metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.43D10, C20⋊3C8⋊6C2, C4.82(C2×D20), C4⋊C4.5Dic5, C20.49(C4⋊C4), (C2×C20).23Q8, C20.84(C2×Q8), (C2×C20).143D4, C20.302(C2×D4), (C2×C4).145D20, C4.5(C4⋊Dic5), C10.60(C8○D4), (C4×C20).18C22, C4.49(C2×Dic10), (C2×C4).33Dic10, C42⋊C2.7D5, C22⋊C4.2Dic5, (C2×C20).846C23, (C22×C4).338D10, C2.4(D4.Dic5), C22.5(C4⋊Dic5), C23.15(C2×Dic5), C5⋊4(C42.6C22), (C22×C20).149C22, C22.43(C22×Dic5), (C5×C4⋊C4).21C4, C10.64(C2×C4⋊C4), C2.9(C2×C4⋊Dic5), (C2×C10).41(C4⋊C4), (C2×C20).263(C2×C4), (C5×C22⋊C4).11C4, (C22×C5⋊2C8).8C2, (C2×C4).43(C2×Dic5), (C5×C42⋊C2).8C2, (C2×C4).788(C22×D5), (C2×C4.Dic5).18C2, (C2×C10).284(C22×C4), (C22×C10).125(C2×C4), (C2×C5⋊2C8).321C22, SmallGroup(320,626)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.43D10
G = < a,b,c,d | a4=b4=c10=1, d2=b, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 254 in 114 conjugacy classes, 71 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C20, C20, C20, C2×C10, C2×C10, C2×C10, C4⋊C8, C42⋊C2, C22×C8, C2×M4(2), C5⋊2C8, C2×C20, C2×C20, C22×C10, C42.6C22, C2×C5⋊2C8, C2×C5⋊2C8, C4.Dic5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C20⋊3C8, C22×C5⋊2C8, C2×C4.Dic5, C5×C42⋊C2, C42.43D10
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, D10, C2×C4⋊C4, C8○D4, Dic10, D20, C2×Dic5, C22×D5, C42.6C22, C4⋊Dic5, C2×Dic10, C2×D20, C22×Dic5, C2×C4⋊Dic5, D4.Dic5, C42.43D10
►On 160 points
Generators in S160
(1 113 50 122)(2 119 46 128)(3 115 47 124)(4 111 48 130)(5 117 49 126)(6 92 11 87)(7 98 12 83)(8 94 13 89)(9 100 14 85)(10 96 15 81)(16 116 45 125)(17 112 41 121)(18 118 42 127)(19 114 43 123)(20 120 44 129)(21 153 36 104)(22 159 37 110)(23 155 38 106)(24 151 39 102)(25 157 40 108)(26 158 31 109)(27 154 32 105)(28 160 33 101)(29 156 34 107)(30 152 35 103)(51 131 69 146)(52 137 70 142)(53 133 66 148)(54 139 67 144)(55 135 68 150)(56 138 63 143)(57 134 64 149)(58 140 65 145)(59 136 61 141)(60 132 62 147)(71 93 78 88)(72 99 79 84)(73 95 80 90)(74 91 76 86)(75 97 77 82)
(1 60 18 52)(2 56 19 53)(3 57 20 54)(4 58 16 55)(5 59 17 51)(6 24 75 29)(7 25 71 30)(8 21 72 26)(9 22 73 27)(10 23 74 28)(11 39 77 34)(12 40 78 35)(13 36 79 31)(14 37 80 32)(15 38 76 33)(41 69 49 61)(42 70 50 62)(43 66 46 63)(44 67 47 64)(45 68 48 65)(81 106 86 101)(82 107 87 102)(83 108 88 103)(84 109 89 104)(85 110 90 105)(91 160 96 155)(92 151 97 156)(93 152 98 157)(94 153 99 158)(95 154 100 159)(111 140 116 135)(112 131 117 136)(113 132 118 137)(114 133 119 138)(115 134 120 139)(121 146 126 141)(122 147 127 142)(123 148 128 143)(124 149 129 144)(125 150 130 145)
(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 71 60 30 18 7 52 25)(2 75 56 29 19 6 53 24)(3 74 57 28 20 10 54 23)(4 73 58 27 16 9 55 22)(5 72 59 26 17 8 51 21)(11 66 39 46 77 63 34 43)(12 70 40 50 78 62 35 42)(13 69 36 49 79 61 31 41)(14 68 37 48 80 65 32 45)(15 67 38 47 76 64 33 44)(81 139 106 115 86 134 101 120)(82 138 107 114 87 133 102 119)(83 137 108 113 88 132 103 118)(84 136 109 112 89 131 104 117)(85 135 110 111 90 140 105 116)(91 149 160 129 96 144 155 124)(92 148 151 128 97 143 156 123)(93 147 152 127 98 142 157 122)(94 146 153 126 99 141 158 121)(95 145 154 125 100 150 159 130)
