direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C4.F5, Dic5.15C24, C5⋊C8⋊1C23, C2.4(C23×F5), (C2×C10)⋊4M4(2), C10⋊1(C2×M4(2)), C10.2(C23×C4), C5⋊1(C22×M4(2)), (C22×C4).24F5, C4.43(C22×F5), C23.65(C2×F5), (C22×C20).26C4, C20.83(C22×C4), (C4×D5).83C23, (C23×D5).18C4, D10.44(C22×C4), C22.55(C22×F5), Dic5.44(C22×C4), (C2×Dic5).362C23, (C22×Dic5).282C22, (C22×C5⋊C8)⋊8C2, (C2×C5⋊C8)⋊9C22, (C2×C4×D5).38C4, (C4×D5).90(C2×C4), (C2×C4).146(C2×F5), (D5×C22×C4).30C2, (C2×C20).133(C2×C4), (C2×C4×D5).399C22, (C22×C10).78(C2×C4), (C2×C10).96(C22×C4), (C2×Dic5).198(C2×C4), (C22×D5).132(C2×C4), SmallGroup(320,1588)
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, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d3 >
Subgroups: 906 in 298 conjugacy classes, 156 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C23, D5, C10, C10, C2×C8, M4(2), C22×C4, C22×C4, C24, Dic5, Dic5, C20, D10, D10, C2×C10, C22×C8, C2×M4(2), C23×C4, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×D5, C22×C10, C22×M4(2), C4.F5, C2×C5⋊C8, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, C2×C4.F5, C22×C5⋊C8, D5×C22×C4, C22×C4.F5
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, C24, F5, C2×M4(2), C23×C4, C2×F5, C22×M4(2), C4.F5, C22×F5, C2×C4.F5, C23×F5, C22×C4.F5
►On 160 points
Generators in S160
(1 139)(2 140)(3 141)(4 142)(5 143)(6 144)(7 137)(8 138)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 119)(18 120)(19 113)(20 114)(21 115)(22 116)(23 117)(24 118)(25 123)(26 124)(27 125)(28 126)(29 127)(30 128)(31 121)(32 122)(41 68)(42 69)(43 70)(44 71)(45 72)(46 65)(47 66)(48 67)(49 100)(50 101)(51 102)(52 103)(53 104)(54 97)(55 98)(56 99)(57 95)(58 96)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(73 83)(74 84)(75 85)(76 86)(77 87)(78 88)(79 81)(80 82)(105 160)(106 153)(107 154)(108 155)(109 156)(110 157)(111 158)(112 159)(129 147)(130 148)(131 149)(132 150)(133 151)(134 152)(135 145)(136 146)
(1 95)(2 96)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(17 153)(18 154)(19 155)(20 156)(21 157)(22 158)(23 159)(24 160)(25 78)(26 79)(27 80)(28 73)(29 74)(30 75)(31 76)(32 77)(33 68)(34 69)(35 70)(36 71)(37 72)(38 65)(39 66)(40 67)(49 136)(50 129)(51 130)(52 131)(53 132)(54 133)(55 134)(56 135)(57 139)(58 140)(59 141)(60 142)(61 143)(62 144)(63 137)(64 138)(81 124)(82 125)(83 126)(84 127)(85 128)(86 121)(87 122)(88 123)(97 151)(98 152)(99 145)(100 146)(101 147)(102 148)(103 149)(104 150)(105 118)(106 119)(107 120)(108 113)(109 114)(110 115)(111 116)(112 117)
(1 89 5 93)(2 94 6 90)(3 91 7 95)(4 96 8 92)(9 17 13 21)(10 22 14 18)(11 19 15 23)(12 24 16 20)(25 103 29 99)(26 100 30 104)(27 97 31 101)(28 102 32 98)(33 114 37 118)(34 119 38 115)(35 116 39 120)(36 113 40 117)(41 156 45 160)(42 153 46 157)(43 158 47 154)(44 155 48 159)(49 128 53 124)(50 125 54 121)(51 122 55 126)(52 127 56 123)(57 141 61 137)(58 138 62 142)(59 143 63 139)(60 140 64 144)(65 110 69 106)(66 107 70 111)(67 112 71 108)(68 109 72 105)(73 148 77 152)(74 145 78 149)(75 150 79 146)(76 147 80 151)(81 136 85 132)(82 133 86 129)(83 130 87 134)(84 135 88 131)
(1 50 21 48 88)(2 41 51 81 22)(3 82 42 23 52)(4 24 83 53 43)(5 54 17 44 84)(6 45 55 85 18)(7 86 46 19 56)(8 20 87 49 47)(9 159 131 89 125)(10 90 160 126 132)(11 127 91 133 153)(12 134 128 154 92)(13 155 135 93 121)(14 94 156 122 136)(15 123 95 129 157)(16 130 124 158 96)(25 57 147 110 40)(26 111 58 33 148)(27 34 112 149 59)(28 150 35 60 105)(29 61 151 106 36)(30 107 62 37 152)(31 38 108 145 63)(32 146 39 64 109)(65 113 99 137 76)(66 138 114 77 100)(67 78 139 101 115)(68 102 79 116 140)(69 117 103 141 80)(70 142 118 73 104)(71 74 143 97 119)(72 98 75 120 144)
(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)| (1,139)(2,140)(3,141)(4,142)(5,143)(6,144)(7,137)(8,138)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(17,119)(18,120)(19,113)(20,114)(21,115)(22,116)(23,117)(24,118)(25,123)(26,124)(27,125)(28,126)(29,127)(30,128)(31,121)(32,122)(41,68)(42,69)(43,70)(44,71)(45,72)(46,65)(47,66)(48,67)(49,100)(50,101)(51,102)(52,103)(53,104)(54,97)(55,98)(56,99)(57,95)(58,96)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(73,83)(74,84)(75,85)(76,86)(77,87)(78,88)(79,81)(80,82)(105,160)(106,153)(107,154)(108,155)(109,156)(110,157)(111,158)(112,159)(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)(135,145)(136,146), (1,95)(2,96)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(17,153)(18,154)(19,155)(20,156)(21,157)(22,158)(23,159)(24,160)(25,78)(26,79)(27,80)(28,73)(29,74)(30,75)(31,76)(32,77)(33,68)(34,69)(35,70)(36,71)(37,72)(38,65)(39,66)(40,67)(49,136)(50,129)(51,130)(52,131)(53,132)(54,133)(55,134)(56,135)(57,139)(58,140)(59,141)(60,142)(61,143)(62,144)(63,137)(64,138)(81,124)(82,125)(83,126)(84,127)(85,128)(86,121)(87,122)(88,123)(97,151)(98,152)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,118)(106,119)(107,120)(108,113)(109,114)(110,115)(111,116)(112,117), (1,89,5,93)(2,94,6,90)(3,91,7,95)(4,96,8,92)(9,17,13,21)(10,22,14,18)(11,19,15,23)(12,24,16,20)(25,103,29,99)(26,100,30,104)(27,97,31,101)(28,102,32,98)(33,114,37,118)(34,119,38,115)(35,116,39,120)(36,113,40,117)(41,156,45,160)(42,153,46,157)(43,158,47,154)(44,155,48,159)(49,128,53,124)(50,125,54,121)(51,122,55,126)(52,127,56,123)(57,141,61,137)(58,138,62,142)(59,143,63,139)(60,140,64,144)(65,110,69,106)(66,107,70,111)(67,112,71,108)(68,109,72,105)(73,148,77,152)(74,145,78,149)(75,150,79,146)(76,147,80,151)(81,136,85,132)(82,133,86,129)(83,130,87,134)(84,135,88,131), (1,50,21,48,88)(2,41,51,81,22)(3,82,42,23,52)(4,24,83,53,43)(5,54,17,44,84)(6,45,55,85,18)(7,86,46,19,56)(8,20,87,49,47)(9,159,131,89,125)(10,90,160,126,132)(11,127,91,133,153)(12,134,128,154,92)(13,155,135,93,121)(14,94,156,122,136)(15,123,95,129,157)(16,130,124,158,96)(25,57,147,110,40)(26,111,58,33,148)(27,34,112,149,59)(28,150,35,60,105)(29,61,151,106,36)(30,107,62,37,152)(31,38,108,145,63)(32,146,39,64,109)(65,113,99,137,76)(66,138,114,77,100)(67,78,139,101,115)(68,102,79,116,140)(69,117,103,141,80)(70,142,118,73,104)(71,74,143,97,119)(72,98,75,120,144), (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( (1,139)(2,140)(3,141)(4,142)(5,143)(6,144)(7,137)(8,138)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(17,119)(18,120)(19,113)(20,114)(21,115)(22,116)(23,117)(24,118)(25,123)(26,124)(27,125)(28,126)(29,127)(30,128)(31,121)(32,122)(41,68)(42,69)(43,70)(44,71)(45,72)(46,65)(47,66)(48,67)(49,100)(50,101)(51,102)(52,103)(53,104)(54,97)(55,98)(56,99)(57,95)(58,96)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(73,83)(74,84)(75,85)(76,86)(77,87)(78,88)(79,81)(80,82)(105,160)(106,153)(107,154)(108,155)(109,156)(110,157)(111,158)(112,159)(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)(135,145)(136,146), (1,95)(2,96)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(17,153)(18,154)(19,155)(20,156)(21,157)(22,158)(23,159)(24,160)(25,78)(26,79)(27,80)(28,73)(29,74)(30,75)(31,76)(32,77)(33,68)(34,69)(35,70)(36,71)(37,72)(38,65)(39,66)(40,67)(49,136)(50,129)(51,130)(52,131)(53,132)(54,133)(55,134)(56,135)(57,139)(58,140)(59,141)(60,142)(61,143)(62,144)(63,137)(64,138)(81,124)(82,125)(83,126)(84,127)(85,128)(86,121)(87,122)(88,123)(97,151)(98,152)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,118)(106,119)(107,120)(108,113)(109,114)(110,115)(111,116)(112,117), (1,89,5,93)(2,94,6,90)(3,91,7,95)(4,96,8,92)(9,17,13,21)(10,22,14,18)(11,19,15,23)(12,24,16,20)(25,103,29,99)(26,100,30,104)(27,97,31,101)(28,102,32,98)(33,114,37,118)(34,119,38,115)(35,116,39,120)(36,113,40,117)(41,156,45,160)(42,153,46,157)(43,158,47,154)(44,155,48,159)(49,128,53,124)(50,125,54,121)(51,122,55,126)(52,127,56,123)(57,141,61,137)(58,138,62,142)(59,143,63,139)(60,140,64,144)(65,110,69,106)(66,107,70,111)(67,112,71,108)(68,109,72,105)(73,148,77,152)(74,145,78,149)(75,150,79,146)(76,147,80,151)(81,136,85,132)(82,133,86,129)(83,130,87,134)(84,135,88,131), (1,50,21,48,88)(2,41,51,81,22)(3,82,42,23,52)(4,24,83,53,43)(5,54,17,44,84)(6,45,55,85,18)(7,86,46,19,56)(8,20,87,49,47)(9,159,131,89,125)(10,90,160,126,132)(11,127,91,133,153)(12,134,128,154,92)(13,155,135,93,121)(14,94,156,122,136)(15,123,95,129,157)(16,130,124,158,96)(25,57,147,110,40)(26,111,58,33,148)(27,34,112,149,59)(28,150,35,60,105)(29,61,151,106,36)(30,107,62,37,152)(31,38,108,145,63)(32,146,39,64,109)(65,113,99,137,76)(66,138,114,77,100)(67,78,139,101,115)(68,102,79,116,140)(69,117,103,141,80)(70,142,118,73,104)(71,74,143,97,119)(72,98,75,120,144), (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([[(1,139),(2,140),(3,141),(4,142),(5,143),(6,144),(7,137),(8,138),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,33),(17,119),(18,120),(19,113),(20,114),(21,115),(22,116),(23,117),(24,118),(25,123),(26,124),(27,125),(28,126),(29,127),(30,128),(31,121),(32,122),(41,68),(42,69),(43,70),(44,71),(45,72),(46,65),(47,66),(48,67),(49,100),(50,101),(51,102),(52,103),(53,104),(54,97),(55,98),(56,99),(57,95),(58,96),(59,89),(60,90),(61,91),(62,92),(63,93),(64,94),(73,83),(74,84),(75,85),(76,86),(77,87),(78,88),(79,81),(80,82),(105,160),(106,153),(107,154),(108,155),(109,156),(110,157),(111,158),(112,159),(129,147),(130,148),(131,149),(132,150),(133,151),(134,152),(135,145),(136,146)], [(1,95),(2,96),(3,89),(4,90),(5,91),(6,92),(7,93),(8,94),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(17,153),(18,154),(19,155),(20,156),(21,157),(22,158),(23,159),(24,160),(25,78),(26,79),(27,80),(28,73),(29,74),(30,75),(31,76),(32,77),(33,68),(34,69),(35,70),(36,71),(37,72),(38,65),(39,66),(40,67),(49,136),(50,129),(51,130),(52,131),(53,132),(54,133),(55,134),(56,135),(57,139),(58,140),(59,141),(60,142),(61,143),(62,144),(63,137),(64,138),(81,124),(82,125),(83,126),(84,127),(85,128),(86,121),(87,122),(88,123),(97,151),(98,152),(99,145),(100,146),(101,147),(102,148),(103,149),(104,150),(105,118),(106,119),(107,120),(108,113),(109,114),(110,115),(111,116),(112,117)], [(1,89,5,93),(2,94,6,90),(3,91,7,95),(4,96,8,92),(9,17,13,21),(10,22,14,18),(11,19,15,23),(12,24,16,20),(25,103,29,99),(26,100,30,104),(27,97,31,101),(28,102,32,98),(33,114,37,118),(34,119,38,115),(35,116,39,120),(36,113,40,117),(41,156,45,160),(42,153,46,157),(43,158,47,154),(44,155,48,159),(49,128,53,124),(50,125,54,121),(51,122,55,126),(52,127,56,123),(57,141,61,137),(58,138,62,142),(59,143,63,139),(60,140,64,144),(65,110,69,106),(66,107,70,111),(67,112,71,108),(68,109,72,105),(73,148,77,152),(74,145,78,149),(75,150,79,146),(76,147,80,151),(81,136,85,132),(82,133,86,129),(83,130,87,134),(84,135,88,131)], [(1,50,21,48,88),(2,41,51,81,22),(3,82,42,23,52),(4,24,83,53,43),(5,54,17,44,84),(6,45,55,85,18),(7,86,46,19,56),(8,20,87,49,47),(9,159,131,89,125),(10,90,160,126,132),(11,127,91,133,153),(12,134,128,154,92),(13,155,135,93,121),(14,94,156,122,136),(15,123,95,129,157),(16,130,124,158,96),(25,57,147,110,40),(26,111,58,33,148),(27,34,112,149,59),(28,150,35,60,105),(29,61,151,106,36),(30,107,62,37,152),(31,38,108,145,63),(32,146,39,64,109),(65,113,99,137,76),(66,138,114,77,100),(67,78,139,101,115),(68,102,79,116,140),(69,117,103,141,80),(70,142,118,73,104),(71,74,143,97,119),(72,98,75,120,144)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5 | 8A | ··· | 8P | 10A | ··· | 10G | 20A | ··· | 20H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 4 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | M4(2) | F5 | C2×F5 | C2×F5 | C4.F5 |
| kernel | C22×C4.F5 | C2×C4.F5 | C22×C5⋊C8 | D5×C22×C4 | C2×C4×D5 | C22×C20 | C23×D5 | C2×C10 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 12 | 2 | 1 | 12 | 2 | 2 | 8 | 1 | 6 | 1 | 8 |
Matrix representation of C22×C4.F5 ►in GL8(𝔽41)
| 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 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 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 |
| 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 27 | 0 | 14 |
| 0 | 0 | 0 | 0 | 0 | 34 | 27 | 14 |
| 0 | 0 | 0 | 0 | 14 | 27 | 34 | 0 |
| 0 | 0 | 0 | 0 | 14 | 0 | 27 | 7 |
| 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 |
| 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 39 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 25 | 15 | 2 |
| 0 | 0 | 0 | 0 | 40 | 27 | 30 | 27 |
| 0 | 0 | 0 | 0 | 14 | 11 | 14 | 1 |
| 0 | 0 | 0 | 0 | 39 | 26 | 16 | 16 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,9,9,0,0,0,0,0,0,0,0,7,0,14,14,0,0,0,0,27,34,27,0,0,0,0,0,0,27,34,27,0,0,0,0,14,14,0,7],[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],[0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,40,39,0,0,0,0,0,0,5,1,0,0,0,0,0,0,0,0,25,40,14,39,0,0,0,0,25,27,11,26,0,0,0,0,15,30,14,16,0,0,0,0,2,27,1,16] >;
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(320,1588);
// by ID
G=gap.SmallGroup(320,1588);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,1123,136,102,6278,818]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^5=1,e^4=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^3>;
// generators/relations