metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.93D10, (C4×D20)⋊9C2, C4⋊C4.270D10, (C4×Dic10)⋊9C2, (D5×C42)⋊18C2, D10⋊Q8⋊50C2, D10.7(C4○D4), D10⋊D4.6C2, C42⋊C2⋊12D5, C42⋊D5⋊30C2, (C2×C10).72C24, C22⋊C4.96D10, C20.255(C4○D4), C4.139(C4○D20), (C2×C20).147C23, (C4×C20).233C22, Dic5.8(C4○D4), (C22×C4).193D10, D10.13D4⋊48C2, D10.12D4⋊52C2, C23.84(C22×D5), Dic5.Q8⋊44C2, Dic5.5D4⋊48C2, (C2×D20).215C22, C23.D10⋊48C2, C4⋊Dic5.292C22, (C2×Dic5).25C23, C22.101(C23×D5), C23.D5.95C22, (C22×C20).377C22, (C22×C10).142C23, C5⋊2(C23.36C23), (C4×Dic5).216C22, (C22×D5).175C23, D10⋊C4.143C22, (C2×Dic10).238C22, C10.D4.152C22, (C4×C5⋊D4)⋊52C2, C2.11(D5×C4○D4), C4⋊C4⋊D5⋊49C2, C10.29(C2×C4○D4), C2.31(C2×C4○D20), (C2×C4×D5).373C22, (C5×C42⋊C2)⋊14C2, (C5×C4⋊C4).308C22, (C2×C4).150(C22×D5), (C2×C5⋊D4).110C22, (C5×C22⋊C4).112C22, SmallGroup(320,1200)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.93D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, ac=ca, ad=da, cbc-1=a2b, dbd-1=b-1, dcd-1=c9 >
Subgroups: 782 in 234 conjugacy classes, 99 normal (91 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, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C23.36C23, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C2×C5⋊D4, C22×C20, C4×Dic10, D5×C42, C42⋊D5, C4×D20, C23.D10, D10.12D4, D10⋊D4, Dic5.5D4, Dic5.Q8, D10.13D4, D10⋊Q8, C4⋊C4⋊D5, C4×C5⋊D4, C5×C42⋊C2, C42.93D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, C22×D5, C23.36C23, C4○D20, C23×D5, C2×C4○D20, D5×C4○D4, C42.93D10
►On 160 points
Generators in S160
(1 87 139 145)(2 88 140 146)(3 89 121 147)(4 90 122 148)(5 91 123 149)(6 92 124 150)(7 93 125 151)(8 94 126 152)(9 95 127 153)(10 96 128 154)(11 97 129 155)(12 98 130 156)(13 99 131 157)(14 100 132 158)(15 81 133 159)(16 82 134 160)(17 83 135 141)(18 84 136 142)(19 85 137 143)(20 86 138 144)(21 110 74 57)(22 111 75 58)(23 112 76 59)(24 113 77 60)(25 114 78 41)(26 115 79 42)(27 116 80 43)(28 117 61 44)(29 118 62 45)(30 119 63 46)(31 120 64 47)(32 101 65 48)(33 102 66 49)(34 103 67 50)(35 104 68 51)(36 105 69 52)(37 106 70 53)(38 107 71 54)(39 108 72 55)(40 109 73 56)
(1 114 129 51)(2 42 130 105)(3 116 131 53)(4 44 132 107)(5 118 133 55)(6 46 134 109)(7 120 135 57)(8 48 136 111)(9 102 137 59)(10 50 138 113)(11 104 139 41)(12 52 140 115)(13 106 121 43)(14 54 122 117)(15 108 123 45)(16 56 124 119)(17 110 125 47)(18 58 126 101)(19 112 127 49)(20 60 128 103)(21 93 64 141)(22 152 65 84)(23 95 66 143)(24 154 67 86)(25 97 68 145)(26 156 69 88)(27 99 70 147)(28 158 71 90)(29 81 72 149)(30 160 73 92)(31 83 74 151)(32 142 75 94)(33 85 76 153)(34 144 77 96)(35 87 78 155)(36 146 79 98)(37 89 80 157)(38 148 61 100)(39 91 62 159)(40 150 63 82)
(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 77 11 67)(2 66 12 76)(3 75 13 65)(4 64 14 74)(5 73 15 63)(6 62 16 72)(7 71 17 61)(8 80 18 70)(9 69 19 79)(10 78 20 68)(21 122 31 132)(22 131 32 121)(23 140 33 130)(24 129 34 139)(25 138 35 128)(26 127 36 137)(27 136 37 126)(28 125 38 135)(29 134 39 124)(30 123 40 133)(41 86 51 96)(42 95 52 85)(43 84 53 94)(44 93 54 83)(45 82 55 92)(46 91 56 81)(47 100 57 90)(48 89 58 99)(49 98 59 88)(50 87 60 97)(101 147 111 157)(102 156 112 146)(103 145 113 155)(104 154 114 144)(105 143 115 153)(106 152 116 142)(107 141 117 151)(108 150 118 160)(109 159 119 149)(110 148 120 158)
G:=sub<Sym(160)| (1,87,139,145)(2,88,140,146)(3,89,121,147)(4,90,122,148)(5,91,123,149)(6,92,124,150)(7,93,125,151)(8,94,126,152)(9,95,127,153)(10,96,128,154)(11,97,129,155)(12,98,130,156)(13,99,131,157)(14,100,132,158)(15,81,133,159)(16,82,134,160)(17,83,135,141)(18,84,136,142)(19,85,137,143)(20,86,138,144)(21,110,74,57)(22,111,75,58)(23,112,76,59)(24,113,77,60)(25,114,78,41)(26,115,79,42)(27,116,80,43)(28,117,61,44)(29,118,62,45)(30,119,63,46)(31,120,64,47)(32,101,65,48)(33,102,66,49)(34,103,67,50)(35,104,68,51)(36,105,69,52)(37,106,70,53)(38,107,71,54)(39,108,72,55)(40,109,73,56), (1,114,129,51)(2,42,130,105)(3,116,131,53)(4,44,132,107)(5,118,133,55)(6,46,134,109)(7,120,135,57)(8,48,136,111)(9,102,137,59)(10,50,138,113)(11,104,139,41)(12,52,140,115)(13,106,121,43)(14,54,122,117)(15,108,123,45)(16,56,124,119)(17,110,125,47)(18,58,126,101)(19,112,127,49)(20,60,128,103)(21,93,64,141)(22,152,65,84)(23,95,66,143)(24,154,67,86)(25,97,68,145)(26,156,69,88)(27,99,70,147)(28,158,71,90)(29,81,72,149)(30,160,73,92)(31,83,74,151)(32,142,75,94)(33,85,76,153)(34,144,77,96)(35,87,78,155)(36,146,79,98)(37,89,80,157)(38,148,61,100)(39,91,62,159)(40,150,63,82), (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), (1,77,11,67)(2,66,12,76)(3,75,13,65)(4,64,14,74)(5,73,15,63)(6,62,16,72)(7,71,17,61)(8,80,18,70)(9,69,19,79)(10,78,20,68)(21,122,31,132)(22,131,32,121)(23,140,33,130)(24,129,34,139)(25,138,35,128)(26,127,36,137)(27,136,37,126)(28,125,38,135)(29,134,39,124)(30,123,40,133)(41,86,51,96)(42,95,52,85)(43,84,53,94)(44,93,54,83)(45,82,55,92)(46,91,56,81)(47,100,57,90)(48,89,58,99)(49,98,59,88)(50,87,60,97)(101,147,111,157)(102,156,112,146)(103,145,113,155)(104,154,114,144)(105,143,115,153)(106,152,116,142)(107,141,117,151)(108,150,118,160)(109,159,119,149)(110,148,120,158)>;
G:=Group( (1,87,139,145)(2,88,140,146)(3,89,121,147)(4,90,122,148)(5,91,123,149)(6,92,124,150)(7,93,125,151)(8,94,126,152)(9,95,127,153)(10,96,128,154)(11,97,129,155)(12,98,130,156)(13,99,131,157)(14,100,132,158)(15,81,133,159)(16,82,134,160)(17,83,135,141)(18,84,136,142)(19,85,137,143)(20,86,138,144)(21,110,74,57)(22,111,75,58)(23,112,76,59)(24,113,77,60)(25,114,78,41)(26,115,79,42)(27,116,80,43)(28,117,61,44)(29,118,62,45)(30,119,63,46)(31,120,64,47)(32,101,65,48)(33,102,66,49)(34,103,67,50)(35,104,68,51)(36,105,69,52)(37,106,70,53)(38,107,71,54)(39,108,72,55)(40,109,73,56), (1,114,129,51)(2,42,130,105)(3,116,131,53)(4,44,132,107)(5,118,133,55)(6,46,134,109)(7,120,135,57)(8,48,136,111)(9,102,137,59)(10,50,138,113)(11,104,139,41)(12,52,140,115)(13,106,121,43)(14,54,122,117)(15,108,123,45)(16,56,124,119)(17,110,125,47)(18,58,126,101)(19,112,127,49)(20,60,128,103)(21,93,64,141)(22,152,65,84)(23,95,66,143)(24,154,67,86)(25,97,68,145)(26,156,69,88)(27,99,70,147)(28,158,71,90)(29,81,72,149)(30,160,73,92)(31,83,74,151)(32,142,75,94)(33,85,76,153)(34,144,77,96)(35,87,78,155)(36,146,79,98)(37,89,80,157)(38,148,61,100)(39,91,62,159)(40,150,63,82), (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), (1,77,11,67)(2,66,12,76)(3,75,13,65)(4,64,14,74)(5,73,15,63)(6,62,16,72)(7,71,17,61)(8,80,18,70)(9,69,19,79)(10,78,20,68)(21,122,31,132)(22,131,32,121)(23,140,33,130)(24,129,34,139)(25,138,35,128)(26,127,36,137)(27,136,37,126)(28,125,38,135)(29,134,39,124)(30,123,40,133)(41,86,51,96)(42,95,52,85)(43,84,53,94)(44,93,54,83)(45,82,55,92)(46,91,56,81)(47,100,57,90)(48,89,58,99)(49,98,59,88)(50,87,60,97)(101,147,111,157)(102,156,112,146)(103,145,113,155)(104,154,114,144)(105,143,115,153)(106,152,116,142)(107,141,117,151)(108,150,118,160)(109,159,119,149)(110,148,120,158) );
G=PermutationGroup([[(1,87,139,145),(2,88,140,146),(3,89,121,147),(4,90,122,148),(5,91,123,149),(6,92,124,150),(7,93,125,151),(8,94,126,152),(9,95,127,153),(10,96,128,154),(11,97,129,155),(12,98,130,156),(13,99,131,157),(14,100,132,158),(15,81,133,159),(16,82,134,160),(17,83,135,141),(18,84,136,142),(19,85,137,143),(20,86,138,144),(21,110,74,57),(22,111,75,58),(23,112,76,59),(24,113,77,60),(25,114,78,41),(26,115,79,42),(27,116,80,43),(28,117,61,44),(29,118,62,45),(30,119,63,46),(31,120,64,47),(32,101,65,48),(33,102,66,49),(34,103,67,50),(35,104,68,51),(36,105,69,52),(37,106,70,53),(38,107,71,54),(39,108,72,55),(40,109,73,56)], [(1,114,129,51),(2,42,130,105),(3,116,131,53),(4,44,132,107),(5,118,133,55),(6,46,134,109),(7,120,135,57),(8,48,136,111),(9,102,137,59),(10,50,138,113),(11,104,139,41),(12,52,140,115),(13,106,121,43),(14,54,122,117),(15,108,123,45),(16,56,124,119),(17,110,125,47),(18,58,126,101),(19,112,127,49),(20,60,128,103),(21,93,64,141),(22,152,65,84),(23,95,66,143),(24,154,67,86),(25,97,68,145),(26,156,69,88),(27,99,70,147),(28,158,71,90),(29,81,72,149),(30,160,73,92),(31,83,74,151),(32,142,75,94),(33,85,76,153),(34,144,77,96),(35,87,78,155),(36,146,79,98),(37,89,80,157),(38,148,61,100),(39,91,62,159),(40,150,63,82)], [(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)], [(1,77,11,67),(2,66,12,76),(3,75,13,65),(4,64,14,74),(5,73,15,63),(6,62,16,72),(7,71,17,61),(8,80,18,70),(9,69,19,79),(10,78,20,68),(21,122,31,132),(22,131,32,121),(23,140,33,130),(24,129,34,139),(25,138,35,128),(26,127,36,137),(27,136,37,126),(28,125,38,135),(29,134,39,124),(30,123,40,133),(41,86,51,96),(42,95,52,85),(43,84,53,94),(44,93,54,83),(45,82,55,92),(46,91,56,81),(47,100,57,90),(48,89,58,99),(49,98,59,88),(50,87,60,97),(101,147,111,157),(102,156,112,146),(103,145,113,155),(104,154,114,144),(105,143,115,153),(106,152,116,142),(107,141,117,151),(108,150,118,160),(109,159,119,149),(110,148,120,158)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 4R | 4S | 4T | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 10 | 10 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | C4○D20 | D5×C4○D4 |
| kernel | C42.93D10 | C4×Dic10 | D5×C42 | C42⋊D5 | C4×D20 | C23.D10 | D10.12D4 | D10⋊D4 | Dic5.5D4 | Dic5.Q8 | D10.13D4 | D10⋊Q8 | C4⋊C4⋊D5 | C4×C5⋊D4 | C5×C42⋊C2 | C42⋊C2 | Dic5 | C20 | D10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 16 | 8 |
Matrix representation of C42.93D10 ►in GL4(𝔽41) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 32 | 2 | 0 | 0 |
| 1 | 9 | 0 | 0 |
| 0 | 0 | 11 | 32 |
| 0 | 0 | 9 | 30 |
| 9 | 39 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 9 | 19 |
| 0 | 0 | 22 | 19 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 24 | 40 |
| 0 | 0 | 3 | 17 |
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,40,0,0,0,0,40],[32,1,0,0,2,9,0,0,0,0,11,9,0,0,32,30],[9,0,0,0,39,32,0,0,0,0,9,22,0,0,19,19],[9,0,0,0,0,9,0,0,0,0,24,3,0,0,40,17] >;
C42.93D10 in GAP, Magma, Sage, TeX
C_4^2._{93}D_{10} % in TeX
G:=Group("C4^2.93D10"); // GroupNames label
G:=SmallGroup(320,1200);
// by ID
G=gap.SmallGroup(320,1200);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,675,297,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2*b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations