metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.99D10, C10.542- 1+4, C10.992+ 1+4, (C4×D20)⋊11C2, C20⋊2Q8⋊8C2, C4⋊C4.274D10, D10⋊Q8⋊6C2, C4.D20⋊5C2, (C4×Dic10)⋊12C2, C20⋊7D4.18C2, C42⋊C2⋊18D5, (C2×C10).78C24, (C4×C20).29C22, D10.12D4⋊5C2, C4.120(C4○D20), C20.236(C4○D4), C20.48D4⋊42C2, (C2×C20).151C23, C22⋊C4.102D10, Dic5.5D4⋊5C2, (C22×C4).199D10, C4⋊Dic5.36C22, C2.11(D4⋊8D10), C23.89(C22×D5), C23.D5.6C22, (C2×D20).216C22, D10⋊C4.4C22, (C2×Dic5).31C23, (C22×D5).26C23, C22.107(C23×D5), (C22×C10).148C23, (C22×C20).308C22, C5⋊1(C22.36C24), (C4×Dic5).220C22, C10.D4.75C22, C2.12(D4.10D10), (C2×Dic10).241C22, C4⋊C4⋊D5⋊6C2, C2.37(C2×C4○D20), C10.34(C2×C4○D4), (C2×C4×D5).246C22, (C5×C42⋊C2)⋊20C2, (C5×C4⋊C4).314C22, (C2×C4).151(C22×D5), (C2×C5⋊D4).11C22, (C5×C22⋊C4).117C22, SmallGroup(320,1206)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.99D10
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=b2c-1 >
Subgroups: 782 in 216 conjugacy classes, 95 normal (51 characteristic)
C1, C2, C2, C4, 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, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4⋊Q8, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C22.36C24, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C2×C4×D5, C2×D20, C2×C5⋊D4, C22×C20, C4×Dic10, C20⋊2Q8, C4×D20, C4.D20, D10.12D4, Dic5.5D4, D10⋊Q8, C4⋊C4⋊D5, C20.48D4, C20⋊7D4, C5×C42⋊C2, C42.99D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5, C22.36C24, C4○D20, C23×D5, C2×C4○D20, D4⋊8D10, D4.10D10, C42.99D10
►On 160 points
Generators in S160
(1 113 33 105)(2 114 34 106)(3 115 35 107)(4 116 36 108)(5 117 37 109)(6 118 38 110)(7 119 39 101)(8 120 40 102)(9 111 31 103)(10 112 32 104)(11 125 45 91)(12 126 46 92)(13 127 47 93)(14 128 48 94)(15 129 49 95)(16 130 50 96)(17 121 41 97)(18 122 42 98)(19 123 43 99)(20 124 44 100)(21 85 135 55)(22 86 136 56)(23 87 137 57)(24 88 138 58)(25 89 139 59)(26 90 140 60)(27 81 131 51)(28 82 132 52)(29 83 133 53)(30 84 134 54)(61 156 76 141)(62 157 77 142)(63 158 78 143)(64 159 79 144)(65 160 80 145)(66 151 71 146)(67 152 72 147)(68 153 73 148)(69 154 74 149)(70 155 75 150)
(1 90 11 73)(2 51 12 69)(3 82 13 75)(4 53 14 61)(5 84 15 77)(6 55 16 63)(7 86 17 79)(8 57 18 65)(9 88 19 71)(10 59 20 67)(21 130 158 118)(22 97 159 101)(23 122 160 120)(24 99 151 103)(25 124 152 112)(26 91 153 105)(27 126 154 114)(28 93 155 107)(29 128 156 116)(30 95 157 109)(31 58 43 66)(32 89 44 72)(33 60 45 68)(34 81 46 74)(35 52 47 70)(36 83 48 76)(37 54 49 62)(38 85 50 78)(39 56 41 64)(40 87 42 80)(92 149 106 131)(94 141 108 133)(96 143 110 135)(98 145 102 137)(100 147 104 139)(111 138 123 146)(113 140 125 148)(115 132 127 150)(117 134 129 142)(119 136 121 144)
(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 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 142 158 134)(22 133 159 141)(23 150 160 132)(24 131 151 149)(25 148 152 140)(26 139 153 147)(27 146 154 138)(28 137 155 145)(29 144 156 136)(30 135 157 143)(31 34 43 46)(32 45 44 33)(35 40 47 42)(36 41 48 39)(37 38 49 50)(51 66 69 58)(52 57 70 65)(53 64 61 56)(54 55 62 63)(59 68 67 60)(71 74 88 81)(72 90 89 73)(75 80 82 87)(76 86 83 79)(77 78 84 85)(91 124 105 112)(92 111 106 123)(93 122 107 120)(94 119 108 121)(95 130 109 118)(96 117 110 129)(97 128 101 116)(98 115 102 127)(99 126 103 114)(100 113 104 125)
G:=sub<Sym(160)| (1,113,33,105)(2,114,34,106)(3,115,35,107)(4,116,36,108)(5,117,37,109)(6,118,38,110)(7,119,39,101)(8,120,40,102)(9,111,31,103)(10,112,32,104)(11,125,45,91)(12,126,46,92)(13,127,47,93)(14,128,48,94)(15,129,49,95)(16,130,50,96)(17,121,41,97)(18,122,42,98)(19,123,43,99)(20,124,44,100)(21,85,135,55)(22,86,136,56)(23,87,137,57)(24,88,138,58)(25,89,139,59)(26,90,140,60)(27,81,131,51)(28,82,132,52)(29,83,133,53)(30,84,134,54)(61,156,76,141)(62,157,77,142)(63,158,78,143)(64,159,79,144)(65,160,80,145)(66,151,71,146)(67,152,72,147)(68,153,73,148)(69,154,74,149)(70,155,75,150), (1,90,11,73)(2,51,12,69)(3,82,13,75)(4,53,14,61)(5,84,15,77)(6,55,16,63)(7,86,17,79)(8,57,18,65)(9,88,19,71)(10,59,20,67)(21,130,158,118)(22,97,159,101)(23,122,160,120)(24,99,151,103)(25,124,152,112)(26,91,153,105)(27,126,154,114)(28,93,155,107)(29,128,156,116)(30,95,157,109)(31,58,43,66)(32,89,44,72)(33,60,45,68)(34,81,46,74)(35,52,47,70)(36,83,48,76)(37,54,49,62)(38,85,50,78)(39,56,41,64)(40,87,42,80)(92,149,106,131)(94,141,108,133)(96,143,110,135)(98,145,102,137)(100,147,104,139)(111,138,123,146)(113,140,125,148)(115,132,127,150)(117,134,129,142)(119,136,121,144), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,142,158,134)(22,133,159,141)(23,150,160,132)(24,131,151,149)(25,148,152,140)(26,139,153,147)(27,146,154,138)(28,137,155,145)(29,144,156,136)(30,135,157,143)(31,34,43,46)(32,45,44,33)(35,40,47,42)(36,41,48,39)(37,38,49,50)(51,66,69,58)(52,57,70,65)(53,64,61,56)(54,55,62,63)(59,68,67,60)(71,74,88,81)(72,90,89,73)(75,80,82,87)(76,86,83,79)(77,78,84,85)(91,124,105,112)(92,111,106,123)(93,122,107,120)(94,119,108,121)(95,130,109,118)(96,117,110,129)(97,128,101,116)(98,115,102,127)(99,126,103,114)(100,113,104,125)>;
G:=Group( (1,113,33,105)(2,114,34,106)(3,115,35,107)(4,116,36,108)(5,117,37,109)(6,118,38,110)(7,119,39,101)(8,120,40,102)(9,111,31,103)(10,112,32,104)(11,125,45,91)(12,126,46,92)(13,127,47,93)(14,128,48,94)(15,129,49,95)(16,130,50,96)(17,121,41,97)(18,122,42,98)(19,123,43,99)(20,124,44,100)(21,85,135,55)(22,86,136,56)(23,87,137,57)(24,88,138,58)(25,89,139,59)(26,90,140,60)(27,81,131,51)(28,82,132,52)(29,83,133,53)(30,84,134,54)(61,156,76,141)(62,157,77,142)(63,158,78,143)(64,159,79,144)(65,160,80,145)(66,151,71,146)(67,152,72,147)(68,153,73,148)(69,154,74,149)(70,155,75,150), (1,90,11,73)(2,51,12,69)(3,82,13,75)(4,53,14,61)(5,84,15,77)(6,55,16,63)(7,86,17,79)(8,57,18,65)(9,88,19,71)(10,59,20,67)(21,130,158,118)(22,97,159,101)(23,122,160,120)(24,99,151,103)(25,124,152,112)(26,91,153,105)(27,126,154,114)(28,93,155,107)(29,128,156,116)(30,95,157,109)(31,58,43,66)(32,89,44,72)(33,60,45,68)(34,81,46,74)(35,52,47,70)(36,83,48,76)(37,54,49,62)(38,85,50,78)(39,56,41,64)(40,87,42,80)(92,149,106,131)(94,141,108,133)(96,143,110,135)(98,145,102,137)(100,147,104,139)(111,138,123,146)(113,140,125,148)(115,132,127,150)(117,134,129,142)(119,136,121,144), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,142,158,134)(22,133,159,141)(23,150,160,132)(24,131,151,149)(25,148,152,140)(26,139,153,147)(27,146,154,138)(28,137,155,145)(29,144,156,136)(30,135,157,143)(31,34,43,46)(32,45,44,33)(35,40,47,42)(36,41,48,39)(37,38,49,50)(51,66,69,58)(52,57,70,65)(53,64,61,56)(54,55,62,63)(59,68,67,60)(71,74,88,81)(72,90,89,73)(75,80,82,87)(76,86,83,79)(77,78,84,85)(91,124,105,112)(92,111,106,123)(93,122,107,120)(94,119,108,121)(95,130,109,118)(96,117,110,129)(97,128,101,116)(98,115,102,127)(99,126,103,114)(100,113,104,125) );
G=PermutationGroup([[(1,113,33,105),(2,114,34,106),(3,115,35,107),(4,116,36,108),(5,117,37,109),(6,118,38,110),(7,119,39,101),(8,120,40,102),(9,111,31,103),(10,112,32,104),(11,125,45,91),(12,126,46,92),(13,127,47,93),(14,128,48,94),(15,129,49,95),(16,130,50,96),(17,121,41,97),(18,122,42,98),(19,123,43,99),(20,124,44,100),(21,85,135,55),(22,86,136,56),(23,87,137,57),(24,88,138,58),(25,89,139,59),(26,90,140,60),(27,81,131,51),(28,82,132,52),(29,83,133,53),(30,84,134,54),(61,156,76,141),(62,157,77,142),(63,158,78,143),(64,159,79,144),(65,160,80,145),(66,151,71,146),(67,152,72,147),(68,153,73,148),(69,154,74,149),(70,155,75,150)], [(1,90,11,73),(2,51,12,69),(3,82,13,75),(4,53,14,61),(5,84,15,77),(6,55,16,63),(7,86,17,79),(8,57,18,65),(9,88,19,71),(10,59,20,67),(21,130,158,118),(22,97,159,101),(23,122,160,120),(24,99,151,103),(25,124,152,112),(26,91,153,105),(27,126,154,114),(28,93,155,107),(29,128,156,116),(30,95,157,109),(31,58,43,66),(32,89,44,72),(33,60,45,68),(34,81,46,74),(35,52,47,70),(36,83,48,76),(37,54,49,62),(38,85,50,78),(39,56,41,64),(40,87,42,80),(92,149,106,131),(94,141,108,133),(96,143,110,135),(98,145,102,137),(100,147,104,139),(111,138,123,146),(113,140,125,148),(115,132,127,150),(117,134,129,142),(119,136,121,144)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,142,158,134),(22,133,159,141),(23,150,160,132),(24,131,151,149),(25,148,152,140),(26,139,153,147),(27,146,154,138),(28,137,155,145),(29,144,156,136),(30,135,157,143),(31,34,43,46),(32,45,44,33),(35,40,47,42),(36,41,48,39),(37,38,49,50),(51,66,69,58),(52,57,70,65),(53,64,61,56),(54,55,62,63),(59,68,67,60),(71,74,88,81),(72,90,89,73),(75,80,82,87),(76,86,83,79),(77,78,84,85),(91,124,105,112),(92,111,106,123),(93,122,107,120),(94,119,108,121),(95,130,109,118),(96,117,110,129),(97,128,101,116),(98,115,102,127),(99,126,103,114),(100,113,104,125)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4F | 4G | 4H | 4I | 4J | ··· | 4O | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | D5 | C4○D4 | D10 | D10 | D10 | D10 | C4○D20 | 2+ 1+4 | 2- 1+4 | D4⋊8D10 | D4.10D10 |
| kernel | C42.99D10 | C4×Dic10 | C20⋊2Q8 | C4×D20 | C4.D20 | D10.12D4 | Dic5.5D4 | D10⋊Q8 | C4⋊C4⋊D5 | C20.48D4 | C20⋊7D4 | C5×C42⋊C2 | C42⋊C2 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 2 | 16 | 1 | 1 | 4 | 4 |
Matrix representation of C42.99D10 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 9 | 0 | 0 |
| 0 | 0 | 32 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 9 |
| 0 | 0 | 0 | 0 | 32 | 11 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 0 | 5 | 22 |
| 0 | 0 | 0 | 14 | 19 | 36 |
| 0 | 0 | 36 | 19 | 27 | 0 |
| 0 | 0 | 22 | 5 | 0 | 27 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 7 |
| 0 | 0 | 0 | 0 | 34 | 7 |
| 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 34 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 40 | 0 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,32,0,0,0,0,9,11,0,0,0,0,0,0,30,32,0,0,0,0,9,11],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,14,0,36,22,0,0,0,14,19,5,0,0,5,19,27,0,0,0,22,36,0,27],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,40,34,0,0,0,0,7,7,0,0],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,40,0,0,40,34,0,0,0,0,0,1,0,0] >;
C42.99D10 in GAP, Magma, Sage, TeX
C_4^2._{99}D_{10} % in TeX
G:=Group("C4^2.99D10"); // GroupNames label
G:=SmallGroup(320,1206);
// by ID
G=gap.SmallGroup(320,1206);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,100,675,297,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^-1>;
// generators/relations