direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C8○2M4(2), M4(2)⋊5C20, C20.61C42, (C2×C8)⋊9C20, (C4×C40)⋊30C2, (C2×C40)⋊29C4, (C4×C8)⋊14C10, C4.5(C4×C20), C4⋊C4.10C20, C8.23(C2×C20), C8⋊C4⋊13C10, C40.129(C2×C4), C22⋊C4.6C20, C22.5(C4×C20), C10.68(C8○D4), (C5×M4(2))⋊17C4, (C22×C40).33C2, (C22×C8).15C10, C10.55(C2×C42), C23.16(C2×C20), C4.34(C22×C20), C42.59(C2×C10), (C2×C10).34C42, C20.251(C22×C4), (C2×C40).453C22, (C4×C20).300C22, (C2×C20).980C23, C42⋊C2.14C10, (C10×M4(2)).36C2, (C2×M4(2)).17C10, C22.18(C22×C20), (C22×C20).579C22, C2.7(C2×C4×C20), C2.1(C5×C8○D4), (C5×C4⋊C4).35C4, (C5×C8⋊C4)⋊27C2, (C2×C4).35(C2×C20), (C2×C20).505(C2×C4), (C2×C8).100(C2×C10), (C5×C22⋊C4).21C4, (C2×C10).331(C22×C4), (C22×C10).151(C2×C4), (C22×C4).109(C2×C10), (C2×C4).148(C22×C10), (C5×C42⋊C2).28C2, SmallGroup(320,906)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8○2M4(2)
G = < a,b,c,d | a5=b8=d2=1, c4=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >
Subgroups: 146 in 130 conjugacy classes, 114 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C20, C20, C20, C2×C10, C2×C10, C2×C10, C4×C8, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), C40, C2×C20, C2×C20, C22×C10, C8○2M4(2), C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C2×C40, C5×M4(2), C22×C20, C4×C40, C5×C8⋊C4, C5×C42⋊C2, C22×C40, C10×M4(2), C5×C8○2M4(2)
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C42, C22×C4, C20, C2×C10, C2×C42, C8○D4, C2×C20, C22×C10, C8○2M4(2), C4×C20, C22×C20, C2×C4×C20, C5×C8○D4, C5×C8○2M4(2)
►On 160 points
Generators in S160
(1 105 32 97 24)(2 106 25 98 17)(3 107 26 99 18)(4 108 27 100 19)(5 109 28 101 20)(6 110 29 102 21)(7 111 30 103 22)(8 112 31 104 23)(9 45 118 37 90)(10 46 119 38 91)(11 47 120 39 92)(12 48 113 40 93)(13 41 114 33 94)(14 42 115 34 95)(15 43 116 35 96)(16 44 117 36 89)(49 122 138 57 130)(50 123 139 58 131)(51 124 140 59 132)(52 125 141 60 133)(53 126 142 61 134)(54 127 143 62 135)(55 128 144 63 136)(56 121 137 64 129)(65 87 154 73 146)(66 88 155 74 147)(67 81 156 75 148)(68 82 157 76 149)(69 83 158 77 150)(70 84 159 78 151)(71 85 160 79 152)(72 86 153 80 145)
(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)
(1 65 121 37 5 69 125 33)(2 66 122 38 6 70 126 34)(3 67 123 39 7 71 127 35)(4 68 124 40 8 72 128 36)(9 28 158 60 13 32 154 64)(10 29 159 61 14 25 155 57)(11 30 160 62 15 26 156 58)(12 31 153 63 16 27 157 59)(17 147 49 119 21 151 53 115)(18 148 50 120 22 152 54 116)(19 149 51 113 23 145 55 117)(20 150 52 114 24 146 56 118)(41 97 73 129 45 101 77 133)(42 98 74 130 46 102 78 134)(43 99 75 131 47 103 79 135)(44 100 76 132 48 104 80 136)(81 139 92 111 85 143 96 107)(82 140 93 112 86 144 89 108)(83 141 94 105 87 137 90 109)(84 142 95 106 88 138 91 110)
(1 123)(2 124)(3 125)(4 126)(5 127)(6 128)(7 121)(8 122)(9 156)(10 157)(11 158)(12 159)(13 160)(14 153)(15 154)(16 155)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 49)(24 50)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)(33 71)(34 72)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 79)(42 80)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(81 90)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 89)(97 131)(98 132)(99 133)(100 134)(101 135)(102 136)(103 129)(104 130)(105 139)(106 140)(107 141)(108 142)(109 143)(110 144)(111 137)(112 138)(113 151)(114 152)(115 145)(116 146)(117 147)(118 148)(119 149)(120 150)
G:=sub<Sym(160)| (1,105,32,97,24)(2,106,25,98,17)(3,107,26,99,18)(4,108,27,100,19)(5,109,28,101,20)(6,110,29,102,21)(7,111,30,103,22)(8,112,31,104,23)(9,45,118,37,90)(10,46,119,38,91)(11,47,120,39,92)(12,48,113,40,93)(13,41,114,33,94)(14,42,115,34,95)(15,43,116,35,96)(16,44,117,36,89)(49,122,138,57,130)(50,123,139,58,131)(51,124,140,59,132)(52,125,141,60,133)(53,126,142,61,134)(54,127,143,62,135)(55,128,144,63,136)(56,121,137,64,129)(65,87,154,73,146)(66,88,155,74,147)(67,81,156,75,148)(68,82,157,76,149)(69,83,158,77,150)(70,84,159,78,151)(71,85,160,79,152)(72,86,153,80,145), (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), (1,65,121,37,5,69,125,33)(2,66,122,38,6,70,126,34)(3,67,123,39,7,71,127,35)(4,68,124,40,8,72,128,36)(9,28,158,60,13,32,154,64)(10,29,159,61,14,25,155,57)(11,30,160,62,15,26,156,58)(12,31,153,63,16,27,157,59)(17,147,49,119,21,151,53,115)(18,148,50,120,22,152,54,116)(19,149,51,113,23,145,55,117)(20,150,52,114,24,146,56,118)(41,97,73,129,45,101,77,133)(42,98,74,130,46,102,78,134)(43,99,75,131,47,103,79,135)(44,100,76,132,48,104,80,136)(81,139,92,111,85,143,96,107)(82,140,93,112,86,144,89,108)(83,141,94,105,87,137,90,109)(84,142,95,106,88,138,91,110), (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,121)(8,122)(9,156)(10,157)(11,158)(12,159)(13,160)(14,153)(15,154)(16,155)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,71)(34,72)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,79)(42,80)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(81,90)(82,91)(83,92)(84,93)(85,94)(86,95)(87,96)(88,89)(97,131)(98,132)(99,133)(100,134)(101,135)(102,136)(103,129)(104,130)(105,139)(106,140)(107,141)(108,142)(109,143)(110,144)(111,137)(112,138)(113,151)(114,152)(115,145)(116,146)(117,147)(118,148)(119,149)(120,150)>;
G:=Group( (1,105,32,97,24)(2,106,25,98,17)(3,107,26,99,18)(4,108,27,100,19)(5,109,28,101,20)(6,110,29,102,21)(7,111,30,103,22)(8,112,31,104,23)(9,45,118,37,90)(10,46,119,38,91)(11,47,120,39,92)(12,48,113,40,93)(13,41,114,33,94)(14,42,115,34,95)(15,43,116,35,96)(16,44,117,36,89)(49,122,138,57,130)(50,123,139,58,131)(51,124,140,59,132)(52,125,141,60,133)(53,126,142,61,134)(54,127,143,62,135)(55,128,144,63,136)(56,121,137,64,129)(65,87,154,73,146)(66,88,155,74,147)(67,81,156,75,148)(68,82,157,76,149)(69,83,158,77,150)(70,84,159,78,151)(71,85,160,79,152)(72,86,153,80,145), (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), (1,65,121,37,5,69,125,33)(2,66,122,38,6,70,126,34)(3,67,123,39,7,71,127,35)(4,68,124,40,8,72,128,36)(9,28,158,60,13,32,154,64)(10,29,159,61,14,25,155,57)(11,30,160,62,15,26,156,58)(12,31,153,63,16,27,157,59)(17,147,49,119,21,151,53,115)(18,148,50,120,22,152,54,116)(19,149,51,113,23,145,55,117)(20,150,52,114,24,146,56,118)(41,97,73,129,45,101,77,133)(42,98,74,130,46,102,78,134)(43,99,75,131,47,103,79,135)(44,100,76,132,48,104,80,136)(81,139,92,111,85,143,96,107)(82,140,93,112,86,144,89,108)(83,141,94,105,87,137,90,109)(84,142,95,106,88,138,91,110), (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,121)(8,122)(9,156)(10,157)(11,158)(12,159)(13,160)(14,153)(15,154)(16,155)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,71)(34,72)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,79)(42,80)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(81,90)(82,91)(83,92)(84,93)(85,94)(86,95)(87,96)(88,89)(97,131)(98,132)(99,133)(100,134)(101,135)(102,136)(103,129)(104,130)(105,139)(106,140)(107,141)(108,142)(109,143)(110,144)(111,137)(112,138)(113,151)(114,152)(115,145)(116,146)(117,147)(118,148)(119,149)(120,150) );
G=PermutationGroup([[(1,105,32,97,24),(2,106,25,98,17),(3,107,26,99,18),(4,108,27,100,19),(5,109,28,101,20),(6,110,29,102,21),(7,111,30,103,22),(8,112,31,104,23),(9,45,118,37,90),(10,46,119,38,91),(11,47,120,39,92),(12,48,113,40,93),(13,41,114,33,94),(14,42,115,34,95),(15,43,116,35,96),(16,44,117,36,89),(49,122,138,57,130),(50,123,139,58,131),(51,124,140,59,132),(52,125,141,60,133),(53,126,142,61,134),(54,127,143,62,135),(55,128,144,63,136),(56,121,137,64,129),(65,87,154,73,146),(66,88,155,74,147),(67,81,156,75,148),(68,82,157,76,149),(69,83,158,77,150),(70,84,159,78,151),(71,85,160,79,152),(72,86,153,80,145)], [(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)], [(1,65,121,37,5,69,125,33),(2,66,122,38,6,70,126,34),(3,67,123,39,7,71,127,35),(4,68,124,40,8,72,128,36),(9,28,158,60,13,32,154,64),(10,29,159,61,14,25,155,57),(11,30,160,62,15,26,156,58),(12,31,153,63,16,27,157,59),(17,147,49,119,21,151,53,115),(18,148,50,120,22,152,54,116),(19,149,51,113,23,145,55,117),(20,150,52,114,24,146,56,118),(41,97,73,129,45,101,77,133),(42,98,74,130,46,102,78,134),(43,99,75,131,47,103,79,135),(44,100,76,132,48,104,80,136),(81,139,92,111,85,143,96,107),(82,140,93,112,86,144,89,108),(83,141,94,105,87,137,90,109),(84,142,95,106,88,138,91,110)], [(1,123),(2,124),(3,125),(4,126),(5,127),(6,128),(7,121),(8,122),(9,156),(10,157),(11,158),(12,159),(13,160),(14,153),(15,154),(16,155),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,49),(24,50),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,57),(32,58),(33,71),(34,72),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70),(41,79),(42,80),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(81,90),(82,91),(83,92),(84,93),(85,94),(86,95),(87,96),(88,89),(97,131),(98,132),(99,133),(100,134),(101,135),(102,136),(103,129),(104,130),(105,139),(106,140),(107,141),(108,142),(109,143),(110,144),(111,137),(112,138),(113,151),(114,152),(115,145),(116,146),(117,147),(118,148),(119,149),(120,150)]])
200 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 8I | ··· | 8T | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | ··· | 20BD | 40A | ··· | 40AF | 40AG | ··· | 40CB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C20 | C20 | C20 | C20 | C8○D4 | C5×C8○D4 |
| kernel | C5×C8○2M4(2) | C4×C40 | C5×C8⋊C4 | C5×C42⋊C2 | C22×C40 | C10×M4(2) | C5×C22⋊C4 | C5×C4⋊C4 | C2×C40 | C5×M4(2) | C8○2M4(2) | C4×C8 | C8⋊C4 | C42⋊C2 | C22×C8 | C2×M4(2) | C22⋊C4 | C4⋊C4 | C2×C8 | M4(2) | C10 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 8 | 8 | 4 | 8 | 8 | 4 | 4 | 4 | 16 | 16 | 32 | 32 | 8 | 32 |
Matrix representation of C5×C8○2M4(2) ►in GL3(𝔽41) generated by
| 1 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 18 |
| 32 | 0 | 0 |
| 0 | 27 | 0 |
| 0 | 0 | 27 |
| 40 | 0 | 0 |
| 0 | 3 | 6 |
| 0 | 0 | 38 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 40 | 40 |
G:=sub<GL(3,GF(41))| [1,0,0,0,18,0,0,0,18],[32,0,0,0,27,0,0,0,27],[40,0,0,0,3,0,0,6,38],[1,0,0,0,1,40,0,0,40] >;
C5×C8○2M4(2) in GAP, Magma, Sage, TeX
C_5\times C_8\circ_2M_4(2)
% in TeX
G:=Group("C5xC8o2M4(2)"); // GroupNames label
G:=SmallGroup(320,906);
// by ID
G=gap.SmallGroup(320,906);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,568,1731,172]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=d^2=1,c^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations