metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.137D10, C10.872- 1+4, C10.702+ 1+4, C4.4D4⋊6D5, C42⋊2D5⋊7C2, (C2×Q8).81D10, D10⋊3Q8⋊27C2, (C4×Dic10)⋊43C2, (C2×D4).107D10, C22⋊C4.71D10, Dic5⋊Q8⋊20C2, Dic5⋊4D4⋊28C2, (C2×C10).213C24, (C2×C20).629C23, (C4×C20).183C22, Dic5⋊D4.5C2, D10.12D4⋊40C2, C2.72(D4⋊6D10), C23.35(C22×D5), Dic5.17(C4○D4), Dic5.5D4⋊37C2, (D4×C10).207C22, C23.D10⋊36C2, C4⋊Dic5.232C22, (C22×C10).43C23, (Q8×C10).122C22, (C22×D5).93C23, C22.234(C23×D5), Dic5.14D4⋊37C2, C23.D5.50C22, D10⋊C4.59C22, C23.18D10⋊24C2, C23.11D10⋊16C2, C5⋊8(C22.36C24), (C4×Dic5).233C22, (C2×Dic5).260C23, C10.D4.82C22, C2.48(D4.10D10), (C2×Dic10).180C22, (C22×Dic5).138C22, C2.72(D5×C4○D4), (C5×C4.4D4)⋊7C2, C10.184(C2×C4○D4), (C2×C4×D5).128C22, (C2×C4).191(C22×D5), (C2×C5⋊D4).56C22, (C5×C22⋊C4).60C22, SmallGroup(320,1341)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.137D10
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=c-1 >
Subgroups: 734 in 216 conjugacy classes, 93 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, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42⋊2C2, C4⋊Q8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, C22×C10, C22.36C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C2×Dic10, C2×C4×D5, C22×Dic5, C2×C5⋊D4, D4×C10, Q8×C10, C4×Dic10, C42⋊2D5, C23.11D10, Dic5.14D4, C23.D10, Dic5⋊4D4, D10.12D4, Dic5.5D4, C23.18D10, Dic5⋊D4, Dic5⋊Q8, D10⋊3Q8, C5×C4.4D4, C42.137D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5, C22.36C24, C23×D5, D4⋊6D10, D5×C4○D4, D4.10D10, C42.137D10
►On 160 points
Generators in S160
(1 113 93 40)(2 71 94 144)(3 115 95 32)(4 73 96 146)(5 117 97 34)(6 75 98 148)(7 119 99 36)(8 77 100 150)(9 111 91 38)(10 79 92 142)(11 65 45 138)(12 109 46 26)(13 67 47 140)(14 101 48 28)(15 69 49 132)(16 103 50 30)(17 61 41 134)(18 105 42 22)(19 63 43 136)(20 107 44 24)(21 87 104 154)(23 89 106 156)(25 81 108 158)(27 83 110 160)(29 85 102 152)(31 124 114 51)(33 126 116 53)(35 128 118 55)(37 130 120 57)(39 122 112 59)(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 30)(2 61 124 104)(3 135 125 22)(4 63 126 106)(5 137 127 24)(6 65 128 108)(7 139 129 26)(8 67 130 110)(9 131 121 28)(10 69 122 102)(11 35 81 148)(12 119 82 76)(13 37 83 150)(14 111 84 78)(15 39 85 142)(16 113 86 80)(17 31 87 144)(18 115 88 72)(19 33 89 146)(20 117 90 74)(21 94 134 51)(23 96 136 53)(25 98 138 55)(27 100 140 57)(29 92 132 59)(32 155 145 42)(34 157 147 44)(36 159 149 46)(38 151 141 48)(40 153 143 50)(41 114 154 71)(43 116 156 73)(45 118 158 75)(47 120 160 77)(49 112 152 79)(52 105 95 62)(54 107 97 64)(56 109 99 66)(58 101 91 68)(60 103 93 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 44 123 157)(2 43 124 156)(3 42 125 155)(4 41 126 154)(5 50 127 153)(6 49 128 152)(7 48 129 151)(8 47 130 160)(9 46 121 159)(10 45 122 158)(11 59 81 92)(12 58 82 91)(13 57 83 100)(14 56 84 99)(15 55 85 98)(16 54 86 97)(17 53 87 96)(18 52 88 95)(19 51 89 94)(20 60 90 93)(21 116 134 73)(22 115 135 72)(23 114 136 71)(24 113 137 80)(25 112 138 79)(26 111 139 78)(27 120 140 77)(28 119 131 76)(29 118 132 75)(30 117 133 74)(31 63 144 106)(32 62 145 105)(33 61 146 104)(34 70 147 103)(35 69 148 102)(36 68 149 101)(37 67 150 110)(38 66 141 109)(39 65 142 108)(40 64 143 107)
G:=sub<Sym(160)| (1,113,93,40)(2,71,94,144)(3,115,95,32)(4,73,96,146)(5,117,97,34)(6,75,98,148)(7,119,99,36)(8,77,100,150)(9,111,91,38)(10,79,92,142)(11,65,45,138)(12,109,46,26)(13,67,47,140)(14,101,48,28)(15,69,49,132)(16,103,50,30)(17,61,41,134)(18,105,42,22)(19,63,43,136)(20,107,44,24)(21,87,104,154)(23,89,106,156)(25,81,108,158)(27,83,110,160)(29,85,102,152)(31,124,114,51)(33,126,116,53)(35,128,118,55)(37,130,120,57)(39,122,112,59)(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), (1,133,123,30)(2,61,124,104)(3,135,125,22)(4,63,126,106)(5,137,127,24)(6,65,128,108)(7,139,129,26)(8,67,130,110)(9,131,121,28)(10,69,122,102)(11,35,81,148)(12,119,82,76)(13,37,83,150)(14,111,84,78)(15,39,85,142)(16,113,86,80)(17,31,87,144)(18,115,88,72)(19,33,89,146)(20,117,90,74)(21,94,134,51)(23,96,136,53)(25,98,138,55)(27,100,140,57)(29,92,132,59)(32,155,145,42)(34,157,147,44)(36,159,149,46)(38,151,141,48)(40,153,143,50)(41,114,154,71)(43,116,156,73)(45,118,158,75)(47,120,160,77)(49,112,152,79)(52,105,95,62)(54,107,97,64)(56,109,99,66)(58,101,91,68)(60,103,93,70), (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), (1,44,123,157)(2,43,124,156)(3,42,125,155)(4,41,126,154)(5,50,127,153)(6,49,128,152)(7,48,129,151)(8,47,130,160)(9,46,121,159)(10,45,122,158)(11,59,81,92)(12,58,82,91)(13,57,83,100)(14,56,84,99)(15,55,85,98)(16,54,86,97)(17,53,87,96)(18,52,88,95)(19,51,89,94)(20,60,90,93)(21,116,134,73)(22,115,135,72)(23,114,136,71)(24,113,137,80)(25,112,138,79)(26,111,139,78)(27,120,140,77)(28,119,131,76)(29,118,132,75)(30,117,133,74)(31,63,144,106)(32,62,145,105)(33,61,146,104)(34,70,147,103)(35,69,148,102)(36,68,149,101)(37,67,150,110)(38,66,141,109)(39,65,142,108)(40,64,143,107)>;
G:=Group( (1,113,93,40)(2,71,94,144)(3,115,95,32)(4,73,96,146)(5,117,97,34)(6,75,98,148)(7,119,99,36)(8,77,100,150)(9,111,91,38)(10,79,92,142)(11,65,45,138)(12,109,46,26)(13,67,47,140)(14,101,48,28)(15,69,49,132)(16,103,50,30)(17,61,41,134)(18,105,42,22)(19,63,43,136)(20,107,44,24)(21,87,104,154)(23,89,106,156)(25,81,108,158)(27,83,110,160)(29,85,102,152)(31,124,114,51)(33,126,116,53)(35,128,118,55)(37,130,120,57)(39,122,112,59)(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), (1,133,123,30)(2,61,124,104)(3,135,125,22)(4,63,126,106)(5,137,127,24)(6,65,128,108)(7,139,129,26)(8,67,130,110)(9,131,121,28)(10,69,122,102)(11,35,81,148)(12,119,82,76)(13,37,83,150)(14,111,84,78)(15,39,85,142)(16,113,86,80)(17,31,87,144)(18,115,88,72)(19,33,89,146)(20,117,90,74)(21,94,134,51)(23,96,136,53)(25,98,138,55)(27,100,140,57)(29,92,132,59)(32,155,145,42)(34,157,147,44)(36,159,149,46)(38,151,141,48)(40,153,143,50)(41,114,154,71)(43,116,156,73)(45,118,158,75)(47,120,160,77)(49,112,152,79)(52,105,95,62)(54,107,97,64)(56,109,99,66)(58,101,91,68)(60,103,93,70), (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), (1,44,123,157)(2,43,124,156)(3,42,125,155)(4,41,126,154)(5,50,127,153)(6,49,128,152)(7,48,129,151)(8,47,130,160)(9,46,121,159)(10,45,122,158)(11,59,81,92)(12,58,82,91)(13,57,83,100)(14,56,84,99)(15,55,85,98)(16,54,86,97)(17,53,87,96)(18,52,88,95)(19,51,89,94)(20,60,90,93)(21,116,134,73)(22,115,135,72)(23,114,136,71)(24,113,137,80)(25,112,138,79)(26,111,139,78)(27,120,140,77)(28,119,131,76)(29,118,132,75)(30,117,133,74)(31,63,144,106)(32,62,145,105)(33,61,146,104)(34,70,147,103)(35,69,148,102)(36,68,149,101)(37,67,150,110)(38,66,141,109)(39,65,142,108)(40,64,143,107) );
G=PermutationGroup([[(1,113,93,40),(2,71,94,144),(3,115,95,32),(4,73,96,146),(5,117,97,34),(6,75,98,148),(7,119,99,36),(8,77,100,150),(9,111,91,38),(10,79,92,142),(11,65,45,138),(12,109,46,26),(13,67,47,140),(14,101,48,28),(15,69,49,132),(16,103,50,30),(17,61,41,134),(18,105,42,22),(19,63,43,136),(20,107,44,24),(21,87,104,154),(23,89,106,156),(25,81,108,158),(27,83,110,160),(29,85,102,152),(31,124,114,51),(33,126,116,53),(35,128,118,55),(37,130,120,57),(39,122,112,59),(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)], [(1,133,123,30),(2,61,124,104),(3,135,125,22),(4,63,126,106),(5,137,127,24),(6,65,128,108),(7,139,129,26),(8,67,130,110),(9,131,121,28),(10,69,122,102),(11,35,81,148),(12,119,82,76),(13,37,83,150),(14,111,84,78),(15,39,85,142),(16,113,86,80),(17,31,87,144),(18,115,88,72),(19,33,89,146),(20,117,90,74),(21,94,134,51),(23,96,136,53),(25,98,138,55),(27,100,140,57),(29,92,132,59),(32,155,145,42),(34,157,147,44),(36,159,149,46),(38,151,141,48),(40,153,143,50),(41,114,154,71),(43,116,156,73),(45,118,158,75),(47,120,160,77),(49,112,152,79),(52,105,95,62),(54,107,97,64),(56,109,99,66),(58,101,91,68),(60,103,93,70)], [(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)], [(1,44,123,157),(2,43,124,156),(3,42,125,155),(4,41,126,154),(5,50,127,153),(6,49,128,152),(7,48,129,151),(8,47,130,160),(9,46,121,159),(10,45,122,158),(11,59,81,92),(12,58,82,91),(13,57,83,100),(14,56,84,99),(15,55,85,98),(16,54,86,97),(17,53,87,96),(18,52,88,95),(19,51,89,94),(20,60,90,93),(21,116,134,73),(22,115,135,72),(23,114,136,71),(24,113,137,80),(25,112,138,79),(26,111,139,78),(27,120,140,77),(28,119,131,76),(29,118,132,75),(30,117,133,74),(31,63,144,106),(32,62,145,105),(33,61,146,104),(34,70,147,103),(35,69,148,102),(36,68,149,101),(37,67,150,110),(38,66,141,109),(39,65,142,108),(40,64,143,107)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ 1+4 | 2- 1+4 | D4⋊6D10 | D5×C4○D4 | D4.10D10 |
| kernel | C42.137D10 | C4×Dic10 | C42⋊2D5 | C23.11D10 | Dic5.14D4 | C23.D10 | Dic5⋊4D4 | D10.12D4 | Dic5.5D4 | C23.18D10 | Dic5⋊D4 | Dic5⋊Q8 | D10⋊3Q8 | C5×C4.4D4 | C4.4D4 | Dic5 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C10 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 8 | 2 | 2 | 1 | 1 | 4 | 4 | 4 |
Matrix representation of C42.137D10 ►in GL6(𝔽41)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 35 | 10 | 28 |
| 0 | 0 | 6 | 23 | 10 | 23 |
| 0 | 0 | 37 | 12 | 24 | 6 |
| 0 | 0 | 16 | 25 | 34 | 17 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 38 | 38 |
| 0 | 0 | 0 | 1 | 3 | 0 |
| 0 | 0 | 0 | 13 | 40 | 0 |
| 0 | 0 | 28 | 28 | 0 | 40 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 21 | 0 | 30 |
| 0 | 0 | 20 | 24 | 6 | 11 |
| 0 | 0 | 7 | 33 | 37 | 20 |
| 0 | 0 | 0 | 15 | 4 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 21 | 0 | 0 |
| 0 | 0 | 18 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 23 |
| 0 | 0 | 0 | 0 | 37 | 38 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,18,6,37,16,0,0,35,23,12,25,0,0,10,10,24,34,0,0,28,23,6,17],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,28,0,0,0,1,13,28,0,0,38,3,40,0,0,0,38,0,0,40],[0,9,0,0,0,0,32,0,0,0,0,0,0,0,21,20,7,0,0,0,21,24,33,15,0,0,0,6,37,4,0,0,30,11,20,0],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,20,18,0,0,0,0,21,21,0,0,0,0,0,0,3,37,0,0,0,0,23,38] >;
C42.137D10 in GAP, Magma, Sage, TeX
C_4^2._{137}D_{10} % in TeX
G:=Group("C4^2.137D10"); // GroupNames label
G:=SmallGroup(320,1341);
// by ID
G=gap.SmallGroup(320,1341);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,387,100,1123,346,136,12550]);
// 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=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations