metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×Dic5)⋊2C8, (C22×C4).1F5, (C22×C20).7C4, C2.1(C23⋊F5), C23.32(C2×F5), C10.6(C22⋊C8), C22.3(D5⋊C8), (C2×C10).5M4(2), C2.7(D10⋊C8), C10.12(C23⋊C4), C22.4(C4.F5), (C2×Dic5).102D4, (C22×Dic5).7C4, C23.2F5.2C2, C10.4(C4.10D4), C22.36(C22⋊F5), C2.2(Dic5.D4), C5⋊2(C22.M4(2)), (C22×Dic5).172C22, (C2×C10).8(C2×C8), (C22×C10).43(C2×C4), (C2×C10.D4).1C2, (C2×C10).27(C22⋊C4), SmallGroup(320,252)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C22×C4).F5
G = < a,b,c,d,e | a2=b2=c4=d5=1, e4=c2, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ece-1=abc, ede-1=d3 >
Subgroups: 306 in 78 conjugacy classes, 28 normal (22 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C23, C10, C10, C4⋊C4, C2×C8, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C22⋊C8, C2×C4⋊C4, C5⋊C8, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C22.M4(2), C10.D4, C2×C5⋊C8, C22×Dic5, C22×C20, C23.2F5, C2×C10.D4, (C22×C4).F5
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), F5, C22⋊C8, C23⋊C4, C4.10D4, C2×F5, C22.M4(2), D5⋊C8, C4.F5, C22⋊F5, D10⋊C8, Dic5.D4, C23⋊F5, (C22×C4).F5
►On 160 points
Generators in S160
(2 78)(4 80)(6 74)(8 76)(9 68)(11 70)(13 72)(15 66)(17 131)(19 133)(21 135)(23 129)(25 127)(27 121)(29 123)(31 125)(34 81)(36 83)(38 85)(40 87)(41 160)(43 154)(45 156)(47 158)(49 141)(51 143)(53 137)(55 139)(58 95)(60 89)(62 91)(64 93)(97 146)(99 148)(101 150)(103 152)(105 117)(107 119)(109 113)(111 115)
(1 77)(2 78)(3 79)(4 80)(5 73)(6 74)(7 75)(8 76)(9 68)(10 69)(11 70)(12 71)(13 72)(14 65)(15 66)(16 67)(17 131)(18 132)(19 133)(20 134)(21 135)(22 136)(23 129)(24 130)(25 127)(26 128)(27 121)(28 122)(29 123)(30 124)(31 125)(32 126)(33 88)(34 81)(35 82)(36 83)(37 84)(38 85)(39 86)(40 87)(41 160)(42 153)(43 154)(44 155)(45 156)(46 157)(47 158)(48 159)(49 141)(50 142)(51 143)(52 144)(53 137)(54 138)(55 139)(56 140)(57 94)(58 95)(59 96)(60 89)(61 90)(62 91)(63 92)(64 93)(97 146)(98 147)(99 148)(100 149)(101 150)(102 151)(103 152)(104 145)(105 117)(106 118)(107 119)(108 120)(109 113)(110 114)(111 115)(112 116)
(1 71 5 67)(2 13 6 9)(3 14 7 10)(4 66 8 70)(11 80 15 76)(12 73 16 77)(17 51 21 55)(18 52 22 56)(19 137 23 141)(20 138 24 142)(25 43 29 47)(26 44 30 48)(27 156 31 160)(28 157 32 153)(33 94 37 90)(34 58 38 62)(35 59 39 63)(36 89 40 93)(41 121 45 125)(42 122 46 126)(49 133 53 129)(50 134 54 130)(57 84 61 88)(60 87 64 83)(65 75 69 79)(68 78 72 74)(81 95 85 91)(82 96 86 92)(97 111 101 107)(98 112 102 108)(99 117 103 113)(100 118 104 114)(105 152 109 148)(106 145 110 149)(115 150 119 146)(116 151 120 147)(123 158 127 154)(124 159 128 155)(131 143 135 139)(132 144 136 140)
(1 88 140 155 149)(2 156 81 150 141)(3 151 157 142 82)(4 143 152 83 158)(5 84 144 159 145)(6 160 85 146 137)(7 147 153 138 86)(8 139 148 87 154)(9 27 91 115 23)(10 116 28 24 92)(11 17 117 93 29)(12 94 18 30 118)(13 31 95 119 19)(14 120 32 20 96)(15 21 113 89 25)(16 90 22 26 114)(33 56 44 100 77)(34 101 49 78 45)(35 79 102 46 50)(36 47 80 51 103)(37 52 48 104 73)(38 97 53 74 41)(39 75 98 42 54)(40 43 76 55 99)(57 132 124 106 71)(58 107 133 72 125)(59 65 108 126 134)(60 127 66 135 109)(61 136 128 110 67)(62 111 129 68 121)(63 69 112 122 130)(64 123 70 131 105)
(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)| (2,78)(4,80)(6,74)(8,76)(9,68)(11,70)(13,72)(15,66)(17,131)(19,133)(21,135)(23,129)(25,127)(27,121)(29,123)(31,125)(34,81)(36,83)(38,85)(40,87)(41,160)(43,154)(45,156)(47,158)(49,141)(51,143)(53,137)(55,139)(58,95)(60,89)(62,91)(64,93)(97,146)(99,148)(101,150)(103,152)(105,117)(107,119)(109,113)(111,115), (1,77)(2,78)(3,79)(4,80)(5,73)(6,74)(7,75)(8,76)(9,68)(10,69)(11,70)(12,71)(13,72)(14,65)(15,66)(16,67)(17,131)(18,132)(19,133)(20,134)(21,135)(22,136)(23,129)(24,130)(25,127)(26,128)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,88)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,87)(41,160)(42,153)(43,154)(44,155)(45,156)(46,157)(47,158)(48,159)(49,141)(50,142)(51,143)(52,144)(53,137)(54,138)(55,139)(56,140)(57,94)(58,95)(59,96)(60,89)(61,90)(62,91)(63,92)(64,93)(97,146)(98,147)(99,148)(100,149)(101,150)(102,151)(103,152)(104,145)(105,117)(106,118)(107,119)(108,120)(109,113)(110,114)(111,115)(112,116), (1,71,5,67)(2,13,6,9)(3,14,7,10)(4,66,8,70)(11,80,15,76)(12,73,16,77)(17,51,21,55)(18,52,22,56)(19,137,23,141)(20,138,24,142)(25,43,29,47)(26,44,30,48)(27,156,31,160)(28,157,32,153)(33,94,37,90)(34,58,38,62)(35,59,39,63)(36,89,40,93)(41,121,45,125)(42,122,46,126)(49,133,53,129)(50,134,54,130)(57,84,61,88)(60,87,64,83)(65,75,69,79)(68,78,72,74)(81,95,85,91)(82,96,86,92)(97,111,101,107)(98,112,102,108)(99,117,103,113)(100,118,104,114)(105,152,109,148)(106,145,110,149)(115,150,119,146)(116,151,120,147)(123,158,127,154)(124,159,128,155)(131,143,135,139)(132,144,136,140), (1,88,140,155,149)(2,156,81,150,141)(3,151,157,142,82)(4,143,152,83,158)(5,84,144,159,145)(6,160,85,146,137)(7,147,153,138,86)(8,139,148,87,154)(9,27,91,115,23)(10,116,28,24,92)(11,17,117,93,29)(12,94,18,30,118)(13,31,95,119,19)(14,120,32,20,96)(15,21,113,89,25)(16,90,22,26,114)(33,56,44,100,77)(34,101,49,78,45)(35,79,102,46,50)(36,47,80,51,103)(37,52,48,104,73)(38,97,53,74,41)(39,75,98,42,54)(40,43,76,55,99)(57,132,124,106,71)(58,107,133,72,125)(59,65,108,126,134)(60,127,66,135,109)(61,136,128,110,67)(62,111,129,68,121)(63,69,112,122,130)(64,123,70,131,105), (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( (2,78)(4,80)(6,74)(8,76)(9,68)(11,70)(13,72)(15,66)(17,131)(19,133)(21,135)(23,129)(25,127)(27,121)(29,123)(31,125)(34,81)(36,83)(38,85)(40,87)(41,160)(43,154)(45,156)(47,158)(49,141)(51,143)(53,137)(55,139)(58,95)(60,89)(62,91)(64,93)(97,146)(99,148)(101,150)(103,152)(105,117)(107,119)(109,113)(111,115), (1,77)(2,78)(3,79)(4,80)(5,73)(6,74)(7,75)(8,76)(9,68)(10,69)(11,70)(12,71)(13,72)(14,65)(15,66)(16,67)(17,131)(18,132)(19,133)(20,134)(21,135)(22,136)(23,129)(24,130)(25,127)(26,128)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,88)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,87)(41,160)(42,153)(43,154)(44,155)(45,156)(46,157)(47,158)(48,159)(49,141)(50,142)(51,143)(52,144)(53,137)(54,138)(55,139)(56,140)(57,94)(58,95)(59,96)(60,89)(61,90)(62,91)(63,92)(64,93)(97,146)(98,147)(99,148)(100,149)(101,150)(102,151)(103,152)(104,145)(105,117)(106,118)(107,119)(108,120)(109,113)(110,114)(111,115)(112,116), (1,71,5,67)(2,13,6,9)(3,14,7,10)(4,66,8,70)(11,80,15,76)(12,73,16,77)(17,51,21,55)(18,52,22,56)(19,137,23,141)(20,138,24,142)(25,43,29,47)(26,44,30,48)(27,156,31,160)(28,157,32,153)(33,94,37,90)(34,58,38,62)(35,59,39,63)(36,89,40,93)(41,121,45,125)(42,122,46,126)(49,133,53,129)(50,134,54,130)(57,84,61,88)(60,87,64,83)(65,75,69,79)(68,78,72,74)(81,95,85,91)(82,96,86,92)(97,111,101,107)(98,112,102,108)(99,117,103,113)(100,118,104,114)(105,152,109,148)(106,145,110,149)(115,150,119,146)(116,151,120,147)(123,158,127,154)(124,159,128,155)(131,143,135,139)(132,144,136,140), (1,88,140,155,149)(2,156,81,150,141)(3,151,157,142,82)(4,143,152,83,158)(5,84,144,159,145)(6,160,85,146,137)(7,147,153,138,86)(8,139,148,87,154)(9,27,91,115,23)(10,116,28,24,92)(11,17,117,93,29)(12,94,18,30,118)(13,31,95,119,19)(14,120,32,20,96)(15,21,113,89,25)(16,90,22,26,114)(33,56,44,100,77)(34,101,49,78,45)(35,79,102,46,50)(36,47,80,51,103)(37,52,48,104,73)(38,97,53,74,41)(39,75,98,42,54)(40,43,76,55,99)(57,132,124,106,71)(58,107,133,72,125)(59,65,108,126,134)(60,127,66,135,109)(61,136,128,110,67)(62,111,129,68,121)(63,69,112,122,130)(64,123,70,131,105), (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([[(2,78),(4,80),(6,74),(8,76),(9,68),(11,70),(13,72),(15,66),(17,131),(19,133),(21,135),(23,129),(25,127),(27,121),(29,123),(31,125),(34,81),(36,83),(38,85),(40,87),(41,160),(43,154),(45,156),(47,158),(49,141),(51,143),(53,137),(55,139),(58,95),(60,89),(62,91),(64,93),(97,146),(99,148),(101,150),(103,152),(105,117),(107,119),(109,113),(111,115)], [(1,77),(2,78),(3,79),(4,80),(5,73),(6,74),(7,75),(8,76),(9,68),(10,69),(11,70),(12,71),(13,72),(14,65),(15,66),(16,67),(17,131),(18,132),(19,133),(20,134),(21,135),(22,136),(23,129),(24,130),(25,127),(26,128),(27,121),(28,122),(29,123),(30,124),(31,125),(32,126),(33,88),(34,81),(35,82),(36,83),(37,84),(38,85),(39,86),(40,87),(41,160),(42,153),(43,154),(44,155),(45,156),(46,157),(47,158),(48,159),(49,141),(50,142),(51,143),(52,144),(53,137),(54,138),(55,139),(56,140),(57,94),(58,95),(59,96),(60,89),(61,90),(62,91),(63,92),(64,93),(97,146),(98,147),(99,148),(100,149),(101,150),(102,151),(103,152),(104,145),(105,117),(106,118),(107,119),(108,120),(109,113),(110,114),(111,115),(112,116)], [(1,71,5,67),(2,13,6,9),(3,14,7,10),(4,66,8,70),(11,80,15,76),(12,73,16,77),(17,51,21,55),(18,52,22,56),(19,137,23,141),(20,138,24,142),(25,43,29,47),(26,44,30,48),(27,156,31,160),(28,157,32,153),(33,94,37,90),(34,58,38,62),(35,59,39,63),(36,89,40,93),(41,121,45,125),(42,122,46,126),(49,133,53,129),(50,134,54,130),(57,84,61,88),(60,87,64,83),(65,75,69,79),(68,78,72,74),(81,95,85,91),(82,96,86,92),(97,111,101,107),(98,112,102,108),(99,117,103,113),(100,118,104,114),(105,152,109,148),(106,145,110,149),(115,150,119,146),(116,151,120,147),(123,158,127,154),(124,159,128,155),(131,143,135,139),(132,144,136,140)], [(1,88,140,155,149),(2,156,81,150,141),(3,151,157,142,82),(4,143,152,83,158),(5,84,144,159,145),(6,160,85,146,137),(7,147,153,138,86),(8,139,148,87,154),(9,27,91,115,23),(10,116,28,24,92),(11,17,117,93,29),(12,94,18,30,118),(13,31,95,119,19),(14,120,32,20,96),(15,21,113,89,25),(16,90,22,26,114),(33,56,44,100,77),(34,101,49,78,45),(35,79,102,46,50),(36,47,80,51,103),(37,52,48,104,73),(38,97,53,74,41),(39,75,98,42,54),(40,43,76,55,99),(57,132,124,106,71),(58,107,133,72,125),(59,65,108,126,134),(60,127,66,135,109),(61,136,128,110,67),(62,111,129,68,121),(63,69,112,122,130),(64,123,70,131,105)], [(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)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 8A | ··· | 8H | 10A | ··· | 10G | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 4 | 20 | ··· | 20 | 4 | ··· | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | M4(2) | F5 | C23⋊C4 | C4.10D4 | C2×F5 | D5⋊C8 | C4.F5 | C22⋊F5 | Dic5.D4 | C23⋊F5 |
| kernel | (C22×C4).F5 | C23.2F5 | C2×C10.D4 | C22×Dic5 | C22×C20 | C2×Dic5 | C2×Dic5 | C2×C10 | C22×C4 | C10 | C10 | C23 | C22 | C22 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of (C22×C4).F5 ►in GL10(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 20 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 38 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 40 |
| 0 | 38 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 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 | 0 | 0 | 0 | 0 | 0 | 1 | 10 | 4 | 24 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 34 | 6 | 25 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 35 | 16 | 29 |
| 0 | 0 | 0 | 0 | 0 | 0 | 17 | 39 | 40 | 31 |
G:=sub<GL(10,GF(41))| [40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,3,20,0,0,0,0,0,0,0,0,20,38,0,0,0,0,0,0,0,0,0,0,3,20,0,0,0,0,0,0,0,0,20,38,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,40,40],[0,3,0,0,0,0,0,0,0,0,38,0,0,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,5,7,17,0,0,0,0,0,0,10,34,35,39,0,0,0,0,0,0,4,6,16,40,0,0,0,0,0,0,24,25,29,31] >;
(C22×C4).F5 in GAP, Magma, Sage, TeX
(C_2^2\times C_4).F_5
% in TeX
G:=Group("(C2^2xC4).F5"); // GroupNames label
G:=SmallGroup(320,252);
// by ID
G=gap.SmallGroup(320,252);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,387,100,1123,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^5=1,e^4=c^2,e*a*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a*b*c,e*d*e^-1=d^3>;
// generators/relations