metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊6M4(2), C42.204D10, C4⋊C8⋊14D5, C5⋊2C8⋊17D4, C5⋊6(C8⋊6D4), C4⋊1(C8⋊D5), (C4×D20).9C2, C4.208(D4×D5), C10.78(C4×D4), (C2×D20).25C4, C20.367(C2×D4), (C2×C8).183D10, C4⋊Dic5.31C4, D10⋊1C8⋊24C2, C10.53(C8○D4), (C4×C20).64C22, D10⋊C4.22C4, C20.336(C4○D4), C2.8(D20⋊8C4), (C2×C40).214C22, (C2×C20).835C23, C4.56(Q8⋊2D5), C10.43(C2×M4(2)), C2.13(D20.2C4), (C5×C4⋊C8)⋊19C2, (C4×C5⋊2C8)⋊5C2, (C2×C4).73(C4×D5), (C2×C8⋊D5)⋊22C2, C2.14(C2×C8⋊D5), C22.113(C2×C4×D5), (C2×C20).332(C2×C4), (C2×C4×D5).232C22, (C2×Dic5).25(C2×C4), (C22×D5).21(C2×C4), (C2×C4).777(C22×D5), (C2×C10).191(C22×C4), (C2×C5⋊2C8).315C22, SmallGroup(320,465)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊6M4(2)
G = < a,b,c | a20=b8=c2=1, bab-1=a9, cac=a-1, cbc=b5 >
Subgroups: 446 in 122 conjugacy classes, 53 normal (31 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, Dic5, C20, C20, C20, D10, C2×C10, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2), C5⋊2C8, C5⋊2C8, C40, C4×D5, D20, C2×Dic5, C2×C20, C22×D5, C8⋊6D4, C8⋊D5, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C2×D20, C4×C5⋊2C8, D10⋊1C8, C5×C4⋊C8, C4×D20, C2×C8⋊D5, C20⋊6M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, M4(2), C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×M4(2), C8○D4, C4×D5, C22×D5, C8⋊6D4, C8⋊D5, C2×C4×D5, D4×D5, Q8⋊2D5, D20⋊8C4, C2×C8⋊D5, D20.2C4, C20⋊6M4(2)
►On 160 points
Generators in S160
(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 48 104 35 81 146 73 121)(2 57 105 24 82 155 74 130)(3 46 106 33 83 144 75 139)(4 55 107 22 84 153 76 128)(5 44 108 31 85 142 77 137)(6 53 109 40 86 151 78 126)(7 42 110 29 87 160 79 135)(8 51 111 38 88 149 80 124)(9 60 112 27 89 158 61 133)(10 49 113 36 90 147 62 122)(11 58 114 25 91 156 63 131)(12 47 115 34 92 145 64 140)(13 56 116 23 93 154 65 129)(14 45 117 32 94 143 66 138)(15 54 118 21 95 152 67 127)(16 43 119 30 96 141 68 136)(17 52 120 39 97 150 69 125)(18 41 101 28 98 159 70 134)(19 50 102 37 99 148 71 123)(20 59 103 26 100 157 72 132)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 135)(22 134)(23 133)(24 132)(25 131)(26 130)(27 129)(28 128)(29 127)(30 126)(31 125)(32 124)(33 123)(34 122)(35 121)(36 140)(37 139)(38 138)(39 137)(40 136)(41 153)(42 152)(43 151)(44 150)(45 149)(46 148)(47 147)(48 146)(49 145)(50 144)(51 143)(52 142)(53 141)(54 160)(55 159)(56 158)(57 157)(58 156)(59 155)(60 154)(61 65)(62 64)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(82 100)(83 99)(84 98)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)(101 107)(102 106)(103 105)(108 120)(109 119)(110 118)(111 117)(112 116)(113 115)
G:=sub<Sym(160)| (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), (1,48,104,35,81,146,73,121)(2,57,105,24,82,155,74,130)(3,46,106,33,83,144,75,139)(4,55,107,22,84,153,76,128)(5,44,108,31,85,142,77,137)(6,53,109,40,86,151,78,126)(7,42,110,29,87,160,79,135)(8,51,111,38,88,149,80,124)(9,60,112,27,89,158,61,133)(10,49,113,36,90,147,62,122)(11,58,114,25,91,156,63,131)(12,47,115,34,92,145,64,140)(13,56,116,23,93,154,65,129)(14,45,117,32,94,143,66,138)(15,54,118,21,95,152,67,127)(16,43,119,30,96,141,68,136)(17,52,120,39,97,150,69,125)(18,41,101,28,98,159,70,134)(19,50,102,37,99,148,71,123)(20,59,103,26,100,157,72,132), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,135)(22,134)(23,133)(24,132)(25,131)(26,130)(27,129)(28,128)(29,127)(30,126)(31,125)(32,124)(33,123)(34,122)(35,121)(36,140)(37,139)(38,138)(39,137)(40,136)(41,153)(42,152)(43,151)(44,150)(45,149)(46,148)(47,147)(48,146)(49,145)(50,144)(51,143)(52,142)(53,141)(54,160)(55,159)(56,158)(57,157)(58,156)(59,155)(60,154)(61,65)(62,64)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(82,100)(83,99)(84,98)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(101,107)(102,106)(103,105)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115)>;
G:=Group( (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), (1,48,104,35,81,146,73,121)(2,57,105,24,82,155,74,130)(3,46,106,33,83,144,75,139)(4,55,107,22,84,153,76,128)(5,44,108,31,85,142,77,137)(6,53,109,40,86,151,78,126)(7,42,110,29,87,160,79,135)(8,51,111,38,88,149,80,124)(9,60,112,27,89,158,61,133)(10,49,113,36,90,147,62,122)(11,58,114,25,91,156,63,131)(12,47,115,34,92,145,64,140)(13,56,116,23,93,154,65,129)(14,45,117,32,94,143,66,138)(15,54,118,21,95,152,67,127)(16,43,119,30,96,141,68,136)(17,52,120,39,97,150,69,125)(18,41,101,28,98,159,70,134)(19,50,102,37,99,148,71,123)(20,59,103,26,100,157,72,132), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,135)(22,134)(23,133)(24,132)(25,131)(26,130)(27,129)(28,128)(29,127)(30,126)(31,125)(32,124)(33,123)(34,122)(35,121)(36,140)(37,139)(38,138)(39,137)(40,136)(41,153)(42,152)(43,151)(44,150)(45,149)(46,148)(47,147)(48,146)(49,145)(50,144)(51,143)(52,142)(53,141)(54,160)(55,159)(56,158)(57,157)(58,156)(59,155)(60,154)(61,65)(62,64)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(82,100)(83,99)(84,98)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(101,107)(102,106)(103,105)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115) );
G=PermutationGroup([[(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)], [(1,48,104,35,81,146,73,121),(2,57,105,24,82,155,74,130),(3,46,106,33,83,144,75,139),(4,55,107,22,84,153,76,128),(5,44,108,31,85,142,77,137),(6,53,109,40,86,151,78,126),(7,42,110,29,87,160,79,135),(8,51,111,38,88,149,80,124),(9,60,112,27,89,158,61,133),(10,49,113,36,90,147,62,122),(11,58,114,25,91,156,63,131),(12,47,115,34,92,145,64,140),(13,56,116,23,93,154,65,129),(14,45,117,32,94,143,66,138),(15,54,118,21,95,152,67,127),(16,43,119,30,96,141,68,136),(17,52,120,39,97,150,69,125),(18,41,101,28,98,159,70,134),(19,50,102,37,99,148,71,123),(20,59,103,26,100,157,72,132)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,135),(22,134),(23,133),(24,132),(25,131),(26,130),(27,129),(28,128),(29,127),(30,126),(31,125),(32,124),(33,123),(34,122),(35,121),(36,140),(37,139),(38,138),(39,137),(40,136),(41,153),(42,152),(43,151),(44,150),(45,149),(46,148),(47,147),(48,146),(49,145),(50,144),(51,143),(52,142),(53,141),(54,160),(55,159),(56,158),(57,157),(58,156),(59,155),(60,154),(61,65),(62,64),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74),(82,100),(83,99),(84,98),(85,97),(86,96),(87,95),(88,94),(89,93),(90,92),(101,107),(102,106),(103,105),(108,120),(109,119),(110,118),(111,117),(112,116),(113,115)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D5 | M4(2) | C4○D4 | D10 | D10 | C8○D4 | C4×D5 | C8⋊D5 | D4×D5 | Q8⋊2D5 | D20.2C4 |
| kernel | C20⋊6M4(2) | C4×C5⋊2C8 | D10⋊1C8 | C5×C4⋊C8 | C4×D20 | C2×C8⋊D5 | C4⋊Dic5 | D10⋊C4 | C2×D20 | C5⋊2C8 | C4⋊C8 | C20 | C20 | C42 | C2×C8 | C10 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 8 | 16 | 2 | 2 | 4 |
Matrix representation of C20⋊6M4(2) ►in GL6(𝔽41)
| 1 | 1 | 0 | 0 | 0 | 0 |
| 5 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 18 | 0 | 0 |
| 0 | 0 | 21 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 6 | 7 | 0 | 0 | 0 | 0 |
| 36 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 5 |
| 0 | 0 | 0 | 0 | 18 | 40 |
| 6 | 7 | 0 | 0 | 0 | 0 |
| 36 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 25 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 | 40 |
G:=sub<GL(6,GF(41))| [1,5,0,0,0,0,1,6,0,0,0,0,0,0,21,21,0,0,0,0,18,20,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,36,0,0,0,0,7,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,18,0,0,0,0,5,40],[6,36,0,0,0,0,7,35,0,0,0,0,0,0,1,25,0,0,0,0,0,40,0,0,0,0,0,0,1,16,0,0,0,0,0,40] >;
C20⋊6M4(2) in GAP, Magma, Sage, TeX
C_{20}\rtimes_6M_4(2) % in TeX
G:=Group("C20:6M4(2)"); // GroupNames label
G:=SmallGroup(320,465);
// by ID
G=gap.SmallGroup(320,465);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,758,219,58,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^9,c*a*c=a^-1,c*b*c=b^5>;
// generators/relations