122)(40 123)(49 159)(50 160)(51 153)(52 154)(53 155)(54 156)(55 157)(56 158)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)(71 97)(72 98)(73 81)(74 82)(75 83)(76 84)(77 85)(78 86)(79 87)(80 88)(89 113)(90 114)(91 115)(92 116)(93 117)(94 118)(95 119)(96 120)(105 149)(106 150)(107 151)(108 152)(109 145)(110 146)(111 147)(112 148)(129 137)(130 138)(131 139)(132 140)(133 141)(134 142)(135 143)(136 144)
(1 97 18 133 111)(2 134 98 112 19)(3 105 135 20 99)(4 21 106 100 136)(5 101 22 129 107)(6 130 102 108 23)(7 109 131 24 103)(8 17 110 104 132)(9 125 85 93 49)(10 94 126 50 86)(11 51 95 87 127)(12 88 52 128 96)(13 121 81 89 53)(14 90 122 54 82)(15 55 91 83 123)(16 84 56 124 92)(25 137 151 57 67)(26 58 138 68 152)(27 69 59 145 139)(28 146 70 140 60)(29 141 147 61 71)(30 62 142 72 148)(31 65 63 149 143)(32 150 66 144 64)(33 116 48 76 158)(34 77 117 159 41)(35 160 78 42 118)(36 43 153 119 79)(37 120 44 80 154)(38 73 113 155 45)(39 156 74 46 114)(40 47 157 115 75)
(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)| (1,159)(2,160)(3,153)(4,154)(5,155)(6,156)(7,157)(8,158)(9,71)(10,72)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,127)(32,128)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60)(73,129)(74,130)(75,131)(76,132)(77,133)(78,134)(79,135)(80,136)(81,137)(82,138)(83,139)(84,140)(85,141)(86,142)(87,143)(88,144)(89,151)(90,152)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(105,119)(106,120)(107,113)(108,114)(109,115)(110,116)(111,117)(112,118), (1,7,5,3)(2,8,6,4)(9,95,13,91)(10,96,14,92)(11,89,15,93)(12,90,16,94)(17,130,21,134)(18,131,22,135)(19,132,23,136)(20,133,24,129)(25,139,29,143)(26,140,30,144)(27,141,31,137)(28,142,32,138)(33,74,37,78)(34,75,38,79)(35,76,39,80)(36,77,40,73)(41,115,45,119)(42,116,46,120)(43,117,47,113)(44,118,48,114)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,151,69,147)(66,152,70,148)(67,145,71,149)(68,146,72,150)(81,123,85,127)(82,124,86,128)(83,125,87,121)(84,126,88,122)(97,109,101,105)(98,110,102,106)(99,111,103,107)(100,112,104,108)(153,159,157,155)(154,160,158,156), (1,61)(2,62)(3,63)(4,64)(5,57)(6,58)(7,59)(8,60)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,28)(18,29)(19,30)(20,31)(21,32)(22,25)(23,26)(24,27)(33,124)(34,125)(35,126)(36,127)(37,128)(38,121)(39,122)(40,123)(49,159)(50,160)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,97)(72,98)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(89,113)(90,114)(91,115)(92,116)(93,117)(94,118)(95,119)(96,120)(105,149)(106,150)(107,151)(108,152)(109,145)(110,146)(111,147)(112,148)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144), (1,97,18,133,111)(2,134,98,112,19)(3,105,135,20,99)(4,21,106,100,136)(5,101,22,129,107)(6,130,102,108,23)(7,109,131,24,103)(8,17,110,104,132)(9,125,85,93,49)(10,94,126,50,86)(11,51,95,87,127)(12,88,52,128,96)(13,121,81,89,53)(14,90,122,54,82)(15,55,91,83,123)(16,84,56,124,92)(25,137,151,57,67)(26,58,138,68,152)(27,69,59,145,139)(28,146,70,140,60)(29,141,147,61,71)(30,62,142,72,148)(31,65,63,149,143)(32,150,66,144,64)(33,116,48,76,158)(34,77,117,159,41)(35,160,78,42,118)(36,43,153,119,79)(37,120,44,80,154)(38,73,113,155,45)(39,156,74,46,114)(40,47,157,115,75), (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( (1,159)(2,160)(3,153)(4,154)(5,155)(6,156)(7,157)(8,158)(9,71)(10,72)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,127)(32,128)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60)(73,129)(74,130)(75,131)(76,132)(77,133)(78,134)(79,135)(80,136)(81,137)(82,138)(83,139)(84,140)(85,141)(86,142)(87,143)(88,144)(89,151)(90,152)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(105,119)(106,120)(107,113)(108,114)(109,115)(110,116)(111,117)(112,118), (1,7,5,3)(2,8,6,4)(9,95,13,91)(10,96,14,92)(11,89,15,93)(12,90,16,94)(17,130,21,134)(18,131,22,135)(19,132,23,136)(20,133,24,129)(25,139,29,143)(26,140,30,144)(27,141,31,137)(28,142,32,138)(33,74,37,78)(34,75,38,79)(35,76,39,80)(36,77,40,73)(41,115,45,119)(42,116,46,120)(43,117,47,113)(44,118,48,114)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,151,69,147)(66,152,70,148)(67,145,71,149)(68,146,72,150)(81,123,85,127)(82,124,86,128)(83,125,87,121)(84,126,88,122)(97,109,101,105)(98,110,102,106)(99,111,103,107)(100,112,104,108)(153,159,157,155)(154,160,158,156), (1,61)(2,62)(3,63)(4,64)(5,57)(6,58)(7,59)(8,60)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,28)(18,29)(19,30)(20,31)(21,32)(22,25)(23,26)(24,27)(33,124)(34,125)(35,126)(36,127)(37,128)(38,121)(39,122)(40,123)(49,159)(50,160)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,97)(72,98)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(89,113)(90,114)(91,115)(92,116)(93,117)(94,118)(95,119)(96,120)(105,149)(106,150)(107,151)(108,152)(109,145)(110,146)(111,147)(112,148)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144), (1,97,18,133,111)(2,134,98,112,19)(3,105,135,20,99)(4,21,106,100,136)(5,101,22,129,107)(6,130,102,108,23)(7,109,131,24,103)(8,17,110,104,132)(9,125,85,93,49)(10,94,126,50,86)(11,51,95,87,127)(12,88,52,128,96)(13,121,81,89,53)(14,90,122,54,82)(15,55,91,83,123)(16,84,56,124,92)(25,137,151,57,67)(26,58,138,68,152)(27,69,59,145,139)(28,146,70,140,60)(29,141,147,61,71)(30,62,142,72,148)(31,65,63,149,143)(32,150,66,144,64)(33,116,48,76,158)(34,77,117,159,41)(35,160,78,42,118)(36,43,153,119,79)(37,120,44,80,154)(38,73,113,155,45)(39,156,74,46,114)(40,47,157,115,75), (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([(1,159),(2,160),(3,153),(4,154),(5,155),(6,156),(7,157),(8,158),(9,71),(10,72),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40),(25,121),(26,122),(27,123),(28,124),(29,125),(30,126),(31,127),(32,128),(41,97),(42,98),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,61),(50,62),(51,63),(52,64),(53,57),(54,58),(55,59),(56,60),(73,129),(74,130),(75,131),(76,132),(77,133),(78,134),(79,135),(80,136),(81,137),(82,138),(83,139),(84,140),(85,141),(86,142),(87,143),(88,144),(89,151),(90,152),(91,145),(92,146),(93,147),(94,148),(95,149),(96,150),(105,119),(106,120),(107,113),(108,114),(109,115),(110,116),(111,117),(112,118)], [(1,7,5,3),(2,8,6,4),(9,95,13,91),(10,96,14,92),(11,89,15,93),(12,90,16,94),(17,130,21,134),(18,131,22,135),(19,132,23,136),(20,133,24,129),(25,139,29,143),(26,140,30,144),(27,141,31,137),(28,142,32,138),(33,74,37,78),(34,75,38,79),(35,76,39,80),(36,77,40,73),(41,115,45,119),(42,116,46,120),(43,117,47,113),(44,118,48,114),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,151,69,147),(66,152,70,148),(67,145,71,149),(68,146,72,150),(81,123,85,127),(82,124,86,128),(83,125,87,121),(84,126,88,122),(97,109,101,105),(98,110,102,106),(99,111,103,107),(100,112,104,108),(153,159,157,155),(154,160,158,156)], [(1,61),(2,62),(3,63),(4,64),(5,57),(6,58),(7,59),(8,60),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,28),(18,29),(19,30),(20,31),(21,32),(22,25),(23,26),(24,27),(33,124),(34,125),(35,126),(36,127),(37,128),(38,121),(39,122),(40,123),(49,159),(50,160),(51,153),(52,154),(53,155),(54,156),(55,157),(56,158),(65,99),(66,100),(67,101),(68,102),(69,103),(70,104),(71,97),(72,98),(73,81),(74,82),(75,83),(76,84),(77,85),(78,86),(79,87),(80,88),(89,113),(90,114),(91,115),(92,116),(93,117),(94,118),(95,119),(96,120),(105,149),(106,150),(107,151),(108,152),(109,145),(110,146),(111,147),(112,148),(129,137),(130,138),(131,139),(132,140),(133,141),(134,142),(135,143),(136,144)], [(1,97,18,133,111),(2,134,98,112,19),(3,105,135,20,99),(4,21,106,100,136),(5,101,22,129,107),(6,130,102,108,23),(7,109,131,24,103),(8,17,110,104,132),(9,125,85,93,49),(10,94,126,50,86),(11,51,95,87,127),(12,88,52,128,96),(13,121,81,89,53),(14,90,122,54,82),(15,55,91,83,123),(16,84,56,124,92),(25,137,151,57,67),(26,58,138,68,152),(27,69,59,145,139),(28,146,70,140,60),(29,141,147,61,71),(30,62,142,72,148),(31,65,63,149,143),(32,150,66,144,64),(33,116,48,76,158),(34,77,117,159,41),(35,160,78,42,118),(36,43,153,119,79),(37,120,44,80,154),(38,73,113,155,45),(39,156,74,46,114),(40,47,157,115,75)], [(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 ⊆ GL8(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 | 40 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 27 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 | 8 | 8 |
| 0 | 0 | 0 | 0 | 8 | 8 | 0 | 36 |
| 0 | 0 | 0 | 0 | 33 | 28 | 33 | 0 |
| 0 | 0 | 0 | 0 | 5 | 13 | 13 | 5 |
G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,32,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,9,9,0,0,0,0,0,0,23,32,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0],[27,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,38,0,0,0,0,0,0,0,0,38,0,0,0,0,0,0,0,0,36,8,33,5,0,0,0,0,0,8,28,13,0,0,0,0,8,0,33,13,0,0,0,0,8,36,0,5] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5 | 8A | ··· | 8H | 8I | ··· | 8T | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 20A | 20B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8○D4 | F5 | C2×F5 | C2×F5 | C2×F5 | D4.F5 |
| kernel | C2×D4.F5 | C2×D5⋊C8 | C2×C4.F5 | D4.F5 | C22×C5⋊C8 | C2×C22.F5 | C2×D4⋊2D5 | C2×Dic10 | D4⋊2D5 | C2×C5⋊D4 | D4×C10 | C10 | C2×D4 | C2×C4 | D4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 2 | 1 | 2 | 8 | 4 | 2 | 8 | 1 | 1 | 4 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_2\times D_4.F_5
% in TeX
G:=Group("C2xD4.F5"); // GroupNames label
G:=SmallGroup(320,1593);
// by ID
G=gap.SmallGroup(320,1593);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,297,102,6278,818]);
// 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,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