metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.91D10, C10.492- 1+4, C10.942+ 1+4, (C4×D20)⋊6C2, C4○D20⋊17C4, D20⋊37(C2×C4), C4⋊C4.310D10, (C4×Dic10)⋊8C2, C42⋊C2⋊7D5, Dic10⋊34(C2×C4), (C4×C20).23C22, C10.40(C23×C4), (C2×C10).67C24, Dic5⋊4D4⋊42C2, C2.2(D4⋊8D10), C20.149(C22×C4), (C2×C20).489C23, C22⋊C4.127D10, D10.15(C22×C4), (C22×C4).189D10, C22.29(C23×D5), (C2×D20).294C22, C4⋊Dic5.397C22, Dic5.16(C22×C4), C23.155(C22×D5), C2.2(D4.10D10), (C22×C10).137C23, (C22×C20).227C22, C5⋊3(C23.33C23), (C4×Dic5).215C22, (C2×Dic5).206C23, (C22×D5).174C23, D10⋊C4.119C22, (C2×Dic10).323C22, C10.D4.132C22, (C22×Dic5).86C22, (C2×C4)⋊7(C4×D5), C4.94(C2×C4×D5), (D5×C4⋊C4)⋊11C2, (C4×D5)⋊2(C2×C4), (C2×C20)⋊26(C2×C4), C5⋊D4⋊11(C2×C4), C22.7(C2×C4×D5), C4⋊C4⋊7D5⋊11C2, C2.21(D5×C22×C4), (C2×C4⋊Dic5)⋊39C2, (C2×C4×D5).67C22, (C5×C42⋊C2)⋊9C2, (C2×C4○D20).18C2, (C5×C4⋊C4).306C22, (C2×C4).273(C22×D5), (C2×C10).124(C22×C4), (C2×C5⋊D4).106C22, (C5×C22⋊C4).137C22, SmallGroup(320,1195)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.91D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=a2b-1, bd=db, dcd-1=c9 >
Subgroups: 926 in 294 conjugacy classes, 151 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, 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, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C23.33C23, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C22×Dic5, C2×C5⋊D4, C22×C20, C4×Dic10, C4×D20, Dic5⋊4D4, D5×C4⋊C4, C4⋊C4⋊7D5, C2×C4⋊Dic5, C5×C42⋊C2, C2×C4○D20, C42.91D10
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, D10, C23×C4, 2+ 1+4, 2- 1+4, C4×D5, C22×D5, C23.33C23, C2×C4×D5, C23×D5, D5×C22×C4, D4⋊8D10, D4.10D10, C42.91D10
►On 160 points
Generators in S160
(1 102 124 100)(2 113 125 91)(3 104 126 82)(4 115 127 93)(5 106 128 84)(6 117 129 95)(7 108 130 86)(8 119 131 97)(9 110 132 88)(10 101 133 99)(11 112 134 90)(12 103 135 81)(13 114 136 92)(14 105 137 83)(15 116 138 94)(16 107 139 85)(17 118 140 96)(18 109 121 87)(19 120 122 98)(20 111 123 89)(21 51 65 143)(22 42 66 154)(23 53 67 145)(24 44 68 156)(25 55 69 147)(26 46 70 158)(27 57 71 149)(28 48 72 160)(29 59 73 151)(30 50 74 142)(31 41 75 153)(32 52 76 144)(33 43 77 155)(34 54 78 146)(35 45 79 157)(36 56 80 148)(37 47 61 159)(38 58 62 150)(39 49 63 141)(40 60 64 152)
(1 51 134 153)(2 42 135 144)(3 53 136 155)(4 44 137 146)(5 55 138 157)(6 46 139 148)(7 57 140 159)(8 48 121 150)(9 59 122 141)(10 50 123 152)(11 41 124 143)(12 52 125 154)(13 43 126 145)(14 54 127 156)(15 45 128 147)(16 56 129 158)(17 47 130 149)(18 58 131 160)(19 49 132 151)(20 60 133 142)(21 112 75 100)(22 103 76 91)(23 114 77 82)(24 105 78 93)(25 116 79 84)(26 107 80 95)(27 118 61 86)(28 109 62 97)(29 120 63 88)(30 111 64 99)(31 102 65 90)(32 113 66 81)(33 104 67 92)(34 115 68 83)(35 106 69 94)(36 117 70 85)(37 108 71 96)(38 119 72 87)(39 110 73 98)(40 101 74 89)
(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 74 11 64)(2 63 12 73)(3 72 13 62)(4 61 14 71)(5 70 15 80)(6 79 16 69)(7 68 17 78)(8 77 18 67)(9 66 19 76)(10 75 20 65)(21 133 31 123)(22 122 32 132)(23 131 33 121)(24 140 34 130)(25 129 35 139)(26 138 36 128)(27 127 37 137)(28 136 38 126)(29 125 39 135)(30 134 40 124)(41 99 51 89)(42 88 52 98)(43 97 53 87)(44 86 54 96)(45 95 55 85)(46 84 56 94)(47 93 57 83)(48 82 58 92)(49 91 59 81)(50 100 60 90)(101 143 111 153)(102 152 112 142)(103 141 113 151)(104 150 114 160)(105 159 115 149)(106 148 116 158)(107 157 117 147)(108 146 118 156)(109 155 119 145)(110 144 120 154)
G:=sub<Sym(160)| (1,102,124,100)(2,113,125,91)(3,104,126,82)(4,115,127,93)(5,106,128,84)(6,117,129,95)(7,108,130,86)(8,119,131,97)(9,110,132,88)(10,101,133,99)(11,112,134,90)(12,103,135,81)(13,114,136,92)(14,105,137,83)(15,116,138,94)(16,107,139,85)(17,118,140,96)(18,109,121,87)(19,120,122,98)(20,111,123,89)(21,51,65,143)(22,42,66,154)(23,53,67,145)(24,44,68,156)(25,55,69,147)(26,46,70,158)(27,57,71,149)(28,48,72,160)(29,59,73,151)(30,50,74,142)(31,41,75,153)(32,52,76,144)(33,43,77,155)(34,54,78,146)(35,45,79,157)(36,56,80,148)(37,47,61,159)(38,58,62,150)(39,49,63,141)(40,60,64,152), (1,51,134,153)(2,42,135,144)(3,53,136,155)(4,44,137,146)(5,55,138,157)(6,46,139,148)(7,57,140,159)(8,48,121,150)(9,59,122,141)(10,50,123,152)(11,41,124,143)(12,52,125,154)(13,43,126,145)(14,54,127,156)(15,45,128,147)(16,56,129,158)(17,47,130,149)(18,58,131,160)(19,49,132,151)(20,60,133,142)(21,112,75,100)(22,103,76,91)(23,114,77,82)(24,105,78,93)(25,116,79,84)(26,107,80,95)(27,118,61,86)(28,109,62,97)(29,120,63,88)(30,111,64,99)(31,102,65,90)(32,113,66,81)(33,104,67,92)(34,115,68,83)(35,106,69,94)(36,117,70,85)(37,108,71,96)(38,119,72,87)(39,110,73,98)(40,101,74,89), (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), (1,74,11,64)(2,63,12,73)(3,72,13,62)(4,61,14,71)(5,70,15,80)(6,79,16,69)(7,68,17,78)(8,77,18,67)(9,66,19,76)(10,75,20,65)(21,133,31,123)(22,122,32,132)(23,131,33,121)(24,140,34,130)(25,129,35,139)(26,138,36,128)(27,127,37,137)(28,136,38,126)(29,125,39,135)(30,134,40,124)(41,99,51,89)(42,88,52,98)(43,97,53,87)(44,86,54,96)(45,95,55,85)(46,84,56,94)(47,93,57,83)(48,82,58,92)(49,91,59,81)(50,100,60,90)(101,143,111,153)(102,152,112,142)(103,141,113,151)(104,150,114,160)(105,159,115,149)(106,148,116,158)(107,157,117,147)(108,146,118,156)(109,155,119,145)(110,144,120,154)>;
G:=Group( (1,102,124,100)(2,113,125,91)(3,104,126,82)(4,115,127,93)(5,106,128,84)(6,117,129,95)(7,108,130,86)(8,119,131,97)(9,110,132,88)(10,101,133,99)(11,112,134,90)(12,103,135,81)(13,114,136,92)(14,105,137,83)(15,116,138,94)(16,107,139,85)(17,118,140,96)(18,109,121,87)(19,120,122,98)(20,111,123,89)(21,51,65,143)(22,42,66,154)(23,53,67,145)(24,44,68,156)(25,55,69,147)(26,46,70,158)(27,57,71,149)(28,48,72,160)(29,59,73,151)(30,50,74,142)(31,41,75,153)(32,52,76,144)(33,43,77,155)(34,54,78,146)(35,45,79,157)(36,56,80,148)(37,47,61,159)(38,58,62,150)(39,49,63,141)(40,60,64,152), (1,51,134,153)(2,42,135,144)(3,53,136,155)(4,44,137,146)(5,55,138,157)(6,46,139,148)(7,57,140,159)(8,48,121,150)(9,59,122,141)(10,50,123,152)(11,41,124,143)(12,52,125,154)(13,43,126,145)(14,54,127,156)(15,45,128,147)(16,56,129,158)(17,47,130,149)(18,58,131,160)(19,49,132,151)(20,60,133,142)(21,112,75,100)(22,103,76,91)(23,114,77,82)(24,105,78,93)(25,116,79,84)(26,107,80,95)(27,118,61,86)(28,109,62,97)(29,120,63,88)(30,111,64,99)(31,102,65,90)(32,113,66,81)(33,104,67,92)(34,115,68,83)(35,106,69,94)(36,117,70,85)(37,108,71,96)(38,119,72,87)(39,110,73,98)(40,101,74,89), (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), (1,74,11,64)(2,63,12,73)(3,72,13,62)(4,61,14,71)(5,70,15,80)(6,79,16,69)(7,68,17,78)(8,77,18,67)(9,66,19,76)(10,75,20,65)(21,133,31,123)(22,122,32,132)(23,131,33,121)(24,140,34,130)(25,129,35,139)(26,138,36,128)(27,127,37,137)(28,136,38,126)(29,125,39,135)(30,134,40,124)(41,99,51,89)(42,88,52,98)(43,97,53,87)(44,86,54,96)(45,95,55,85)(46,84,56,94)(47,93,57,83)(48,82,58,92)(49,91,59,81)(50,100,60,90)(101,143,111,153)(102,152,112,142)(103,141,113,151)(104,150,114,160)(105,159,115,149)(106,148,116,158)(107,157,117,147)(108,146,118,156)(109,155,119,145)(110,144,120,154) );
G=PermutationGroup([[(1,102,124,100),(2,113,125,91),(3,104,126,82),(4,115,127,93),(5,106,128,84),(6,117,129,95),(7,108,130,86),(8,119,131,97),(9,110,132,88),(10,101,133,99),(11,112,134,90),(12,103,135,81),(13,114,136,92),(14,105,137,83),(15,116,138,94),(16,107,139,85),(17,118,140,96),(18,109,121,87),(19,120,122,98),(20,111,123,89),(21,51,65,143),(22,42,66,154),(23,53,67,145),(24,44,68,156),(25,55,69,147),(26,46,70,158),(27,57,71,149),(28,48,72,160),(29,59,73,151),(30,50,74,142),(31,41,75,153),(32,52,76,144),(33,43,77,155),(34,54,78,146),(35,45,79,157),(36,56,80,148),(37,47,61,159),(38,58,62,150),(39,49,63,141),(40,60,64,152)], [(1,51,134,153),(2,42,135,144),(3,53,136,155),(4,44,137,146),(5,55,138,157),(6,46,139,148),(7,57,140,159),(8,48,121,150),(9,59,122,141),(10,50,123,152),(11,41,124,143),(12,52,125,154),(13,43,126,145),(14,54,127,156),(15,45,128,147),(16,56,129,158),(17,47,130,149),(18,58,131,160),(19,49,132,151),(20,60,133,142),(21,112,75,100),(22,103,76,91),(23,114,77,82),(24,105,78,93),(25,116,79,84),(26,107,80,95),(27,118,61,86),(28,109,62,97),(29,120,63,88),(30,111,64,99),(31,102,65,90),(32,113,66,81),(33,104,67,92),(34,115,68,83),(35,106,69,94),(36,117,70,85),(37,108,71,96),(38,119,72,87),(39,110,73,98),(40,101,74,89)], [(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)], [(1,74,11,64),(2,63,12,73),(3,72,13,62),(4,61,14,71),(5,70,15,80),(6,79,16,69),(7,68,17,78),(8,77,18,67),(9,66,19,76),(10,75,20,65),(21,133,31,123),(22,122,32,132),(23,131,33,121),(24,140,34,130),(25,129,35,139),(26,138,36,128),(27,127,37,137),(28,136,38,126),(29,125,39,135),(30,134,40,124),(41,99,51,89),(42,88,52,98),(43,97,53,87),(44,86,54,96),(45,95,55,85),(46,84,56,94),(47,93,57,83),(48,82,58,92),(49,91,59,81),(50,100,60,90),(101,143,111,153),(102,152,112,142),(103,141,113,151),(104,150,114,160),(105,159,115,149),(106,148,116,158),(107,157,117,147),(108,146,118,156),(109,155,119,145),(110,144,120,154)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4L | 4M | ··· | 4X | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D5 | D10 | D10 | D10 | D10 | C4×D5 | 2+ 1+4 | 2- 1+4 | D4⋊8D10 | D4.10D10 |
| kernel | C42.91D10 | C4×Dic10 | C4×D20 | Dic5⋊4D4 | D5×C4⋊C4 | C4⋊C4⋊7D5 | C2×C4⋊Dic5 | C5×C42⋊C2 | C2×C4○D20 | C4○D20 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 4 | 2 | 2 | 1 | 1 | 1 | 16 | 2 | 4 | 4 | 4 | 2 | 16 | 1 | 1 | 4 | 4 |
Matrix representation of C42.91D10 ►in GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 34 | 34 |
| 0 | 0 | 0 | 4 | 14 | 1 |
| 0 | 0 | 30 | 5 | 37 | 0 |
| 0 | 0 | 31 | 36 | 0 | 37 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 30 | 18 | 23 |
| 0 | 0 | 6 | 3 | 0 | 21 |
| 0 | 0 | 26 | 34 | 33 | 11 |
| 0 | 0 | 0 | 34 | 30 | 8 |
| 7 | 6 | 0 | 0 | 0 | 0 |
| 34 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 11 | 3 | 1 |
| 0 | 0 | 35 | 0 | 37 | 37 |
| 0 | 0 | 17 | 21 | 5 | 30 |
| 0 | 0 | 13 | 34 | 11 | 30 |
| 34 | 40 | 0 | 0 | 0 | 0 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 10 | 22 | 9 |
| 0 | 0 | 25 | 29 | 19 | 10 |
| 0 | 0 | 29 | 19 | 10 | 17 |
| 0 | 0 | 31 | 31 | 39 | 31 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,4,0,30,31,0,0,0,4,5,36,0,0,34,14,37,0,0,0,34,1,0,37],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,38,6,26,0,0,0,30,3,34,34,0,0,18,0,33,30,0,0,23,21,11,8],[7,34,0,0,0,0,6,0,0,0,0,0,0,0,6,35,17,13,0,0,11,0,21,34,0,0,3,37,5,11,0,0,1,37,30,30],[34,7,0,0,0,0,40,7,0,0,0,0,0,0,12,25,29,31,0,0,10,29,19,31,0,0,22,19,10,39,0,0,9,10,17,31] >;
C42.91D10 in GAP, Magma, Sage, TeX
C_4^2._{91}D_{10} % in TeX
G:=Group("C4^2.91D10"); // GroupNames label
G:=SmallGroup(320,1195);
// by ID
G=gap.SmallGroup(320,1195);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,570,297,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=c^9>;
// generators/relations