metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.139D10, C10.882- 1+4, (Q8×Dic5)⋊18C2, C4.4D4.8D5, (C4×Dic10)⋊44C2, (C2×D4).169D10, (C2×C20).77C23, (C2×Q8).135D10, C22⋊C4.33D10, (D4×Dic5).15C2, Dic5⋊Q8⋊22C2, C20.124(C4○D4), C4.15(D4⋊2D5), (C4×C20).184C22, (C2×C10).215C24, C23.37(C22×D5), Dic5.44(C4○D4), C20.17D4.11C2, (D4×C10).151C22, C23.D10⋊37C2, C4⋊Dic5.233C22, (C22×C10).45C23, (Q8×C10).124C22, C22.236(C23×D5), Dic5.14D4⋊38C2, C23.D5.52C22, C23.11D10⋊18C2, C5⋊6(C22.50C24), (C2×Dic5).262C23, (C4×Dic5).139C22, C10.D4.48C22, C2.49(D4.10D10), (C2×Dic10).304C22, (C22×Dic5).140C22, C2.74(D5×C4○D4), C10.93(C2×C4○D4), C2.55(C2×D4⋊2D5), (C5×C4.4D4).6C2, (C2×C4).299(C22×D5), (C5×C22⋊C4).62C22, SmallGroup(320,1343)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.139D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, bd=db, dcd-1=c-1 >
Subgroups: 614 in 212 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C2×C10, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C4.4D4, C4.4D4, C42⋊2C2, C4⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22.50C24, C4×Dic5, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C4⋊Dic5, C23.D5, C4×C20, C5×C22⋊C4, C2×Dic10, C22×Dic5, D4×C10, Q8×C10, C4×Dic10, C23.11D10, Dic5.14D4, C23.D10, D4×Dic5, C20.17D4, Dic5⋊Q8, Q8×Dic5, C5×C4.4D4, C42.139D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.50C24, D4⋊2D5, C23×D5, C2×D4⋊2D5, D5×C4○D4, D4.10D10, C42.139D10
►On 160 points
Generators in S160
(1 118 15 113)(2 114 11 119)(3 120 12 115)(4 116 13 111)(5 112 14 117)(6 100 21 95)(7 96 22 91)(8 92 23 97)(9 98 24 93)(10 94 25 99)(16 124 34 129)(17 130 35 125)(18 126 31 121)(19 122 32 127)(20 128 33 123)(26 44 36 49)(27 50 37 45)(28 46 38 41)(29 42 39 47)(30 48 40 43)(51 68 106 71)(52 72 107 69)(53 70 108 73)(54 74 109 61)(55 62 110 75)(56 76 101 63)(57 64 102 77)(58 78 103 65)(59 66 104 79)(60 80 105 67)(81 137 152 150)(82 141 153 138)(83 139 154 142)(84 143 155 140)(85 131 156 144)(86 145 157 132)(87 133 158 146)(88 147 159 134)(89 135 160 148)(90 149 151 136)
(1 141 20 133)(2 147 16 139)(3 143 17 135)(4 149 18 131)(5 145 19 137)(6 60 30 110)(7 56 26 106)(8 52 27 102)(9 58 28 108)(10 54 29 104)(11 134 34 142)(12 140 35 148)(13 136 31 144)(14 132 32 150)(15 138 33 146)(21 105 40 55)(22 101 36 51)(23 107 37 57)(24 103 38 53)(25 109 39 59)(41 70 93 65)(42 79 94 74)(43 62 95 67)(44 71 96 76)(45 64 97 69)(46 73 98 78)(47 66 99 61)(48 75 100 80)(49 68 91 63)(50 77 92 72)(81 117 86 127)(82 123 87 113)(83 119 88 129)(84 125 89 115)(85 111 90 121)(112 157 122 152)(114 159 124 154)(116 151 126 156)(118 153 128 158)(120 155 130 160)
(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 7 33 36)(2 6 34 40)(3 10 35 39)(4 9 31 38)(5 8 32 37)(11 21 16 30)(12 25 17 29)(13 24 18 28)(14 23 19 27)(15 22 20 26)(41 111 98 126)(42 120 99 125)(43 119 100 124)(44 118 91 123)(45 117 92 122)(46 116 93 121)(47 115 94 130)(48 114 95 129)(49 113 96 128)(50 112 97 127)(51 141 56 146)(52 150 57 145)(53 149 58 144)(54 148 59 143)(55 147 60 142)(61 89 79 155)(62 88 80 154)(63 87 71 153)(64 86 72 152)(65 85 73 151)(66 84 74 160)(67 83 75 159)(68 82 76 158)(69 81 77 157)(70 90 78 156)(101 133 106 138)(102 132 107 137)(103 131 108 136)(104 140 109 135)(105 139 110 134)
G:=sub<Sym(160)| (1,118,15,113)(2,114,11,119)(3,120,12,115)(4,116,13,111)(5,112,14,117)(6,100,21,95)(7,96,22,91)(8,92,23,97)(9,98,24,93)(10,94,25,99)(16,124,34,129)(17,130,35,125)(18,126,31,121)(19,122,32,127)(20,128,33,123)(26,44,36,49)(27,50,37,45)(28,46,38,41)(29,42,39,47)(30,48,40,43)(51,68,106,71)(52,72,107,69)(53,70,108,73)(54,74,109,61)(55,62,110,75)(56,76,101,63)(57,64,102,77)(58,78,103,65)(59,66,104,79)(60,80,105,67)(81,137,152,150)(82,141,153,138)(83,139,154,142)(84,143,155,140)(85,131,156,144)(86,145,157,132)(87,133,158,146)(88,147,159,134)(89,135,160,148)(90,149,151,136), (1,141,20,133)(2,147,16,139)(3,143,17,135)(4,149,18,131)(5,145,19,137)(6,60,30,110)(7,56,26,106)(8,52,27,102)(9,58,28,108)(10,54,29,104)(11,134,34,142)(12,140,35,148)(13,136,31,144)(14,132,32,150)(15,138,33,146)(21,105,40,55)(22,101,36,51)(23,107,37,57)(24,103,38,53)(25,109,39,59)(41,70,93,65)(42,79,94,74)(43,62,95,67)(44,71,96,76)(45,64,97,69)(46,73,98,78)(47,66,99,61)(48,75,100,80)(49,68,91,63)(50,77,92,72)(81,117,86,127)(82,123,87,113)(83,119,88,129)(84,125,89,115)(85,111,90,121)(112,157,122,152)(114,159,124,154)(116,151,126,156)(118,153,128,158)(120,155,130,160), (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), (1,7,33,36)(2,6,34,40)(3,10,35,39)(4,9,31,38)(5,8,32,37)(11,21,16,30)(12,25,17,29)(13,24,18,28)(14,23,19,27)(15,22,20,26)(41,111,98,126)(42,120,99,125)(43,119,100,124)(44,118,91,123)(45,117,92,122)(46,116,93,121)(47,115,94,130)(48,114,95,129)(49,113,96,128)(50,112,97,127)(51,141,56,146)(52,150,57,145)(53,149,58,144)(54,148,59,143)(55,147,60,142)(61,89,79,155)(62,88,80,154)(63,87,71,153)(64,86,72,152)(65,85,73,151)(66,84,74,160)(67,83,75,159)(68,82,76,158)(69,81,77,157)(70,90,78,156)(101,133,106,138)(102,132,107,137)(103,131,108,136)(104,140,109,135)(105,139,110,134)>;
G:=Group( (1,118,15,113)(2,114,11,119)(3,120,12,115)(4,116,13,111)(5,112,14,117)(6,100,21,95)(7,96,22,91)(8,92,23,97)(9,98,24,93)(10,94,25,99)(16,124,34,129)(17,130,35,125)(18,126,31,121)(19,122,32,127)(20,128,33,123)(26,44,36,49)(27,50,37,45)(28,46,38,41)(29,42,39,47)(30,48,40,43)(51,68,106,71)(52,72,107,69)(53,70,108,73)(54,74,109,61)(55,62,110,75)(56,76,101,63)(57,64,102,77)(58,78,103,65)(59,66,104,79)(60,80,105,67)(81,137,152,150)(82,141,153,138)(83,139,154,142)(84,143,155,140)(85,131,156,144)(86,145,157,132)(87,133,158,146)(88,147,159,134)(89,135,160,148)(90,149,151,136), (1,141,20,133)(2,147,16,139)(3,143,17,135)(4,149,18,131)(5,145,19,137)(6,60,30,110)(7,56,26,106)(8,52,27,102)(9,58,28,108)(10,54,29,104)(11,134,34,142)(12,140,35,148)(13,136,31,144)(14,132,32,150)(15,138,33,146)(21,105,40,55)(22,101,36,51)(23,107,37,57)(24,103,38,53)(25,109,39,59)(41,70,93,65)(42,79,94,74)(43,62,95,67)(44,71,96,76)(45,64,97,69)(46,73,98,78)(47,66,99,61)(48,75,100,80)(49,68,91,63)(50,77,92,72)(81,117,86,127)(82,123,87,113)(83,119,88,129)(84,125,89,115)(85,111,90,121)(112,157,122,152)(114,159,124,154)(116,151,126,156)(118,153,128,158)(120,155,130,160), (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), (1,7,33,36)(2,6,34,40)(3,10,35,39)(4,9,31,38)(5,8,32,37)(11,21,16,30)(12,25,17,29)(13,24,18,28)(14,23,19,27)(15,22,20,26)(41,111,98,126)(42,120,99,125)(43,119,100,124)(44,118,91,123)(45,117,92,122)(46,116,93,121)(47,115,94,130)(48,114,95,129)(49,113,96,128)(50,112,97,127)(51,141,56,146)(52,150,57,145)(53,149,58,144)(54,148,59,143)(55,147,60,142)(61,89,79,155)(62,88,80,154)(63,87,71,153)(64,86,72,152)(65,85,73,151)(66,84,74,160)(67,83,75,159)(68,82,76,158)(69,81,77,157)(70,90,78,156)(101,133,106,138)(102,132,107,137)(103,131,108,136)(104,140,109,135)(105,139,110,134) );
G=PermutationGroup([[(1,118,15,113),(2,114,11,119),(3,120,12,115),(4,116,13,111),(5,112,14,117),(6,100,21,95),(7,96,22,91),(8,92,23,97),(9,98,24,93),(10,94,25,99),(16,124,34,129),(17,130,35,125),(18,126,31,121),(19,122,32,127),(20,128,33,123),(26,44,36,49),(27,50,37,45),(28,46,38,41),(29,42,39,47),(30,48,40,43),(51,68,106,71),(52,72,107,69),(53,70,108,73),(54,74,109,61),(55,62,110,75),(56,76,101,63),(57,64,102,77),(58,78,103,65),(59,66,104,79),(60,80,105,67),(81,137,152,150),(82,141,153,138),(83,139,154,142),(84,143,155,140),(85,131,156,144),(86,145,157,132),(87,133,158,146),(88,147,159,134),(89,135,160,148),(90,149,151,136)], [(1,141,20,133),(2,147,16,139),(3,143,17,135),(4,149,18,131),(5,145,19,137),(6,60,30,110),(7,56,26,106),(8,52,27,102),(9,58,28,108),(10,54,29,104),(11,134,34,142),(12,140,35,148),(13,136,31,144),(14,132,32,150),(15,138,33,146),(21,105,40,55),(22,101,36,51),(23,107,37,57),(24,103,38,53),(25,109,39,59),(41,70,93,65),(42,79,94,74),(43,62,95,67),(44,71,96,76),(45,64,97,69),(46,73,98,78),(47,66,99,61),(48,75,100,80),(49,68,91,63),(50,77,92,72),(81,117,86,127),(82,123,87,113),(83,119,88,129),(84,125,89,115),(85,111,90,121),(112,157,122,152),(114,159,124,154),(116,151,126,156),(118,153,128,158),(120,155,130,160)], [(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)], [(1,7,33,36),(2,6,34,40),(3,10,35,39),(4,9,31,38),(5,8,32,37),(11,21,16,30),(12,25,17,29),(13,24,18,28),(14,23,19,27),(15,22,20,26),(41,111,98,126),(42,120,99,125),(43,119,100,124),(44,118,91,123),(45,117,92,122),(46,116,93,121),(47,115,94,130),(48,114,95,129),(49,113,96,128),(50,112,97,127),(51,141,56,146),(52,150,57,145),(53,149,58,144),(54,148,59,143),(55,147,60,142),(61,89,79,155),(62,88,80,154),(63,87,71,153),(64,86,72,152),(65,85,73,151),(66,84,74,160),(67,83,75,159),(68,82,76,158),(69,81,77,157),(70,90,78,156),(101,133,106,138),(102,132,107,137),(103,131,108,136),(104,140,109,135),(105,139,110,134)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4O | 4P | 4Q | 4R | 4S | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 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 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
53 irreducible representations
| dim | 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 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | 2- 1+4 | D4⋊2D5 | D5×C4○D4 | D4.10D10 |
| kernel | C42.139D10 | C4×Dic10 | C23.11D10 | Dic5.14D4 | C23.D10 | D4×Dic5 | C20.17D4 | Dic5⋊Q8 | Q8×Dic5 | C5×C4.4D4 | C4.4D4 | Dic5 | C20 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 8 | 2 | 2 | 1 | 4 | 4 | 4 |
Matrix representation of C42.139D10 ►in GL6(𝔽41)
| 40 | 9 | 0 | 0 | 0 | 0 |
| 18 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 9 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 23 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 39 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 23 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 30 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
G:=sub<GL(6,GF(41))| [40,18,0,0,0,0,9,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,9,32],[1,23,0,0,0,0,0,40,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,1,39,0,0,0,0,0,40],[1,23,0,0,0,0,0,40,0,0,0,0,0,0,14,30,0,0,0,0,14,27,0,0,0,0,0,0,32,0,0,0,0,0,0,32] >;
C42.139D10 in GAP, Magma, Sage, TeX
C_4^2._{139}D_{10} % in TeX
G:=Group("C4^2.139D10"); // GroupNames label
G:=SmallGroup(320,1343);
// by ID
G=gap.SmallGroup(320,1343);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,387,100,794,297,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations