metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.114D10, C10.202+ 1+4, (C4×D4)⋊21D5, (D4×C20)⋊23C2, C4⋊C4.319D10, C20⋊D4.9C2, D20⋊8C4⋊16C2, (C4×Dic10)⋊34C2, (C2×D4).220D10, C4.16(C4○D20), C4.D20⋊19C2, C20.17D4⋊9C2, (C22×C4).48D10, Dic5⋊3Q8⋊16C2, D10.12D4⋊8C2, C20.111(C4○D4), (C4×C20).158C22, (C2×C20).701C23, (C2×C10).103C24, C22⋊C4.116D10, Dic5.5D4⋊8C2, C2.21(D4⋊6D10), Dic5.61(C4○D4), (C2×D20).145C22, (D4×C10).263C22, C4⋊Dic5.301C22, (C2×Dic5).44C23, (C4×Dic5).84C22, (C22×D5).37C23, C22.128(C23×D5), C23.100(C22×D5), D10⋊C4.87C22, C23.23D10⋊18C2, (C22×C20).365C22, (C22×C10).173C23, C5⋊1(C22.53C24), C10.D4.66C22, C23.D5.107C22, (C2×Dic10).151C22, (C4×C5⋊D4)⋊45C2, C2.26(D5×C4○D4), C10.45(C2×C4○D4), C2.52(C2×C4○D20), (C2×C4×D5).253C22, (C5×C4⋊C4).332C22, (C2×C4).286(C22×D5), (C2×C5⋊D4).124C22, (C5×C22⋊C4).127C22, SmallGroup(320,1231)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.114D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=c9 >
Subgroups: 838 in 236 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C4, C4, 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×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C4×D4, C4×D4, C4×Q8, C22.D4, C4.4D4, C4⋊1D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22.53C24, C4×Dic5, C4×Dic5, C10.D4, 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, D4×C10, C4×Dic10, C4.D20, D10.12D4, Dic5.5D4, Dic5⋊3Q8, D20⋊8C4, C4×C5⋊D4, C23.23D10, C20.17D4, C20⋊D4, D4×C20, C42.114D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.53C24, C4○D20, C23×D5, C2×C4○D20, D4⋊6D10, D5×C4○D4, C42.114D10
►On 160 points
Generators in S160
(1 94 39 70)(2 61 40 85)(3 96 21 72)(4 63 22 87)(5 98 23 74)(6 65 24 89)(7 100 25 76)(8 67 26 91)(9 82 27 78)(10 69 28 93)(11 84 29 80)(12 71 30 95)(13 86 31 62)(14 73 32 97)(15 88 33 64)(16 75 34 99)(17 90 35 66)(18 77 36 81)(19 92 37 68)(20 79 38 83)(41 148 128 108)(42 119 129 159)(43 150 130 110)(44 101 131 141)(45 152 132 112)(46 103 133 143)(47 154 134 114)(48 105 135 145)(49 156 136 116)(50 107 137 147)(51 158 138 118)(52 109 139 149)(53 160 140 120)(54 111 121 151)(55 142 122 102)(56 113 123 153)(57 144 124 104)(58 115 125 155)(59 146 126 106)(60 117 127 157)
(1 156 11 146)(2 157 12 147)(3 158 13 148)(4 159 14 149)(5 160 15 150)(6 141 16 151)(7 142 17 152)(8 143 18 153)(9 144 19 154)(10 145 20 155)(21 118 31 108)(22 119 32 109)(23 120 33 110)(24 101 34 111)(25 102 35 112)(26 103 36 113)(27 104 37 114)(28 105 38 115)(29 106 39 116)(30 107 40 117)(41 72 51 62)(42 73 52 63)(43 74 53 64)(44 75 54 65)(45 76 55 66)(46 77 56 67)(47 78 57 68)(48 79 58 69)(49 80 59 70)(50 61 60 71)(81 123 91 133)(82 124 92 134)(83 125 93 135)(84 126 94 136)(85 127 95 137)(86 128 96 138)(87 129 97 139)(88 130 98 140)(89 131 99 121)(90 132 100 122)
(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 26 31 36)(22 35 32 25)(23 24 33 34)(27 40 37 30)(28 29 38 39)(41 123 51 133)(42 132 52 122)(43 121 53 131)(44 130 54 140)(45 139 55 129)(46 128 56 138)(47 137 57 127)(48 126 58 136)(49 135 59 125)(50 124 60 134)(61 68 71 78)(62 77 72 67)(63 66 73 76)(64 75 74 65)(69 80 79 70)(81 96 91 86)(82 85 92 95)(83 94 93 84)(87 90 97 100)(88 99 98 89)(101 150 111 160)(102 159 112 149)(103 148 113 158)(104 157 114 147)(105 146 115 156)(106 155 116 145)(107 144 117 154)(108 153 118 143)(109 142 119 152)(110 151 120 141)
G:=sub<Sym(160)| (1,94,39,70)(2,61,40,85)(3,96,21,72)(4,63,22,87)(5,98,23,74)(6,65,24,89)(7,100,25,76)(8,67,26,91)(9,82,27,78)(10,69,28,93)(11,84,29,80)(12,71,30,95)(13,86,31,62)(14,73,32,97)(15,88,33,64)(16,75,34,99)(17,90,35,66)(18,77,36,81)(19,92,37,68)(20,79,38,83)(41,148,128,108)(42,119,129,159)(43,150,130,110)(44,101,131,141)(45,152,132,112)(46,103,133,143)(47,154,134,114)(48,105,135,145)(49,156,136,116)(50,107,137,147)(51,158,138,118)(52,109,139,149)(53,160,140,120)(54,111,121,151)(55,142,122,102)(56,113,123,153)(57,144,124,104)(58,115,125,155)(59,146,126,106)(60,117,127,157), (1,156,11,146)(2,157,12,147)(3,158,13,148)(4,159,14,149)(5,160,15,150)(6,141,16,151)(7,142,17,152)(8,143,18,153)(9,144,19,154)(10,145,20,155)(21,118,31,108)(22,119,32,109)(23,120,33,110)(24,101,34,111)(25,102,35,112)(26,103,36,113)(27,104,37,114)(28,105,38,115)(29,106,39,116)(30,107,40,117)(41,72,51,62)(42,73,52,63)(43,74,53,64)(44,75,54,65)(45,76,55,66)(46,77,56,67)(47,78,57,68)(48,79,58,69)(49,80,59,70)(50,61,60,71)(81,123,91,133)(82,124,92,134)(83,125,93,135)(84,126,94,136)(85,127,95,137)(86,128,96,138)(87,129,97,139)(88,130,98,140)(89,131,99,121)(90,132,100,122), (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)(21,26,31,36)(22,35,32,25)(23,24,33,34)(27,40,37,30)(28,29,38,39)(41,123,51,133)(42,132,52,122)(43,121,53,131)(44,130,54,140)(45,139,55,129)(46,128,56,138)(47,137,57,127)(48,126,58,136)(49,135,59,125)(50,124,60,134)(61,68,71,78)(62,77,72,67)(63,66,73,76)(64,75,74,65)(69,80,79,70)(81,96,91,86)(82,85,92,95)(83,94,93,84)(87,90,97,100)(88,99,98,89)(101,150,111,160)(102,159,112,149)(103,148,113,158)(104,157,114,147)(105,146,115,156)(106,155,116,145)(107,144,117,154)(108,153,118,143)(109,142,119,152)(110,151,120,141)>;
G:=Group( (1,94,39,70)(2,61,40,85)(3,96,21,72)(4,63,22,87)(5,98,23,74)(6,65,24,89)(7,100,25,76)(8,67,26,91)(9,82,27,78)(10,69,28,93)(11,84,29,80)(12,71,30,95)(13,86,31,62)(14,73,32,97)(15,88,33,64)(16,75,34,99)(17,90,35,66)(18,77,36,81)(19,92,37,68)(20,79,38,83)(41,148,128,108)(42,119,129,159)(43,150,130,110)(44,101,131,141)(45,152,132,112)(46,103,133,143)(47,154,134,114)(48,105,135,145)(49,156,136,116)(50,107,137,147)(51,158,138,118)(52,109,139,149)(53,160,140,120)(54,111,121,151)(55,142,122,102)(56,113,123,153)(57,144,124,104)(58,115,125,155)(59,146,126,106)(60,117,127,157), (1,156,11,146)(2,157,12,147)(3,158,13,148)(4,159,14,149)(5,160,15,150)(6,141,16,151)(7,142,17,152)(8,143,18,153)(9,144,19,154)(10,145,20,155)(21,118,31,108)(22,119,32,109)(23,120,33,110)(24,101,34,111)(25,102,35,112)(26,103,36,113)(27,104,37,114)(28,105,38,115)(29,106,39,116)(30,107,40,117)(41,72,51,62)(42,73,52,63)(43,74,53,64)(44,75,54,65)(45,76,55,66)(46,77,56,67)(47,78,57,68)(48,79,58,69)(49,80,59,70)(50,61,60,71)(81,123,91,133)(82,124,92,134)(83,125,93,135)(84,126,94,136)(85,127,95,137)(86,128,96,138)(87,129,97,139)(88,130,98,140)(89,131,99,121)(90,132,100,122), (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)(21,26,31,36)(22,35,32,25)(23,24,33,34)(27,40,37,30)(28,29,38,39)(41,123,51,133)(42,132,52,122)(43,121,53,131)(44,130,54,140)(45,139,55,129)(46,128,56,138)(47,137,57,127)(48,126,58,136)(49,135,59,125)(50,124,60,134)(61,68,71,78)(62,77,72,67)(63,66,73,76)(64,75,74,65)(69,80,79,70)(81,96,91,86)(82,85,92,95)(83,94,93,84)(87,90,97,100)(88,99,98,89)(101,150,111,160)(102,159,112,149)(103,148,113,158)(104,157,114,147)(105,146,115,156)(106,155,116,145)(107,144,117,154)(108,153,118,143)(109,142,119,152)(110,151,120,141) );
G=PermutationGroup([[(1,94,39,70),(2,61,40,85),(3,96,21,72),(4,63,22,87),(5,98,23,74),(6,65,24,89),(7,100,25,76),(8,67,26,91),(9,82,27,78),(10,69,28,93),(11,84,29,80),(12,71,30,95),(13,86,31,62),(14,73,32,97),(15,88,33,64),(16,75,34,99),(17,90,35,66),(18,77,36,81),(19,92,37,68),(20,79,38,83),(41,148,128,108),(42,119,129,159),(43,150,130,110),(44,101,131,141),(45,152,132,112),(46,103,133,143),(47,154,134,114),(48,105,135,145),(49,156,136,116),(50,107,137,147),(51,158,138,118),(52,109,139,149),(53,160,140,120),(54,111,121,151),(55,142,122,102),(56,113,123,153),(57,144,124,104),(58,115,125,155),(59,146,126,106),(60,117,127,157)], [(1,156,11,146),(2,157,12,147),(3,158,13,148),(4,159,14,149),(5,160,15,150),(6,141,16,151),(7,142,17,152),(8,143,18,153),(9,144,19,154),(10,145,20,155),(21,118,31,108),(22,119,32,109),(23,120,33,110),(24,101,34,111),(25,102,35,112),(26,103,36,113),(27,104,37,114),(28,105,38,115),(29,106,39,116),(30,107,40,117),(41,72,51,62),(42,73,52,63),(43,74,53,64),(44,75,54,65),(45,76,55,66),(46,77,56,67),(47,78,57,68),(48,79,58,69),(49,80,59,70),(50,61,60,71),(81,123,91,133),(82,124,92,134),(83,125,93,135),(84,126,94,136),(85,127,95,137),(86,128,96,138),(87,129,97,139),(88,130,98,140),(89,131,99,121),(90,132,100,122)], [(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),(21,26,31,36),(22,35,32,25),(23,24,33,34),(27,40,37,30),(28,29,38,39),(41,123,51,133),(42,132,52,122),(43,121,53,131),(44,130,54,140),(45,139,55,129),(46,128,56,138),(47,137,57,127),(48,126,58,136),(49,135,59,125),(50,124,60,134),(61,68,71,78),(62,77,72,67),(63,66,73,76),(64,75,74,65),(69,80,79,70),(81,96,91,86),(82,85,92,95),(83,94,93,84),(87,90,97,100),(88,99,98,89),(101,150,111,160),(102,159,112,149),(103,148,113,158),(104,157,114,147),(105,146,115,156),(106,155,116,145),(107,144,117,154),(108,153,118,143),(109,142,119,152),(110,151,120,141)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | 2+ 1+4 | D4⋊6D10 | D5×C4○D4 |
| kernel | C42.114D10 | C4×Dic10 | C4.D20 | D10.12D4 | Dic5.5D4 | Dic5⋊3Q8 | D20⋊8C4 | C4×C5⋊D4 | C23.23D10 | C20.17D4 | C20⋊D4 | D4×C20 | C4×D4 | Dic5 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of C42.114D10 ►in GL6(𝔽41)
| 32 | 4 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 29 |
| 0 | 0 | 0 | 0 | 5 | 15 |
| 9 | 37 | 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 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 13 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 21 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,4,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,26,5,0,0,0,0,29,15],[9,0,0,0,0,0,37,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,0,0,0,0,9,0,0,0,0,0,13,32],[32,21,0,0,0,0,0,9,0,0,0,0,0,0,34,1,0,0,0,0,34,7,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;
C42.114D10 in GAP, Magma, Sage, TeX
C_4^2._{114}D_{10} % in TeX
G:=Group("C4^2.114D10"); // GroupNames label
G:=SmallGroup(320,1231);
// by ID
G=gap.SmallGroup(320,1231);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,136,12550]);
// 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,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations