metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.160D10, C10.992- 1+4, C20⋊Q8⋊40C2, C4⋊C4.117D10, C42⋊2C2⋊1D5, C42⋊D5⋊5C2, D10⋊Q8⋊39C2, (C4×Dic10)⋊14C2, (C4×C20).32C22, C22⋊C4.75D10, Dic5⋊3Q8⋊40C2, (C2×C10).246C24, (C2×C20).192C23, Dic5⋊4D4.4C2, C23.52(C22×D5), Dic5.47(C4○D4), C23.D10⋊43C2, C4⋊Dic5.317C22, (C22×C10).60C23, Dic5.5D4.4C2, C22.267(C23×D5), Dic5.14D4⋊44C2, C23.D5.62C22, C23.11D10⋊20C2, C5⋊7(C22.50C24), (C4×Dic5).157C22, (C2×Dic5).273C23, (C22×D5).110C23, C2.63(D4.10D10), D10⋊C4.139C22, (C2×Dic10).262C22, C10.D4.145C22, (C22×Dic5).149C22, C2.93(D5×C4○D4), C4⋊C4⋊D5⋊39C2, (C5×C42⋊2C2)⋊1C2, C10.204(C2×C4○D4), (C2×C4×D5).269C22, (C2×C4).83(C22×D5), (C5×C4⋊C4).201C22, (C2×C5⋊D4).67C22, (C5×C22⋊C4).71C22, SmallGroup(320,1374)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.160D10
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd-1=a2b-1, dcd-1=c-1 >
Subgroups: 678 in 212 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C4.4D4, C42⋊2C2, C42⋊2C2, C4⋊Q8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C22.50C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C4×Dic10, C42⋊D5, C23.11D10, Dic5.14D4, C23.D10, Dic5⋊4D4, Dic5.5D4, Dic5⋊3Q8, C20⋊Q8, D10⋊Q8, C4⋊C4⋊D5, C5×C42⋊2C2, C42.160D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.50C24, C23×D5, D5×C4○D4, D4.10D10, C42.160D10
►On 160 points
Generators in S160
(1 119 99 36)(2 71 100 144)(3 111 91 38)(4 73 92 146)(5 113 93 40)(6 75 94 148)(7 115 95 32)(8 77 96 150)(9 117 97 34)(10 79 98 142)(11 101 48 28)(12 63 49 136)(13 103 50 30)(14 65 41 138)(15 105 42 22)(16 67 43 140)(17 107 44 24)(18 69 45 132)(19 109 46 26)(20 61 47 134)(21 81 104 158)(23 83 106 160)(25 85 108 152)(27 87 110 154)(29 89 102 156)(31 128 114 55)(33 130 116 57)(35 122 118 59)(37 124 120 51)(39 126 112 53)(52 145 125 72)(54 147 127 74)(56 149 129 76)(58 141 121 78)(60 143 123 80)(62 155 135 88)(64 157 137 90)(66 159 139 82)(68 151 131 84)(70 153 133 86)
(1 133 123 26)(2 110 124 61)(3 135 125 28)(4 102 126 63)(5 137 127 30)(6 104 128 65)(7 139 129 22)(8 106 130 67)(9 131 121 24)(10 108 122 69)(11 111 88 72)(12 146 89 39)(13 113 90 74)(14 148 81 31)(15 115 82 76)(16 150 83 33)(17 117 84 78)(18 142 85 35)(19 119 86 80)(20 144 87 37)(21 55 138 94)(23 57 140 96)(25 59 132 98)(27 51 134 100)(29 53 136 92)(32 159 149 42)(34 151 141 44)(36 153 143 46)(38 155 145 48)(40 157 147 50)(41 75 158 114)(43 77 160 116)(45 79 152 118)(47 71 154 120)(49 73 156 112)(52 101 91 62)(54 103 93 64)(56 105 95 66)(58 107 97 68)(60 109 99 70)
(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 25 123 132)(2 24 124 131)(3 23 125 140)(4 22 126 139)(5 21 127 138)(6 30 128 137)(7 29 129 136)(8 28 130 135)(9 27 121 134)(10 26 122 133)(11 116 88 77)(12 115 89 76)(13 114 90 75)(14 113 81 74)(15 112 82 73)(16 111 83 72)(17 120 84 71)(18 119 85 80)(19 118 86 79)(20 117 87 78)(31 157 148 50)(32 156 149 49)(33 155 150 48)(34 154 141 47)(35 153 142 46)(36 152 143 45)(37 151 144 44)(38 160 145 43)(39 159 146 42)(40 158 147 41)(51 68 100 107)(52 67 91 106)(53 66 92 105)(54 65 93 104)(55 64 94 103)(56 63 95 102)(57 62 96 101)(58 61 97 110)(59 70 98 109)(60 69 99 108)
G:=sub<Sym(160)| (1,119,99,36)(2,71,100,144)(3,111,91,38)(4,73,92,146)(5,113,93,40)(6,75,94,148)(7,115,95,32)(8,77,96,150)(9,117,97,34)(10,79,98,142)(11,101,48,28)(12,63,49,136)(13,103,50,30)(14,65,41,138)(15,105,42,22)(16,67,43,140)(17,107,44,24)(18,69,45,132)(19,109,46,26)(20,61,47,134)(21,81,104,158)(23,83,106,160)(25,85,108,152)(27,87,110,154)(29,89,102,156)(31,128,114,55)(33,130,116,57)(35,122,118,59)(37,124,120,51)(39,126,112,53)(52,145,125,72)(54,147,127,74)(56,149,129,76)(58,141,121,78)(60,143,123,80)(62,155,135,88)(64,157,137,90)(66,159,139,82)(68,151,131,84)(70,153,133,86), (1,133,123,26)(2,110,124,61)(3,135,125,28)(4,102,126,63)(5,137,127,30)(6,104,128,65)(7,139,129,22)(8,106,130,67)(9,131,121,24)(10,108,122,69)(11,111,88,72)(12,146,89,39)(13,113,90,74)(14,148,81,31)(15,115,82,76)(16,150,83,33)(17,117,84,78)(18,142,85,35)(19,119,86,80)(20,144,87,37)(21,55,138,94)(23,57,140,96)(25,59,132,98)(27,51,134,100)(29,53,136,92)(32,159,149,42)(34,151,141,44)(36,153,143,46)(38,155,145,48)(40,157,147,50)(41,75,158,114)(43,77,160,116)(45,79,152,118)(47,71,154,120)(49,73,156,112)(52,101,91,62)(54,103,93,64)(56,105,95,66)(58,107,97,68)(60,109,99,70), (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), (1,25,123,132)(2,24,124,131)(3,23,125,140)(4,22,126,139)(5,21,127,138)(6,30,128,137)(7,29,129,136)(8,28,130,135)(9,27,121,134)(10,26,122,133)(11,116,88,77)(12,115,89,76)(13,114,90,75)(14,113,81,74)(15,112,82,73)(16,111,83,72)(17,120,84,71)(18,119,85,80)(19,118,86,79)(20,117,87,78)(31,157,148,50)(32,156,149,49)(33,155,150,48)(34,154,141,47)(35,153,142,46)(36,152,143,45)(37,151,144,44)(38,160,145,43)(39,159,146,42)(40,158,147,41)(51,68,100,107)(52,67,91,106)(53,66,92,105)(54,65,93,104)(55,64,94,103)(56,63,95,102)(57,62,96,101)(58,61,97,110)(59,70,98,109)(60,69,99,108)>;
G:=Group( (1,119,99,36)(2,71,100,144)(3,111,91,38)(4,73,92,146)(5,113,93,40)(6,75,94,148)(7,115,95,32)(8,77,96,150)(9,117,97,34)(10,79,98,142)(11,101,48,28)(12,63,49,136)(13,103,50,30)(14,65,41,138)(15,105,42,22)(16,67,43,140)(17,107,44,24)(18,69,45,132)(19,109,46,26)(20,61,47,134)(21,81,104,158)(23,83,106,160)(25,85,108,152)(27,87,110,154)(29,89,102,156)(31,128,114,55)(33,130,116,57)(35,122,118,59)(37,124,120,51)(39,126,112,53)(52,145,125,72)(54,147,127,74)(56,149,129,76)(58,141,121,78)(60,143,123,80)(62,155,135,88)(64,157,137,90)(66,159,139,82)(68,151,131,84)(70,153,133,86), (1,133,123,26)(2,110,124,61)(3,135,125,28)(4,102,126,63)(5,137,127,30)(6,104,128,65)(7,139,129,22)(8,106,130,67)(9,131,121,24)(10,108,122,69)(11,111,88,72)(12,146,89,39)(13,113,90,74)(14,148,81,31)(15,115,82,76)(16,150,83,33)(17,117,84,78)(18,142,85,35)(19,119,86,80)(20,144,87,37)(21,55,138,94)(23,57,140,96)(25,59,132,98)(27,51,134,100)(29,53,136,92)(32,159,149,42)(34,151,141,44)(36,153,143,46)(38,155,145,48)(40,157,147,50)(41,75,158,114)(43,77,160,116)(45,79,152,118)(47,71,154,120)(49,73,156,112)(52,101,91,62)(54,103,93,64)(56,105,95,66)(58,107,97,68)(60,109,99,70), (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), (1,25,123,132)(2,24,124,131)(3,23,125,140)(4,22,126,139)(5,21,127,138)(6,30,128,137)(7,29,129,136)(8,28,130,135)(9,27,121,134)(10,26,122,133)(11,116,88,77)(12,115,89,76)(13,114,90,75)(14,113,81,74)(15,112,82,73)(16,111,83,72)(17,120,84,71)(18,119,85,80)(19,118,86,79)(20,117,87,78)(31,157,148,50)(32,156,149,49)(33,155,150,48)(34,154,141,47)(35,153,142,46)(36,152,143,45)(37,151,144,44)(38,160,145,43)(39,159,146,42)(40,158,147,41)(51,68,100,107)(52,67,91,106)(53,66,92,105)(54,65,93,104)(55,64,94,103)(56,63,95,102)(57,62,96,101)(58,61,97,110)(59,70,98,109)(60,69,99,108) );
G=PermutationGroup([[(1,119,99,36),(2,71,100,144),(3,111,91,38),(4,73,92,146),(5,113,93,40),(6,75,94,148),(7,115,95,32),(8,77,96,150),(9,117,97,34),(10,79,98,142),(11,101,48,28),(12,63,49,136),(13,103,50,30),(14,65,41,138),(15,105,42,22),(16,67,43,140),(17,107,44,24),(18,69,45,132),(19,109,46,26),(20,61,47,134),(21,81,104,158),(23,83,106,160),(25,85,108,152),(27,87,110,154),(29,89,102,156),(31,128,114,55),(33,130,116,57),(35,122,118,59),(37,124,120,51),(39,126,112,53),(52,145,125,72),(54,147,127,74),(56,149,129,76),(58,141,121,78),(60,143,123,80),(62,155,135,88),(64,157,137,90),(66,159,139,82),(68,151,131,84),(70,153,133,86)], [(1,133,123,26),(2,110,124,61),(3,135,125,28),(4,102,126,63),(5,137,127,30),(6,104,128,65),(7,139,129,22),(8,106,130,67),(9,131,121,24),(10,108,122,69),(11,111,88,72),(12,146,89,39),(13,113,90,74),(14,148,81,31),(15,115,82,76),(16,150,83,33),(17,117,84,78),(18,142,85,35),(19,119,86,80),(20,144,87,37),(21,55,138,94),(23,57,140,96),(25,59,132,98),(27,51,134,100),(29,53,136,92),(32,159,149,42),(34,151,141,44),(36,153,143,46),(38,155,145,48),(40,157,147,50),(41,75,158,114),(43,77,160,116),(45,79,152,118),(47,71,154,120),(49,73,156,112),(52,101,91,62),(54,103,93,64),(56,105,95,66),(58,107,97,68),(60,109,99,70)], [(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)], [(1,25,123,132),(2,24,124,131),(3,23,125,140),(4,22,126,139),(5,21,127,138),(6,30,128,137),(7,29,129,136),(8,28,130,135),(9,27,121,134),(10,26,122,133),(11,116,88,77),(12,115,89,76),(13,114,90,75),(14,113,81,74),(15,112,82,73),(16,111,83,72),(17,120,84,71),(18,119,85,80),(19,118,86,79),(20,117,87,78),(31,157,148,50),(32,156,149,49),(33,155,150,48),(34,154,141,47),(35,153,142,46),(36,152,143,45),(37,151,144,44),(38,160,145,43),(39,159,146,42),(40,158,147,41),(51,68,100,107),(52,67,91,106),(53,66,92,105),(54,65,93,104),(55,64,94,103),(56,63,95,102),(57,62,96,101),(58,61,97,110),(59,70,98,109),(60,69,99,108)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 20A | ··· | 20L | 20M | ··· | 20R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | 2- 1+4 | D5×C4○D4 | D4.10D10 |
| kernel | C42.160D10 | C4×Dic10 | C42⋊D5 | C23.11D10 | Dic5.14D4 | C23.D10 | Dic5⋊4D4 | Dic5.5D4 | Dic5⋊3Q8 | C20⋊Q8 | D10⋊Q8 | C4⋊C4⋊D5 | C5×C42⋊2C2 | C42⋊2C2 | Dic5 | C42 | C22⋊C4 | C4⋊C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 8 | 2 | 6 | 6 | 1 | 8 | 4 |
Matrix representation of C42.160D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 33 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 34 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 2 | 0 | 0 |
| 0 | 0 | 30 | 33 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 |
| 7 | 35 | 0 | 0 | 0 | 0 |
| 8 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 2 | 0 | 0 |
| 0 | 0 | 30 | 33 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,33,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[0,34,0,0,0,0,6,7,0,0,0,0,0,0,8,30,0,0,0,0,2,33,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[7,8,0,0,0,0,35,34,0,0,0,0,0,0,8,30,0,0,0,0,2,33,0,0,0,0,0,0,32,0,0,0,0,0,0,32] >;
C42.160D10 in GAP, Magma, Sage, TeX
C_4^2._{160}D_{10} % in TeX
G:=Group("C4^2.160D10"); // GroupNames label
G:=SmallGroup(320,1374);
// by ID
G=gap.SmallGroup(320,1374);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,387,100,794,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,c*b*c^-1=d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations