metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.145D10, C10.742+ 1+4, C10.932- 1+4, C4.4D4⋊17D5, (C2×Q8).84D10, D10⋊D4⋊43C2, D10⋊3Q8⋊34C2, (C2×D4).113D10, C4.D20⋊31C2, C22⋊C4.38D10, C20.6Q8⋊29C2, Dic5⋊D4⋊35C2, (C2×C20).633C23, (C4×C20).222C22, (C2×C10).228C24, (C2×D20).37C22, C4⋊Dic5.52C22, D10.12D4⋊47C2, C2.54(D4⋊8D10), C2.78(D4⋊6D10), C23.50(C22×D5), (D4×C10).213C22, C22.D20⋊28C2, (C22×C10).58C23, (Q8×C10).131C22, C22.249(C23×D5), Dic5.14D4⋊43C2, C23.D5.60C22, D10⋊C4.73C22, C5⋊4(C22.56C24), (C2×Dic10).41C22, (C2×Dic5).118C23, C10.D4.84C22, (C22×D5).100C23, C2.54(D4.10D10), (C22×Dic5).147C22, (C5×C4.4D4)⋊20C2, (C2×C4×D5).132C22, (C2×C4).201(C22×D5), (C2×C5⋊D4).66C22, (C5×C22⋊C4).69C22, SmallGroup(320,1356)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.145D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c9 >
Subgroups: 854 in 220 conjugacy classes, 91 normal (31 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, Dic5, C20, D10, C2×C10, C2×C10, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×C10, C22.56C24, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C22×Dic5, C2×C5⋊D4, D4×C10, Q8×C10, C20.6Q8, C4.D20, Dic5.14D4, D10.12D4, D10⋊D4, C22.D20, Dic5⋊D4, D10⋊3Q8, C5×C4.4D4, C42.145D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, 2- 1+4, C22×D5, C22.56C24, C23×D5, D4⋊6D10, D4⋊8D10, D4.10D10, C42.145D10
►On 160 points
Generators in S160
(1 117 51 39)(2 30 52 108)(3 119 53 21)(4 32 54 110)(5 101 55 23)(6 34 56 112)(7 103 57 25)(8 36 58 114)(9 105 59 27)(10 38 60 116)(11 107 41 29)(12 40 42 118)(13 109 43 31)(14 22 44 120)(15 111 45 33)(16 24 46 102)(17 113 47 35)(18 26 48 104)(19 115 49 37)(20 28 50 106)(61 146 123 87)(62 98 124 157)(63 148 125 89)(64 100 126 159)(65 150 127 91)(66 82 128 141)(67 152 129 93)(68 84 130 143)(69 154 131 95)(70 86 132 145)(71 156 133 97)(72 88 134 147)(73 158 135 99)(74 90 136 149)(75 160 137 81)(76 92 138 151)(77 142 139 83)(78 94 140 153)(79 144 121 85)(80 96 122 155)
(1 94 11 84)(2 85 12 95)(3 96 13 86)(4 87 14 97)(5 98 15 88)(6 89 16 99)(7 100 17 90)(8 91 18 81)(9 82 19 92)(10 93 20 83)(21 80 31 70)(22 71 32 61)(23 62 33 72)(24 73 34 63)(25 64 35 74)(26 75 36 65)(27 66 37 76)(28 77 38 67)(29 68 39 78)(30 79 40 69)(41 143 51 153)(42 154 52 144)(43 145 53 155)(44 156 54 146)(45 147 55 157)(46 158 56 148)(47 149 57 159)(48 160 58 150)(49 151 59 141)(50 142 60 152)(101 124 111 134)(102 135 112 125)(103 126 113 136)(104 137 114 127)(105 128 115 138)(106 139 116 129)(107 130 117 140)(108 121 118 131)(109 132 119 122)(110 123 120 133)
(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 36 31 26)(22 25 32 35)(23 34 33 24)(27 30 37 40)(28 39 38 29)(41 50 51 60)(42 59 52 49)(43 48 53 58)(44 57 54 47)(45 46 55 56)(61 126 71 136)(62 135 72 125)(63 124 73 134)(64 133 74 123)(65 122 75 132)(66 131 76 121)(67 140 77 130)(68 129 78 139)(69 138 79 128)(70 127 80 137)(81 145 91 155)(82 154 92 144)(83 143 93 153)(84 152 94 142)(85 141 95 151)(86 150 96 160)(87 159 97 149)(88 148 98 158)(89 157 99 147)(90 146 100 156)(101 112 111 102)(103 110 113 120)(104 119 114 109)(105 108 115 118)(106 117 116 107)
G:=sub<Sym(160)| (1,117,51,39)(2,30,52,108)(3,119,53,21)(4,32,54,110)(5,101,55,23)(6,34,56,112)(7,103,57,25)(8,36,58,114)(9,105,59,27)(10,38,60,116)(11,107,41,29)(12,40,42,118)(13,109,43,31)(14,22,44,120)(15,111,45,33)(16,24,46,102)(17,113,47,35)(18,26,48,104)(19,115,49,37)(20,28,50,106)(61,146,123,87)(62,98,124,157)(63,148,125,89)(64,100,126,159)(65,150,127,91)(66,82,128,141)(67,152,129,93)(68,84,130,143)(69,154,131,95)(70,86,132,145)(71,156,133,97)(72,88,134,147)(73,158,135,99)(74,90,136,149)(75,160,137,81)(76,92,138,151)(77,142,139,83)(78,94,140,153)(79,144,121,85)(80,96,122,155), (1,94,11,84)(2,85,12,95)(3,96,13,86)(4,87,14,97)(5,98,15,88)(6,89,16,99)(7,100,17,90)(8,91,18,81)(9,82,19,92)(10,93,20,83)(21,80,31,70)(22,71,32,61)(23,62,33,72)(24,73,34,63)(25,64,35,74)(26,75,36,65)(27,66,37,76)(28,77,38,67)(29,68,39,78)(30,79,40,69)(41,143,51,153)(42,154,52,144)(43,145,53,155)(44,156,54,146)(45,147,55,157)(46,158,56,148)(47,149,57,159)(48,160,58,150)(49,151,59,141)(50,142,60,152)(101,124,111,134)(102,135,112,125)(103,126,113,136)(104,137,114,127)(105,128,115,138)(106,139,116,129)(107,130,117,140)(108,121,118,131)(109,132,119,122)(110,123,120,133), (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)(21,36,31,26)(22,25,32,35)(23,34,33,24)(27,30,37,40)(28,39,38,29)(41,50,51,60)(42,59,52,49)(43,48,53,58)(44,57,54,47)(45,46,55,56)(61,126,71,136)(62,135,72,125)(63,124,73,134)(64,133,74,123)(65,122,75,132)(66,131,76,121)(67,140,77,130)(68,129,78,139)(69,138,79,128)(70,127,80,137)(81,145,91,155)(82,154,92,144)(83,143,93,153)(84,152,94,142)(85,141,95,151)(86,150,96,160)(87,159,97,149)(88,148,98,158)(89,157,99,147)(90,146,100,156)(101,112,111,102)(103,110,113,120)(104,119,114,109)(105,108,115,118)(106,117,116,107)>;
G:=Group( (1,117,51,39)(2,30,52,108)(3,119,53,21)(4,32,54,110)(5,101,55,23)(6,34,56,112)(7,103,57,25)(8,36,58,114)(9,105,59,27)(10,38,60,116)(11,107,41,29)(12,40,42,118)(13,109,43,31)(14,22,44,120)(15,111,45,33)(16,24,46,102)(17,113,47,35)(18,26,48,104)(19,115,49,37)(20,28,50,106)(61,146,123,87)(62,98,124,157)(63,148,125,89)(64,100,126,159)(65,150,127,91)(66,82,128,141)(67,152,129,93)(68,84,130,143)(69,154,131,95)(70,86,132,145)(71,156,133,97)(72,88,134,147)(73,158,135,99)(74,90,136,149)(75,160,137,81)(76,92,138,151)(77,142,139,83)(78,94,140,153)(79,144,121,85)(80,96,122,155), (1,94,11,84)(2,85,12,95)(3,96,13,86)(4,87,14,97)(5,98,15,88)(6,89,16,99)(7,100,17,90)(8,91,18,81)(9,82,19,92)(10,93,20,83)(21,80,31,70)(22,71,32,61)(23,62,33,72)(24,73,34,63)(25,64,35,74)(26,75,36,65)(27,66,37,76)(28,77,38,67)(29,68,39,78)(30,79,40,69)(41,143,51,153)(42,154,52,144)(43,145,53,155)(44,156,54,146)(45,147,55,157)(46,158,56,148)(47,149,57,159)(48,160,58,150)(49,151,59,141)(50,142,60,152)(101,124,111,134)(102,135,112,125)(103,126,113,136)(104,137,114,127)(105,128,115,138)(106,139,116,129)(107,130,117,140)(108,121,118,131)(109,132,119,122)(110,123,120,133), (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)(21,36,31,26)(22,25,32,35)(23,34,33,24)(27,30,37,40)(28,39,38,29)(41,50,51,60)(42,59,52,49)(43,48,53,58)(44,57,54,47)(45,46,55,56)(61,126,71,136)(62,135,72,125)(63,124,73,134)(64,133,74,123)(65,122,75,132)(66,131,76,121)(67,140,77,130)(68,129,78,139)(69,138,79,128)(70,127,80,137)(81,145,91,155)(82,154,92,144)(83,143,93,153)(84,152,94,142)(85,141,95,151)(86,150,96,160)(87,159,97,149)(88,148,98,158)(89,157,99,147)(90,146,100,156)(101,112,111,102)(103,110,113,120)(104,119,114,109)(105,108,115,118)(106,117,116,107) );
G=PermutationGroup([[(1,117,51,39),(2,30,52,108),(3,119,53,21),(4,32,54,110),(5,101,55,23),(6,34,56,112),(7,103,57,25),(8,36,58,114),(9,105,59,27),(10,38,60,116),(11,107,41,29),(12,40,42,118),(13,109,43,31),(14,22,44,120),(15,111,45,33),(16,24,46,102),(17,113,47,35),(18,26,48,104),(19,115,49,37),(20,28,50,106),(61,146,123,87),(62,98,124,157),(63,148,125,89),(64,100,126,159),(65,150,127,91),(66,82,128,141),(67,152,129,93),(68,84,130,143),(69,154,131,95),(70,86,132,145),(71,156,133,97),(72,88,134,147),(73,158,135,99),(74,90,136,149),(75,160,137,81),(76,92,138,151),(77,142,139,83),(78,94,140,153),(79,144,121,85),(80,96,122,155)], [(1,94,11,84),(2,85,12,95),(3,96,13,86),(4,87,14,97),(5,98,15,88),(6,89,16,99),(7,100,17,90),(8,91,18,81),(9,82,19,92),(10,93,20,83),(21,80,31,70),(22,71,32,61),(23,62,33,72),(24,73,34,63),(25,64,35,74),(26,75,36,65),(27,66,37,76),(28,77,38,67),(29,68,39,78),(30,79,40,69),(41,143,51,153),(42,154,52,144),(43,145,53,155),(44,156,54,146),(45,147,55,157),(46,158,56,148),(47,149,57,159),(48,160,58,150),(49,151,59,141),(50,142,60,152),(101,124,111,134),(102,135,112,125),(103,126,113,136),(104,137,114,127),(105,128,115,138),(106,139,116,129),(107,130,117,140),(108,121,118,131),(109,132,119,122),(110,123,120,133)], [(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),(21,36,31,26),(22,25,32,35),(23,34,33,24),(27,30,37,40),(28,39,38,29),(41,50,51,60),(42,59,52,49),(43,48,53,58),(44,57,54,47),(45,46,55,56),(61,126,71,136),(62,135,72,125),(63,124,73,134),(64,133,74,123),(65,122,75,132),(66,131,76,121),(67,140,77,130),(68,129,78,139),(69,138,79,128),(70,127,80,137),(81,145,91,155),(82,154,92,144),(83,143,93,153),(84,152,94,142),(85,141,95,151),(86,150,96,160),(87,159,97,149),(88,148,98,158),(89,157,99,147),(90,146,100,156),(101,112,111,102),(103,110,113,120),(104,119,114,109),(105,108,115,118),(106,117,116,107)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4E | 4F | ··· | 4K | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 20 | 4 | ··· | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | D10 | D10 | 2+ 1+4 | 2- 1+4 | D4⋊6D10 | D4⋊8D10 | D4.10D10 |
| kernel | C42.145D10 | C20.6Q8 | C4.D20 | Dic5.14D4 | D10.12D4 | D10⋊D4 | C22.D20 | Dic5⋊D4 | D10⋊3Q8 | C5×C4.4D4 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C10 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 1 | 4 | 4 | 4 |
Matrix representation of C42.145D10 ►in GL8(𝔽41)
| 39 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 10 | 23 | 7 |
| 0 | 0 | 0 | 0 | 9 | 22 | 9 | 14 |
| 0 | 0 | 0 | 0 | 0 | 16 | 10 | 9 |
| 0 | 0 | 0 | 0 | 25 | 16 | 32 | 38 |
| 40 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 40 | 13 | 13 | 0 | 0 | 0 | 0 |
| 3 | 3 | 1 | 0 | 0 | 0 | 0 | 0 |
| 38 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 36 | 15 | 26 |
| 0 | 0 | 0 | 0 | 40 | 18 | 0 | 26 |
| 0 | 0 | 0 | 0 | 0 | 0 | 17 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 24 |
| 5 | 22 | 30 | 35 | 0 | 0 | 0 | 0 |
| 28 | 33 | 22 | 36 | 0 | 0 | 0 | 0 |
| 0 | 28 | 27 | 19 | 0 | 0 | 0 | 0 |
| 17 | 13 | 22 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 25 | 25 | 38 |
| 0 | 0 | 0 | 0 | 5 | 33 | 18 | 7 |
| 0 | 0 | 0 | 0 | 32 | 37 | 3 | 34 |
| 0 | 0 | 0 | 0 | 32 | 0 | 36 | 26 |
| 21 | 35 | 6 | 11 | 0 | 0 | 0 | 0 |
| 12 | 20 | 5 | 19 | 0 | 0 | 0 | 0 |
| 24 | 12 | 22 | 14 | 0 | 0 | 0 | 0 |
| 39 | 27 | 24 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 25 | 7 | 30 |
| 0 | 0 | 0 | 0 | 3 | 33 | 14 | 2 |
| 0 | 0 | 0 | 0 | 13 | 37 | 38 | 16 |
| 0 | 0 | 0 | 0 | 9 | 0 | 5 | 15 |
G:=sub<GL(8,GF(41))| [39,4,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,11,32,0,0,0,0,0,0,9,30,0,0,0,0,0,0,0,0,12,9,0,25,0,0,0,0,10,22,16,16,0,0,0,0,23,9,10,32,0,0,0,0,7,14,9,38],[40,0,3,38,0,0,0,0,0,40,3,0,0,0,0,0,0,13,1,0,0,0,0,0,28,13,0,1,0,0,0,0,0,0,0,0,23,40,0,0,0,0,0,0,36,18,0,0,0,0,0,0,15,0,17,40,0,0,0,0,26,26,1,24],[5,28,0,17,0,0,0,0,22,33,28,13,0,0,0,0,30,22,27,22,0,0,0,0,35,36,19,17,0,0,0,0,0,0,0,0,20,5,32,32,0,0,0,0,25,33,37,0,0,0,0,0,25,18,3,36,0,0,0,0,38,7,34,26],[21,12,24,39,0,0,0,0,35,20,12,27,0,0,0,0,6,5,22,24,0,0,0,0,11,19,14,19,0,0,0,0,0,0,0,0,37,3,13,9,0,0,0,0,25,33,37,0,0,0,0,0,7,14,38,5,0,0,0,0,30,2,16,15] >;
C42.145D10 in GAP, Magma, Sage, TeX
C_4^2._{145}D_{10} % in TeX
G:=Group("C4^2.145D10"); // GroupNames label
G:=SmallGroup(320,1356);
// by ID
G=gap.SmallGroup(320,1356);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;
// generators/relations