G:=sub<Sym(160)| (1,113,50,122)(2,119,46,128)(3,115,47,124)(4,111,48,130)(5,117,49,126)(6,92,11,87)(7,98,12,83)(8,94,13,89)(9,100,14,85)(10,96,15,81)(16,116,45,125)(17,112,41,121)(18,118,42,127)(19,114,43,123)(20,120,44,129)(21,153,36,104)(22,159,37,110)(23,155,38,106)(24,151,39,102)(25,157,40,108)(26,158,31,109)(27,154,32,105)(28,160,33,101)(29,156,34,107)(30,152,35,103)(51,131,69,146)(52,137,70,142)(53,133,66,148)(54,139,67,144)(55,135,68,150)(56,138,63,143)(57,134,64,149)(58,140,65,145)(59,136,61,141)(60,132,62,147)(71,93,78,88)(72,99,79,84)(73,95,80,90)(74,91,76,86)(75,97,77,82), (1,60,18,52)(2,56,19,53)(3,57,20,54)(4,58,16,55)(5,59,17,51)(6,24,75,29)(7,25,71,30)(8,21,72,26)(9,22,73,27)(10,23,74,28)(11,39,77,34)(12,40,78,35)(13,36,79,31)(14,37,80,32)(15,38,76,33)(41,69,49,61)(42,70,50,62)(43,66,46,63)(44,67,47,64)(45,68,48,65)(81,106,86,101)(82,107,87,102)(83,108,88,103)(84,109,89,104)(85,110,90,105)(91,160,96,155)(92,151,97,156)(93,152,98,157)(94,153,99,158)(95,154,100,159)(111,140,116,135)(112,131,117,136)(113,132,118,137)(114,133,119,138)(115,134,120,139)(121,146,126,141)(122,147,127,142)(123,148,128,143)(124,149,129,144)(125,150,130,145), (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), (1,71,60,30,18,7,52,25)(2,75,56,29,19,6,53,24)(3,74,57,28,20,10,54,23)(4,73,58,27,16,9,55,22)(5,72,59,26,17,8,51,21)(11,66,39,46,77,63,34,43)(12,70,40,50,78,62,35,42)(13,69,36,49,79,61,31,41)(14,68,37,48,80,65,32,45)(15,67,38,47,76,64,33,44)(81,139,106,115,86,134,101,120)(82,138,107,114,87,133,102,119)(83,137,108,113,88,132,103,118)(84,136,109,112,89,131,104,117)(85,135,110,111,90,140,105,116)(91,149,160,129,96,144,155,124)(92,148,151,128,97,143,156,123)(93,147,152,127,98,142,157,122)(94,146,153,126,99,141,158,121)(95,145,154,125,100,150,159,130)>;
G:=Group( (1,113,50,122)(2,119,46,128)(3,115,47,124)(4,111,48,130)(5,117,49,126)(6,92,11,87)(7,98,12,83)(8,94,13,89)(9,100,14,85)(10,96,15,81)(16,116,45,125)(17,112,41,121)(18,118,42,127)(19,114,43,123)(20,120,44,129)(21,153,36,104)(22,159,37,110)(23,155,38,106)(24,151,39,102)(25,157,40,108)(26,158,31,109)(27,154,32,105)(28,160,33,101)(29,156,34,107)(30,152,35,103)(51,131,69,146)(52,137,70,142)(53,133,66,148)(54,139,67,144)(55,135,68,150)(56,138,63,143)(57,134,64,149)(58,140,65,145)(59,136,61,141)(60,132,62,147)(71,93,78,88)(72,99,79,84)(73,95,80,90)(74,91,76,86)(75,97,77,82), (1,60,18,52)(2,56,19,53)(3,57,20,54)(4,58,16,55)(5,59,17,51)(6,24,75,29)(7,25,71,30)(8,21,72,26)(9,22,73,27)(10,23,74,28)(11,39,77,34)(12,40,78,35)(13,36,79,31)(14,37,80,32)(15,38,76,33)(41,69,49,61)(42,70,50,62)(43,66,46,63)(44,67,47,64)(45,68,48,65)(81,106,86,101)(82,107,87,102)(83,108,88,103)(84,109,89,104)(85,110,90,105)(91,160,96,155)(92,151,97,156)(93,152,98,157)(94,153,99,158)(95,154,100,159)(111,140,116,135)(112,131,117,136)(113,132,118,137)(114,133,119,138)(115,134,120,139)(121,146,126,141)(122,147,127,142)(123,148,128,143)(124,149,129,144)(125,150,130,145), (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), (1,71,60,30,18,7,52,25)(2,75,56,29,19,6,53,24)(3,74,57,28,20,10,54,23)(4,73,58,27,16,9,55,22)(5,72,59,26,17,8,51,21)(11,66,39,46,77,63,34,43)(12,70,40,50,78,62,35,42)(13,69,36,49,79,61,31,41)(14,68,37,48,80,65,32,45)(15,67,38,47,76,64,33,44)(81,139,106,115,86,134,101,120)(82,138,107,114,87,133,102,119)(83,137,108,113,88,132,103,118)(84,136,109,112,89,131,104,117)(85,135,110,111,90,140,105,116)(91,149,160,129,96,144,155,124)(92,148,151,128,97,143,156,123)(93,147,152,127,98,142,157,122)(94,146,153,126,99,141,158,121)(95,145,154,125,100,150,159,130) );
G=PermutationGroup([[(1,113,50,122),(2,119,46,128),(3,115,47,124),(4,111,48,130),(5,117,49,126),(6,92,11,87),(7,98,12,83),(8,94,13,89),(9,100,14,85),(10,96,15,81),(16,116,45,125),(17,112,41,121),(18,118,42,127),(19,114,43,123),(20,120,44,129),(21,153,36,104),(22,159,37,110),(23,155,38,106),(24,151,39,102),(25,157,40,108),(26,158,31,109),(27,154,32,105),(28,160,33,101),(29,156,34,107),(30,152,35,103),(51,131,69,146),(52,137,70,142),(53,133,66,148),(54,139,67,144),(55,135,68,150),(56,138,63,143),(57,134,64,149),(58,140,65,145),(59,136,61,141),(60,132,62,147),(71,93,78,88),(72,99,79,84),(73,95,80,90),(74,91,76,86),(75,97,77,82)], [(1,60,18,52),(2,56,19,53),(3,57,20,54),(4,58,16,55),(5,59,17,51),(6,24,75,29),(7,25,71,30),(8,21,72,26),(9,22,73,27),(10,23,74,28),(11,39,77,34),(12,40,78,35),(13,36,79,31),(14,37,80,32),(15,38,76,33),(41,69,49,61),(42,70,50,62),(43,66,46,63),(44,67,47,64),(45,68,48,65),(81,106,86,101),(82,107,87,102),(83,108,88,103),(84,109,89,104),(85,110,90,105),(91,160,96,155),(92,151,97,156),(93,152,98,157),(94,153,99,158),(95,154,100,159),(111,140,116,135),(112,131,117,136),(113,132,118,137),(114,133,119,138),(115,134,120,139),(121,146,126,141),(122,147,127,142),(123,148,128,143),(124,149,129,144),(125,150,130,145)], [(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)], [(1,71,60,30,18,7,52,25),(2,75,56,29,19,6,53,24),(3,74,57,28,20,10,54,23),(4,73,58,27,16,9,55,22),(5,72,59,26,17,8,51,21),(11,66,39,46,77,63,34,43),(12,70,40,50,78,62,35,42),(13,69,36,49,79,61,31,41),(14,68,37,48,80,65,32,45),(15,67,38,47,76,64,33,44),(81,139,106,115,86,134,101,120),(82,138,107,114,87,133,102,119),(83,137,108,113,88,132,103,118),(84,136,109,112,89,131,104,117),(85,135,110,111,90,140,105,116),(91,149,160,129,96,144,155,124),(92,148,151,128,97,143,156,123),(93,147,152,127,98,142,157,122),(94,146,153,126,99,141,158,121),(95,145,154,125,100,150,159,130)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | - | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | D5 | D10 | Dic5 | Dic5 | D10 | C8○D4 | Dic10 | D20 | D4.Dic5 |
| kernel | C42.43D10 | C20⋊3C8 | C22×C5⋊2C8 | C2×C4.Dic5 | C5×C42⋊C2 | C5×C22⋊C4 | C5×C4⋊C4 | C2×C20 | C2×C20 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C10 | C2×C4 | C2×C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 8 | 8 | 8 | 8 |
Matrix representation of C42.43D10 ►in GL4(𝔽41) generated by
| 1 | 23 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 2 | 28 |
| 0 | 0 | 13 | 39 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 32 | 40 | 0 | 0 |
| 0 | 0 | 40 | 35 |
| 0 | 0 | 6 | 35 |
| 14 | 0 | 0 | 0 |
| 0 | 14 | 0 | 0 |
| 0 | 0 | 21 | 21 |
| 0 | 0 | 18 | 20 |
G:=sub<GL(4,GF(41))| [1,0,0,0,23,40,0,0,0,0,2,13,0,0,28,39],[32,0,0,0,0,32,0,0,0,0,40,0,0,0,0,40],[1,32,0,0,0,40,0,0,0,0,40,6,0,0,35,35],[14,0,0,0,0,14,0,0,0,0,21,18,0,0,21,20] >;
C42.43D10 in GAP, Magma, Sage, TeX
C_4^2._{43}D_{10} % in TeX
G:=Group("C4^2.43D10"); // GroupNames label
G:=SmallGroup(320,626);
// by ID
G=gap.SmallGroup(320,626);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,100,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=b,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations