metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.118D10, C10.632- 1+4, C10.232+ 1+4, (C4×D4)⋊26D5, (D4×C20)⋊28C2, C4⋊C4.288D10, D10⋊Q8⋊9C2, (C2×D4).225D10, C42⋊2D5⋊11C2, C20.6Q8⋊26C2, (C22×C4).49D10, C20.48D4⋊13C2, (C2×C10).108C24, (C2×C20).166C23, (C4×C20).220C22, C22⋊C4.120D10, Dic5.Q8⋊8C2, Dic5⋊D4.4C2, C22.7(C4○D20), C22.D20⋊7C2, C4⋊Dic5.41C22, D10.12D4⋊10C2, C2.25(D4⋊6D10), (D4×C10).309C22, C23.D10⋊10C2, C23.23D10⋊5C2, (C2×Dic5).48C23, C10.D4.8C22, (C22×D5).42C23, C22.133(C23×D5), C23.105(C22×D5), Dic5.14D4⋊10C2, C23.D5.17C22, D10⋊C4.67C22, C23.18D10⋊19C2, (C22×C20).366C22, (C22×C10).178C23, C5⋊2(C22.33C24), (C4×Dic5).227C22, (C2×Dic10).31C22, C2.20(D4.10D10), (C22×Dic5).100C22, (C4×C5⋊D4)⋊47C2, C10.50(C2×C4○D4), C2.57(C2×C4○D20), (C2×C4×D5).255C22, (C2×C10).18(C4○D4), (C2×C10.D4)⋊39C2, (C5×C4⋊C4).336C22, (C2×C4).164(C22×D5), (C2×C5⋊D4).125C22, (C5×C22⋊C4).107C22, SmallGroup(320,1236)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.118D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=a2b, dbd-1=b-1, dcd-1=a2c-1 >
Subgroups: 718 in 218 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C4, C22, 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, C22×C4, C2×D4, C2×D4, C2×Q8, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C4×D4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22.33C24, 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, C22×C20, D4×C10, C20.6Q8, C42⋊2D5, Dic5.14D4, C23.D10, D10.12D4, C22.D20, Dic5.Q8, D10⋊Q8, C2×C10.D4, C20.48D4, C4×C5⋊D4, C23.23D10, C23.18D10, Dic5⋊D4, D4×C20, C42.118D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5, C22.33C24, C4○D20, C23×D5, C2×C4○D20, D4⋊6D10, D4.10D10, C42.118D10
►On 160 points
Generators in S160
(1 138 123 27)(2 28 124 139)(3 140 125 29)(4 30 126 131)(5 132 127 21)(6 22 128 133)(7 134 129 23)(8 24 130 135)(9 136 121 25)(10 26 122 137)(11 74 85 119)(12 120 86 75)(13 76 87 111)(14 112 88 77)(15 78 89 113)(16 114 90 79)(17 80 81 115)(18 116 82 71)(19 72 83 117)(20 118 84 73)(31 157 142 46)(32 47 143 158)(33 159 144 48)(34 49 145 160)(35 151 146 50)(36 41 147 152)(37 153 148 42)(38 43 149 154)(39 155 150 44)(40 45 141 156)(51 66 96 101)(52 102 97 67)(53 68 98 103)(54 104 99 69)(55 70 100 105)(56 106 91 61)(57 62 92 107)(58 108 93 63)(59 64 94 109)(60 110 95 65)
(1 86 60 42)(2 13 51 154)(3 88 52 44)(4 15 53 156)(5 90 54 46)(6 17 55 158)(7 82 56 48)(8 19 57 160)(9 84 58 50)(10 11 59 152)(12 95 153 123)(14 97 155 125)(16 99 157 127)(18 91 159 129)(20 93 151 121)(21 114 69 142)(22 80 70 32)(23 116 61 144)(24 72 62 34)(25 118 63 146)(26 74 64 36)(27 120 65 148)(28 76 66 38)(29 112 67 150)(30 78 68 40)(31 132 79 104)(33 134 71 106)(35 136 73 108)(37 138 75 110)(39 140 77 102)(41 122 85 94)(43 124 87 96)(45 126 89 98)(47 128 81 100)(49 130 83 92)(101 149 139 111)(103 141 131 113)(105 143 133 115)(107 145 135 117)(109 147 137 119)
(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 10 123 122)(2 121 124 9)(3 8 125 130)(4 129 126 7)(5 6 127 128)(11 153 85 42)(12 41 86 152)(13 151 87 50)(14 49 88 160)(15 159 89 48)(16 47 90 158)(17 157 81 46)(18 45 82 156)(19 155 83 44)(20 43 84 154)(21 70 132 105)(22 104 133 69)(23 68 134 103)(24 102 135 67)(25 66 136 101)(26 110 137 65)(27 64 138 109)(28 108 139 63)(29 62 140 107)(30 106 131 61)(31 143 142 32)(33 141 144 40)(34 39 145 150)(35 149 146 38)(36 37 147 148)(51 93 96 58)(52 57 97 92)(53 91 98 56)(54 55 99 100)(59 95 94 60)(71 113 116 78)(72 77 117 112)(73 111 118 76)(74 75 119 120)(79 115 114 80)
G:=sub<Sym(160)| (1,138,123,27)(2,28,124,139)(3,140,125,29)(4,30,126,131)(5,132,127,21)(6,22,128,133)(7,134,129,23)(8,24,130,135)(9,136,121,25)(10,26,122,137)(11,74,85,119)(12,120,86,75)(13,76,87,111)(14,112,88,77)(15,78,89,113)(16,114,90,79)(17,80,81,115)(18,116,82,71)(19,72,83,117)(20,118,84,73)(31,157,142,46)(32,47,143,158)(33,159,144,48)(34,49,145,160)(35,151,146,50)(36,41,147,152)(37,153,148,42)(38,43,149,154)(39,155,150,44)(40,45,141,156)(51,66,96,101)(52,102,97,67)(53,68,98,103)(54,104,99,69)(55,70,100,105)(56,106,91,61)(57,62,92,107)(58,108,93,63)(59,64,94,109)(60,110,95,65), (1,86,60,42)(2,13,51,154)(3,88,52,44)(4,15,53,156)(5,90,54,46)(6,17,55,158)(7,82,56,48)(8,19,57,160)(9,84,58,50)(10,11,59,152)(12,95,153,123)(14,97,155,125)(16,99,157,127)(18,91,159,129)(20,93,151,121)(21,114,69,142)(22,80,70,32)(23,116,61,144)(24,72,62,34)(25,118,63,146)(26,74,64,36)(27,120,65,148)(28,76,66,38)(29,112,67,150)(30,78,68,40)(31,132,79,104)(33,134,71,106)(35,136,73,108)(37,138,75,110)(39,140,77,102)(41,122,85,94)(43,124,87,96)(45,126,89,98)(47,128,81,100)(49,130,83,92)(101,149,139,111)(103,141,131,113)(105,143,133,115)(107,145,135,117)(109,147,137,119), (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), (1,10,123,122)(2,121,124,9)(3,8,125,130)(4,129,126,7)(5,6,127,128)(11,153,85,42)(12,41,86,152)(13,151,87,50)(14,49,88,160)(15,159,89,48)(16,47,90,158)(17,157,81,46)(18,45,82,156)(19,155,83,44)(20,43,84,154)(21,70,132,105)(22,104,133,69)(23,68,134,103)(24,102,135,67)(25,66,136,101)(26,110,137,65)(27,64,138,109)(28,108,139,63)(29,62,140,107)(30,106,131,61)(31,143,142,32)(33,141,144,40)(34,39,145,150)(35,149,146,38)(36,37,147,148)(51,93,96,58)(52,57,97,92)(53,91,98,56)(54,55,99,100)(59,95,94,60)(71,113,116,78)(72,77,117,112)(73,111,118,76)(74,75,119,120)(79,115,114,80)>;
G:=Group( (1,138,123,27)(2,28,124,139)(3,140,125,29)(4,30,126,131)(5,132,127,21)(6,22,128,133)(7,134,129,23)(8,24,130,135)(9,136,121,25)(10,26,122,137)(11,74,85,119)(12,120,86,75)(13,76,87,111)(14,112,88,77)(15,78,89,113)(16,114,90,79)(17,80,81,115)(18,116,82,71)(19,72,83,117)(20,118,84,73)(31,157,142,46)(32,47,143,158)(33,159,144,48)(34,49,145,160)(35,151,146,50)(36,41,147,152)(37,153,148,42)(38,43,149,154)(39,155,150,44)(40,45,141,156)(51,66,96,101)(52,102,97,67)(53,68,98,103)(54,104,99,69)(55,70,100,105)(56,106,91,61)(57,62,92,107)(58,108,93,63)(59,64,94,109)(60,110,95,65), (1,86,60,42)(2,13,51,154)(3,88,52,44)(4,15,53,156)(5,90,54,46)(6,17,55,158)(7,82,56,48)(8,19,57,160)(9,84,58,50)(10,11,59,152)(12,95,153,123)(14,97,155,125)(16,99,157,127)(18,91,159,129)(20,93,151,121)(21,114,69,142)(22,80,70,32)(23,116,61,144)(24,72,62,34)(25,118,63,146)(26,74,64,36)(27,120,65,148)(28,76,66,38)(29,112,67,150)(30,78,68,40)(31,132,79,104)(33,134,71,106)(35,136,73,108)(37,138,75,110)(39,140,77,102)(41,122,85,94)(43,124,87,96)(45,126,89,98)(47,128,81,100)(49,130,83,92)(101,149,139,111)(103,141,131,113)(105,143,133,115)(107,145,135,117)(109,147,137,119), (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), (1,10,123,122)(2,121,124,9)(3,8,125,130)(4,129,126,7)(5,6,127,128)(11,153,85,42)(12,41,86,152)(13,151,87,50)(14,49,88,160)(15,159,89,48)(16,47,90,158)(17,157,81,46)(18,45,82,156)(19,155,83,44)(20,43,84,154)(21,70,132,105)(22,104,133,69)(23,68,134,103)(24,102,135,67)(25,66,136,101)(26,110,137,65)(27,64,138,109)(28,108,139,63)(29,62,140,107)(30,106,131,61)(31,143,142,32)(33,141,144,40)(34,39,145,150)(35,149,146,38)(36,37,147,148)(51,93,96,58)(52,57,97,92)(53,91,98,56)(54,55,99,100)(59,95,94,60)(71,113,116,78)(72,77,117,112)(73,111,118,76)(74,75,119,120)(79,115,114,80) );
G=PermutationGroup([[(1,138,123,27),(2,28,124,139),(3,140,125,29),(4,30,126,131),(5,132,127,21),(6,22,128,133),(7,134,129,23),(8,24,130,135),(9,136,121,25),(10,26,122,137),(11,74,85,119),(12,120,86,75),(13,76,87,111),(14,112,88,77),(15,78,89,113),(16,114,90,79),(17,80,81,115),(18,116,82,71),(19,72,83,117),(20,118,84,73),(31,157,142,46),(32,47,143,158),(33,159,144,48),(34,49,145,160),(35,151,146,50),(36,41,147,152),(37,153,148,42),(38,43,149,154),(39,155,150,44),(40,45,141,156),(51,66,96,101),(52,102,97,67),(53,68,98,103),(54,104,99,69),(55,70,100,105),(56,106,91,61),(57,62,92,107),(58,108,93,63),(59,64,94,109),(60,110,95,65)], [(1,86,60,42),(2,13,51,154),(3,88,52,44),(4,15,53,156),(5,90,54,46),(6,17,55,158),(7,82,56,48),(8,19,57,160),(9,84,58,50),(10,11,59,152),(12,95,153,123),(14,97,155,125),(16,99,157,127),(18,91,159,129),(20,93,151,121),(21,114,69,142),(22,80,70,32),(23,116,61,144),(24,72,62,34),(25,118,63,146),(26,74,64,36),(27,120,65,148),(28,76,66,38),(29,112,67,150),(30,78,68,40),(31,132,79,104),(33,134,71,106),(35,136,73,108),(37,138,75,110),(39,140,77,102),(41,122,85,94),(43,124,87,96),(45,126,89,98),(47,128,81,100),(49,130,83,92),(101,149,139,111),(103,141,131,113),(105,143,133,115),(107,145,135,117),(109,147,137,119)], [(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)], [(1,10,123,122),(2,121,124,9),(3,8,125,130),(4,129,126,7),(5,6,127,128),(11,153,85,42),(12,41,86,152),(13,151,87,50),(14,49,88,160),(15,159,89,48),(16,47,90,158),(17,157,81,46),(18,45,82,156),(19,155,83,44),(20,43,84,154),(21,70,132,105),(22,104,133,69),(23,68,134,103),(24,102,135,67),(25,66,136,101),(26,110,137,65),(27,64,138,109),(28,108,139,63),(29,62,140,107),(30,106,131,61),(31,143,142,32),(33,141,144,40),(34,39,145,150),(35,149,146,38),(36,37,147,148),(51,93,96,58),(52,57,97,92),(53,91,98,56),(54,55,99,100),(59,95,94,60),(71,113,116,78),(72,77,117,112),(73,111,118,76),(74,75,119,120),(79,115,114,80)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4N | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | 2+ 1+4 | 2- 1+4 | D4⋊6D10 | D4.10D10 |
| kernel | C42.118D10 | C20.6Q8 | C42⋊2D5 | Dic5.14D4 | C23.D10 | D10.12D4 | C22.D20 | Dic5.Q8 | D10⋊Q8 | C2×C10.D4 | C20.48D4 | C4×C5⋊D4 | C23.23D10 | C23.18D10 | Dic5⋊D4 | D4×C20 | C4×D4 | C2×C10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C22 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 1 | 4 | 4 |
Matrix representation of C42.118D10 ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 16 | 18 | 33 |
| 0 | 0 | 17 | 34 | 0 | 33 |
| 0 | 0 | 0 | 25 | 18 | 25 |
| 0 | 0 | 24 | 30 | 28 | 31 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 37 | 13 |
| 0 | 0 | 28 | 30 | 15 | 28 |
| 0 | 0 | 15 | 15 | 11 | 0 |
| 0 | 0 | 20 | 33 | 4 | 39 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 26 | 28 | 4 |
| 0 | 0 | 15 | 15 | 37 | 0 |
| 0 | 0 | 26 | 0 | 8 | 15 |
| 0 | 0 | 30 | 26 | 16 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 16 | 13 | 17 |
| 0 | 0 | 19 | 34 | 4 | 4 |
| 0 | 0 | 18 | 25 | 33 | 7 |
| 0 | 0 | 14 | 30 | 25 | 32 |
G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,17,0,24,0,0,16,34,25,30,0,0,18,0,18,28,0,0,33,33,25,31],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,2,28,15,20,0,0,0,30,15,33,0,0,37,15,11,4,0,0,13,28,0,39],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,15,26,30,0,0,26,15,0,26,0,0,28,37,8,16,0,0,4,0,15,7],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,24,19,18,14,0,0,16,34,25,30,0,0,13,4,33,25,0,0,17,4,7,32] >;
C42.118D10 in GAP, Magma, Sage, TeX
C_4^2._{118}D_{10} % in TeX
G:=Group("C4^2.118D10"); // GroupNames label
G:=SmallGroup(320,1236);
// by ID
G=gap.SmallGroup(320,1236);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,100,675,570,297,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*c^-1>;
// generators/relations