metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.152D10, C10.962- 1+4, C42.C2⋊8D5, C4⋊C4.208D10, (C4×D20).25C2, D10⋊2Q8⋊38C2, (C4×Dic10)⋊48C2, (C2×C20).90C23, C4.Dic10⋊35C2, D10.39(C4○D4), C20.129(C4○D4), (C2×C10).238C24, (C4×C20).197C22, C4.38(Q8⋊2D5), D10.13D4.3C2, (C2×D20).233C22, C4⋊Dic5.243C22, C22.259(C23×D5), (C4×Dic5).152C22, (C2×Dic5).123C23, (C22×D5).103C23, C2.58(D4.10D10), D10⋊C4.138C22, C5⋊10(C22.46C24), (C2×Dic10).307C22, C10.D4.123C22, (D5×C4⋊C4)⋊38C2, C2.89(D5×C4○D4), C4⋊C4⋊7D5⋊37C2, C4⋊C4⋊D5⋊36C2, C10.200(C2×C4○D4), C2.23(C2×Q8⋊2D5), (C5×C42.C2)⋊11C2, (C2×C4×D5).137C22, (C2×C4).81(C22×D5), (C5×C4⋊C4).193C22, SmallGroup(320,1366)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.152D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c9 >
Subgroups: 710 in 214 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42.C2, C42⋊2C2, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22.46C24, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, D10⋊C4, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C4×Dic10, C4×D20, C4.Dic10, D5×C4⋊C4, C4⋊C4⋊7D5, C4⋊C4⋊7D5, D10.13D4, D10⋊2Q8, C4⋊C4⋊D5, C5×C42.C2, C42.152D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.46C24, Q8⋊2D5, C23×D5, C2×Q8⋊2D5, D5×C4○D4, D4.10D10, C42.152D10
►On 160 points
Generators in S160
(1 118 147 127)(2 128 148 119)(3 120 149 129)(4 130 150 101)(5 102 151 131)(6 132 152 103)(7 104 153 133)(8 134 154 105)(9 106 155 135)(10 136 156 107)(11 108 157 137)(12 138 158 109)(13 110 159 139)(14 140 160 111)(15 112 141 121)(16 122 142 113)(17 114 143 123)(18 124 144 115)(19 116 145 125)(20 126 146 117)(21 50 71 83)(22 84 72 51)(23 52 73 85)(24 86 74 53)(25 54 75 87)(26 88 76 55)(27 56 77 89)(28 90 78 57)(29 58 79 91)(30 92 80 59)(31 60 61 93)(32 94 62 41)(33 42 63 95)(34 96 64 43)(35 44 65 97)(36 98 66 45)(37 46 67 99)(38 100 68 47)(39 48 69 81)(40 82 70 49)
(1 34 11 24)(2 75 12 65)(3 36 13 26)(4 77 14 67)(5 38 15 28)(6 79 16 69)(7 40 17 30)(8 61 18 71)(9 22 19 32)(10 63 20 73)(21 154 31 144)(23 156 33 146)(25 158 35 148)(27 160 37 150)(29 142 39 152)(41 135 51 125)(42 117 52 107)(43 137 53 127)(44 119 54 109)(45 139 55 129)(46 101 56 111)(47 121 57 131)(48 103 58 113)(49 123 59 133)(50 105 60 115)(62 155 72 145)(64 157 74 147)(66 159 76 149)(68 141 78 151)(70 143 80 153)(81 132 91 122)(82 114 92 104)(83 134 93 124)(84 116 94 106)(85 136 95 126)(86 118 96 108)(87 138 97 128)(88 120 98 110)(89 140 99 130)(90 102 100 112)
(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 33 11 23)(2 22 12 32)(3 31 13 21)(4 40 14 30)(5 29 15 39)(6 38 16 28)(7 27 17 37)(8 36 18 26)(9 25 19 35)(10 34 20 24)(41 128 51 138)(42 137 52 127)(43 126 53 136)(44 135 54 125)(45 124 55 134)(46 133 56 123)(47 122 57 132)(48 131 58 121)(49 140 59 130)(50 129 60 139)(61 159 71 149)(62 148 72 158)(63 157 73 147)(64 146 74 156)(65 155 75 145)(66 144 76 154)(67 153 77 143)(68 142 78 152)(69 151 79 141)(70 160 80 150)(81 102 91 112)(82 111 92 101)(83 120 93 110)(84 109 94 119)(85 118 95 108)(86 107 96 117)(87 116 97 106)(88 105 98 115)(89 114 99 104)(90 103 100 113)
G:=sub<Sym(160)| (1,118,147,127)(2,128,148,119)(3,120,149,129)(4,130,150,101)(5,102,151,131)(6,132,152,103)(7,104,153,133)(8,134,154,105)(9,106,155,135)(10,136,156,107)(11,108,157,137)(12,138,158,109)(13,110,159,139)(14,140,160,111)(15,112,141,121)(16,122,142,113)(17,114,143,123)(18,124,144,115)(19,116,145,125)(20,126,146,117)(21,50,71,83)(22,84,72,51)(23,52,73,85)(24,86,74,53)(25,54,75,87)(26,88,76,55)(27,56,77,89)(28,90,78,57)(29,58,79,91)(30,92,80,59)(31,60,61,93)(32,94,62,41)(33,42,63,95)(34,96,64,43)(35,44,65,97)(36,98,66,45)(37,46,67,99)(38,100,68,47)(39,48,69,81)(40,82,70,49), (1,34,11,24)(2,75,12,65)(3,36,13,26)(4,77,14,67)(5,38,15,28)(6,79,16,69)(7,40,17,30)(8,61,18,71)(9,22,19,32)(10,63,20,73)(21,154,31,144)(23,156,33,146)(25,158,35,148)(27,160,37,150)(29,142,39,152)(41,135,51,125)(42,117,52,107)(43,137,53,127)(44,119,54,109)(45,139,55,129)(46,101,56,111)(47,121,57,131)(48,103,58,113)(49,123,59,133)(50,105,60,115)(62,155,72,145)(64,157,74,147)(66,159,76,149)(68,141,78,151)(70,143,80,153)(81,132,91,122)(82,114,92,104)(83,134,93,124)(84,116,94,106)(85,136,95,126)(86,118,96,108)(87,138,97,128)(88,120,98,110)(89,140,99,130)(90,102,100,112), (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), (1,33,11,23)(2,22,12,32)(3,31,13,21)(4,40,14,30)(5,29,15,39)(6,38,16,28)(7,27,17,37)(8,36,18,26)(9,25,19,35)(10,34,20,24)(41,128,51,138)(42,137,52,127)(43,126,53,136)(44,135,54,125)(45,124,55,134)(46,133,56,123)(47,122,57,132)(48,131,58,121)(49,140,59,130)(50,129,60,139)(61,159,71,149)(62,148,72,158)(63,157,73,147)(64,146,74,156)(65,155,75,145)(66,144,76,154)(67,153,77,143)(68,142,78,152)(69,151,79,141)(70,160,80,150)(81,102,91,112)(82,111,92,101)(83,120,93,110)(84,109,94,119)(85,118,95,108)(86,107,96,117)(87,116,97,106)(88,105,98,115)(89,114,99,104)(90,103,100,113)>;
G:=Group( (1,118,147,127)(2,128,148,119)(3,120,149,129)(4,130,150,101)(5,102,151,131)(6,132,152,103)(7,104,153,133)(8,134,154,105)(9,106,155,135)(10,136,156,107)(11,108,157,137)(12,138,158,109)(13,110,159,139)(14,140,160,111)(15,112,141,121)(16,122,142,113)(17,114,143,123)(18,124,144,115)(19,116,145,125)(20,126,146,117)(21,50,71,83)(22,84,72,51)(23,52,73,85)(24,86,74,53)(25,54,75,87)(26,88,76,55)(27,56,77,89)(28,90,78,57)(29,58,79,91)(30,92,80,59)(31,60,61,93)(32,94,62,41)(33,42,63,95)(34,96,64,43)(35,44,65,97)(36,98,66,45)(37,46,67,99)(38,100,68,47)(39,48,69,81)(40,82,70,49), (1,34,11,24)(2,75,12,65)(3,36,13,26)(4,77,14,67)(5,38,15,28)(6,79,16,69)(7,40,17,30)(8,61,18,71)(9,22,19,32)(10,63,20,73)(21,154,31,144)(23,156,33,146)(25,158,35,148)(27,160,37,150)(29,142,39,152)(41,135,51,125)(42,117,52,107)(43,137,53,127)(44,119,54,109)(45,139,55,129)(46,101,56,111)(47,121,57,131)(48,103,58,113)(49,123,59,133)(50,105,60,115)(62,155,72,145)(64,157,74,147)(66,159,76,149)(68,141,78,151)(70,143,80,153)(81,132,91,122)(82,114,92,104)(83,134,93,124)(84,116,94,106)(85,136,95,126)(86,118,96,108)(87,138,97,128)(88,120,98,110)(89,140,99,130)(90,102,100,112), (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), (1,33,11,23)(2,22,12,32)(3,31,13,21)(4,40,14,30)(5,29,15,39)(6,38,16,28)(7,27,17,37)(8,36,18,26)(9,25,19,35)(10,34,20,24)(41,128,51,138)(42,137,52,127)(43,126,53,136)(44,135,54,125)(45,124,55,134)(46,133,56,123)(47,122,57,132)(48,131,58,121)(49,140,59,130)(50,129,60,139)(61,159,71,149)(62,148,72,158)(63,157,73,147)(64,146,74,156)(65,155,75,145)(66,144,76,154)(67,153,77,143)(68,142,78,152)(69,151,79,141)(70,160,80,150)(81,102,91,112)(82,111,92,101)(83,120,93,110)(84,109,94,119)(85,118,95,108)(86,107,96,117)(87,116,97,106)(88,105,98,115)(89,114,99,104)(90,103,100,113) );
G=PermutationGroup([[(1,118,147,127),(2,128,148,119),(3,120,149,129),(4,130,150,101),(5,102,151,131),(6,132,152,103),(7,104,153,133),(8,134,154,105),(9,106,155,135),(10,136,156,107),(11,108,157,137),(12,138,158,109),(13,110,159,139),(14,140,160,111),(15,112,141,121),(16,122,142,113),(17,114,143,123),(18,124,144,115),(19,116,145,125),(20,126,146,117),(21,50,71,83),(22,84,72,51),(23,52,73,85),(24,86,74,53),(25,54,75,87),(26,88,76,55),(27,56,77,89),(28,90,78,57),(29,58,79,91),(30,92,80,59),(31,60,61,93),(32,94,62,41),(33,42,63,95),(34,96,64,43),(35,44,65,97),(36,98,66,45),(37,46,67,99),(38,100,68,47),(39,48,69,81),(40,82,70,49)], [(1,34,11,24),(2,75,12,65),(3,36,13,26),(4,77,14,67),(5,38,15,28),(6,79,16,69),(7,40,17,30),(8,61,18,71),(9,22,19,32),(10,63,20,73),(21,154,31,144),(23,156,33,146),(25,158,35,148),(27,160,37,150),(29,142,39,152),(41,135,51,125),(42,117,52,107),(43,137,53,127),(44,119,54,109),(45,139,55,129),(46,101,56,111),(47,121,57,131),(48,103,58,113),(49,123,59,133),(50,105,60,115),(62,155,72,145),(64,157,74,147),(66,159,76,149),(68,141,78,151),(70,143,80,153),(81,132,91,122),(82,114,92,104),(83,134,93,124),(84,116,94,106),(85,136,95,126),(86,118,96,108),(87,138,97,128),(88,120,98,110),(89,140,99,130),(90,102,100,112)], [(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)], [(1,33,11,23),(2,22,12,32),(3,31,13,21),(4,40,14,30),(5,29,15,39),(6,38,16,28),(7,27,17,37),(8,36,18,26),(9,25,19,35),(10,34,20,24),(41,128,51,138),(42,137,52,127),(43,126,53,136),(44,135,54,125),(45,124,55,134),(46,133,56,123),(47,122,57,132),(48,131,58,121),(49,140,59,130),(50,129,60,139),(61,159,71,149),(62,148,72,158),(63,157,73,147),(64,146,74,156),(65,155,75,145),(66,144,76,154),(67,153,77,143),(68,142,78,152),(69,151,79,141),(70,160,80,150),(81,102,91,112),(82,111,92,101),(83,120,93,110),(84,109,94,119),(85,118,95,108),(86,107,96,117),(87,116,97,106),(88,105,98,115),(89,114,99,104),(90,103,100,113)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4O | 4P | 4Q | 4R | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | 2- 1+4 | Q8⋊2D5 | D5×C4○D4 | D4.10D10 |
| kernel | C42.152D10 | C4×Dic10 | C4×D20 | C4.Dic10 | D5×C4⋊C4 | C4⋊C4⋊7D5 | D10.13D4 | D10⋊2Q8 | C4⋊C4⋊D5 | C5×C42.C2 | C42.C2 | C20 | D10 | C42 | C4⋊C4 | C10 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 3 | 2 | 2 | 2 | 1 | 2 | 4 | 4 | 2 | 12 | 1 | 4 | 4 | 4 |
Matrix representation of C42.152D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 1 | 32 |
| 1 | 39 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 9 | 1 |
| 9 | 23 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 39 |
| 0 | 0 | 0 | 0 | 40 | 9 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 35 | 0 | 0 |
| 0 | 0 | 8 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 2 |
| 0 | 0 | 0 | 0 | 1 | 32 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,1,0,0,0,0,0,32],[1,1,0,0,0,0,39,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,9,0,0,0,0,0,1],[9,0,0,0,0,0,23,32,0,0,0,0,0,0,0,34,0,0,0,0,6,7,0,0,0,0,0,0,32,40,0,0,0,0,39,9],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,7,8,0,0,0,0,35,34,0,0,0,0,0,0,9,1,0,0,0,0,2,32] >;
C42.152D10 in GAP, Magma, Sage, TeX
C_4^2._{152}D_{10} % in TeX
G:=Group("C4^2.152D10"); // GroupNames label
G:=SmallGroup(320,1366);
// by ID
G=gap.SmallGroup(320,1366);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,1571,297,192,12550]);
// 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=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations