metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.153D10, C10.1332+ 1+4, (C4×D20)⋊48C2, C42.C2⋊9D5, C4⋊D20⋊33C2, C4⋊C4.209D10, D20⋊8C4⋊37C2, (C2×C20).91C23, D10.55(C4○D4), C20.130(C4○D4), (C2×C10).239C24, (C4×C20).198C22, C4.39(Q8⋊2D5), D10.13D4⋊35C2, C2.58(D4⋊8D10), (C2×D20).276C22, C4⋊Dic5.315C22, C22.260(C23×D5), D10⋊C4.41C22, (C4×Dic5).153C22, (C2×Dic5).124C23, C10.D4.54C22, (C22×D5).104C23, C5⋊10(C22.47C24), (D5×C4⋊C4)⋊39C2, C2.90(D5×C4○D4), C4⋊C4⋊D5⋊37C2, C4⋊C4⋊7D5⋊38C2, C10.201(C2×C4○D4), C2.24(C2×Q8⋊2D5), (C5×C42.C2)⋊12C2, (C2×C4×D5).138C22, (C2×C4).82(C22×D5), (C5×C4⋊C4).194C22, SmallGroup(320,1367)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.153D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c9 >
Subgroups: 950 in 238 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C42.C2, C42⋊2C2, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22.47C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C4×D20, D5×C4⋊C4, C4⋊C4⋊7D5, D20⋊8C4, D10.13D4, C4⋊D20, C4⋊D20, C4⋊C4⋊D5, C5×C42.C2, C42.153D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.47C24, Q8⋊2D5, C23×D5, C2×Q8⋊2D5, D5×C4○D4, D4⋊8D10, C42.153D10
►On 160 points
Generators in S160
(1 153 63 116)(2 117 64 154)(3 155 65 118)(4 119 66 156)(5 157 67 120)(6 101 68 158)(7 159 69 102)(8 103 70 160)(9 141 71 104)(10 105 72 142)(11 143 73 106)(12 107 74 144)(13 145 75 108)(14 109 76 146)(15 147 77 110)(16 111 78 148)(17 149 79 112)(18 113 80 150)(19 151 61 114)(20 115 62 152)(21 96 130 43)(22 44 131 97)(23 98 132 45)(24 46 133 99)(25 100 134 47)(26 48 135 81)(27 82 136 49)(28 50 137 83)(29 84 138 51)(30 52 139 85)(31 86 140 53)(32 54 121 87)(33 88 122 55)(34 56 123 89)(35 90 124 57)(36 58 125 91)(37 92 126 59)(38 60 127 93)(39 94 128 41)(40 42 129 95)
(1 136 73 37)(2 127 74 28)(3 138 75 39)(4 129 76 30)(5 140 77 21)(6 131 78 32)(7 122 79 23)(8 133 80 34)(9 124 61 25)(10 135 62 36)(11 126 63 27)(12 137 64 38)(13 128 65 29)(14 139 66 40)(15 130 67 31)(16 121 68 22)(17 132 69 33)(18 123 70 24)(19 134 71 35)(20 125 72 26)(41 118 84 145)(42 109 85 156)(43 120 86 147)(44 111 87 158)(45 102 88 149)(46 113 89 160)(47 104 90 151)(48 115 91 142)(49 106 92 153)(50 117 93 144)(51 108 94 155)(52 119 95 146)(53 110 96 157)(54 101 97 148)(55 112 98 159)(56 103 99 150)(57 114 100 141)(58 105 81 152)(59 116 82 143)(60 107 83 154)
(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 96 11 86)(2 85 12 95)(3 94 13 84)(4 83 14 93)(5 92 15 82)(6 81 16 91)(7 90 17 100)(8 99 18 89)(9 88 19 98)(10 97 20 87)(21 143 31 153)(22 152 32 142)(23 141 33 151)(24 150 34 160)(25 159 35 149)(26 148 36 158)(27 157 37 147)(28 146 38 156)(29 155 39 145)(30 144 40 154)(41 75 51 65)(42 64 52 74)(43 73 53 63)(44 62 54 72)(45 71 55 61)(46 80 56 70)(47 69 57 79)(48 78 58 68)(49 67 59 77)(50 76 60 66)(101 135 111 125)(102 124 112 134)(103 133 113 123)(104 122 114 132)(105 131 115 121)(106 140 116 130)(107 129 117 139)(108 138 118 128)(109 127 119 137)(110 136 120 126)
G:=sub<Sym(160)| (1,153,63,116)(2,117,64,154)(3,155,65,118)(4,119,66,156)(5,157,67,120)(6,101,68,158)(7,159,69,102)(8,103,70,160)(9,141,71,104)(10,105,72,142)(11,143,73,106)(12,107,74,144)(13,145,75,108)(14,109,76,146)(15,147,77,110)(16,111,78,148)(17,149,79,112)(18,113,80,150)(19,151,61,114)(20,115,62,152)(21,96,130,43)(22,44,131,97)(23,98,132,45)(24,46,133,99)(25,100,134,47)(26,48,135,81)(27,82,136,49)(28,50,137,83)(29,84,138,51)(30,52,139,85)(31,86,140,53)(32,54,121,87)(33,88,122,55)(34,56,123,89)(35,90,124,57)(36,58,125,91)(37,92,126,59)(38,60,127,93)(39,94,128,41)(40,42,129,95), (1,136,73,37)(2,127,74,28)(3,138,75,39)(4,129,76,30)(5,140,77,21)(6,131,78,32)(7,122,79,23)(8,133,80,34)(9,124,61,25)(10,135,62,36)(11,126,63,27)(12,137,64,38)(13,128,65,29)(14,139,66,40)(15,130,67,31)(16,121,68,22)(17,132,69,33)(18,123,70,24)(19,134,71,35)(20,125,72,26)(41,118,84,145)(42,109,85,156)(43,120,86,147)(44,111,87,158)(45,102,88,149)(46,113,89,160)(47,104,90,151)(48,115,91,142)(49,106,92,153)(50,117,93,144)(51,108,94,155)(52,119,95,146)(53,110,96,157)(54,101,97,148)(55,112,98,159)(56,103,99,150)(57,114,100,141)(58,105,81,152)(59,116,82,143)(60,107,83,154), (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), (1,96,11,86)(2,85,12,95)(3,94,13,84)(4,83,14,93)(5,92,15,82)(6,81,16,91)(7,90,17,100)(8,99,18,89)(9,88,19,98)(10,97,20,87)(21,143,31,153)(22,152,32,142)(23,141,33,151)(24,150,34,160)(25,159,35,149)(26,148,36,158)(27,157,37,147)(28,146,38,156)(29,155,39,145)(30,144,40,154)(41,75,51,65)(42,64,52,74)(43,73,53,63)(44,62,54,72)(45,71,55,61)(46,80,56,70)(47,69,57,79)(48,78,58,68)(49,67,59,77)(50,76,60,66)(101,135,111,125)(102,124,112,134)(103,133,113,123)(104,122,114,132)(105,131,115,121)(106,140,116,130)(107,129,117,139)(108,138,118,128)(109,127,119,137)(110,136,120,126)>;
G:=Group( (1,153,63,116)(2,117,64,154)(3,155,65,118)(4,119,66,156)(5,157,67,120)(6,101,68,158)(7,159,69,102)(8,103,70,160)(9,141,71,104)(10,105,72,142)(11,143,73,106)(12,107,74,144)(13,145,75,108)(14,109,76,146)(15,147,77,110)(16,111,78,148)(17,149,79,112)(18,113,80,150)(19,151,61,114)(20,115,62,152)(21,96,130,43)(22,44,131,97)(23,98,132,45)(24,46,133,99)(25,100,134,47)(26,48,135,81)(27,82,136,49)(28,50,137,83)(29,84,138,51)(30,52,139,85)(31,86,140,53)(32,54,121,87)(33,88,122,55)(34,56,123,89)(35,90,124,57)(36,58,125,91)(37,92,126,59)(38,60,127,93)(39,94,128,41)(40,42,129,95), (1,136,73,37)(2,127,74,28)(3,138,75,39)(4,129,76,30)(5,140,77,21)(6,131,78,32)(7,122,79,23)(8,133,80,34)(9,124,61,25)(10,135,62,36)(11,126,63,27)(12,137,64,38)(13,128,65,29)(14,139,66,40)(15,130,67,31)(16,121,68,22)(17,132,69,33)(18,123,70,24)(19,134,71,35)(20,125,72,26)(41,118,84,145)(42,109,85,156)(43,120,86,147)(44,111,87,158)(45,102,88,149)(46,113,89,160)(47,104,90,151)(48,115,91,142)(49,106,92,153)(50,117,93,144)(51,108,94,155)(52,119,95,146)(53,110,96,157)(54,101,97,148)(55,112,98,159)(56,103,99,150)(57,114,100,141)(58,105,81,152)(59,116,82,143)(60,107,83,154), (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), (1,96,11,86)(2,85,12,95)(3,94,13,84)(4,83,14,93)(5,92,15,82)(6,81,16,91)(7,90,17,100)(8,99,18,89)(9,88,19,98)(10,97,20,87)(21,143,31,153)(22,152,32,142)(23,141,33,151)(24,150,34,160)(25,159,35,149)(26,148,36,158)(27,157,37,147)(28,146,38,156)(29,155,39,145)(30,144,40,154)(41,75,51,65)(42,64,52,74)(43,73,53,63)(44,62,54,72)(45,71,55,61)(46,80,56,70)(47,69,57,79)(48,78,58,68)(49,67,59,77)(50,76,60,66)(101,135,111,125)(102,124,112,134)(103,133,113,123)(104,122,114,132)(105,131,115,121)(106,140,116,130)(107,129,117,139)(108,138,118,128)(109,127,119,137)(110,136,120,126) );
G=PermutationGroup([[(1,153,63,116),(2,117,64,154),(3,155,65,118),(4,119,66,156),(5,157,67,120),(6,101,68,158),(7,159,69,102),(8,103,70,160),(9,141,71,104),(10,105,72,142),(11,143,73,106),(12,107,74,144),(13,145,75,108),(14,109,76,146),(15,147,77,110),(16,111,78,148),(17,149,79,112),(18,113,80,150),(19,151,61,114),(20,115,62,152),(21,96,130,43),(22,44,131,97),(23,98,132,45),(24,46,133,99),(25,100,134,47),(26,48,135,81),(27,82,136,49),(28,50,137,83),(29,84,138,51),(30,52,139,85),(31,86,140,53),(32,54,121,87),(33,88,122,55),(34,56,123,89),(35,90,124,57),(36,58,125,91),(37,92,126,59),(38,60,127,93),(39,94,128,41),(40,42,129,95)], [(1,136,73,37),(2,127,74,28),(3,138,75,39),(4,129,76,30),(5,140,77,21),(6,131,78,32),(7,122,79,23),(8,133,80,34),(9,124,61,25),(10,135,62,36),(11,126,63,27),(12,137,64,38),(13,128,65,29),(14,139,66,40),(15,130,67,31),(16,121,68,22),(17,132,69,33),(18,123,70,24),(19,134,71,35),(20,125,72,26),(41,118,84,145),(42,109,85,156),(43,120,86,147),(44,111,87,158),(45,102,88,149),(46,113,89,160),(47,104,90,151),(48,115,91,142),(49,106,92,153),(50,117,93,144),(51,108,94,155),(52,119,95,146),(53,110,96,157),(54,101,97,148),(55,112,98,159),(56,103,99,150),(57,114,100,141),(58,105,81,152),(59,116,82,143),(60,107,83,154)], [(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)], [(1,96,11,86),(2,85,12,95),(3,94,13,84),(4,83,14,93),(5,92,15,82),(6,81,16,91),(7,90,17,100),(8,99,18,89),(9,88,19,98),(10,97,20,87),(21,143,31,153),(22,152,32,142),(23,141,33,151),(24,150,34,160),(25,159,35,149),(26,148,36,158),(27,157,37,147),(28,146,38,156),(29,155,39,145),(30,144,40,154),(41,75,51,65),(42,64,52,74),(43,73,53,63),(44,62,54,72),(45,71,55,61),(46,80,56,70),(47,69,57,79),(48,78,58,68),(49,67,59,77),(50,76,60,66),(101,135,111,125),(102,124,112,134),(103,133,113,123),(104,122,114,132),(105,131,115,121),(106,140,116,130),(107,129,117,139),(108,138,118,128),(109,127,119,137),(110,136,120,126)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | ··· | 10 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | 2+ 1+4 | Q8⋊2D5 | D5×C4○D4 | D4⋊8D10 |
| kernel | C42.153D10 | C4×D20 | D5×C4⋊C4 | C4⋊C4⋊7D5 | D20⋊8C4 | D10.13D4 | C4⋊D20 | C4⋊C4⋊D5 | C5×C42.C2 | C42.C2 | C20 | D10 | C42 | C4⋊C4 | C10 | C4 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 2 | 4 | 4 | 2 | 12 | 1 | 4 | 4 | 4 |
Matrix representation of C42.153D10 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 12 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 29 | 18 | 0 | 0 | 0 | 0 |
| 17 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 34 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,0,9,0],[9,12,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[29,17,0,0,0,0,18,12,0,0,0,0,0,0,35,6,0,0,0,0,34,0,0,0,0,0,0,0,32,0,0,0,0,0,0,9],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,35,6,0,0,0,0,1,6,0,0,0,0,0,0,9,0,0,0,0,0,0,32] >;
C42.153D10 in GAP, Magma, Sage, TeX
C_4^2._{153}D_{10} % in TeX
G:=Group("C4^2.153D10"); // GroupNames label
G:=SmallGroup(320,1367);
// by ID
G=gap.SmallGroup(320,1367);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,100,1571,185,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations