metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.31D10, C4⋊C8⋊16D5, C20⋊3C8⋊16C2, C40⋊8C4⋊22C2, (C8×Dic5)⋊24C2, (C2×C8).218D10, D10⋊1C8.9C2, C10.38(C8○D4), (C4×C20).66C22, C42⋊D5.2C2, D10⋊C4.25C4, C20.337(C4○D4), (C2×C20).837C23, (C2×C40).216C22, C4.57(Q8⋊2D5), C10.D4.25C4, C4.132(D4⋊2D5), C5⋊6(C42.7C22), C10.52(C42⋊C2), C2.16(D20.3C4), C2.15(D20.2C4), (C4×Dic5).309C22, (C5×C4⋊C8)⋊21C2, (C2×C4).37(C4×D5), C22.115(C2×C4×D5), (C2×C20).355(C2×C4), (C2×C4×D5).233C22, C2.10(C4⋊C4⋊7D5), (C22×D5).22(C2×C4), (C2×C4).779(C22×D5), (C2×C10).193(C22×C4), (C2×C5⋊2C8).201C22, (C2×Dic5).100(C2×C4), SmallGroup(320,467)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.31D10
G = < a,b,c,d | a4=b4=1, c10=b, d2=a2b, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, bd=db, dcd-1=a2c9 >
Subgroups: 302 in 96 conjugacy classes, 47 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, Dic5, C20, C20, D10, C2×C10, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C4⋊C8, C42⋊C2, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C42.7C22, C2×C5⋊2C8, C4×Dic5, C10.D4, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C20⋊3C8, C8×Dic5, C40⋊8C4, D10⋊1C8, C5×C4⋊C8, C42⋊D5, C42.31D10
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C4○D4, D10, C42⋊C2, C8○D4, C4×D5, C22×D5, C42.7C22, C2×C4×D5, D4⋊2D5, Q8⋊2D5, C4⋊C4⋊7D5, D20.3C4, D20.2C4, C42.31D10
►On 160 points
Generators in S160
(1 156 103 42)(2 43 104 157)(3 158 105 44)(4 45 106 159)(5 160 107 46)(6 47 108 121)(7 122 109 48)(8 49 110 123)(9 124 111 50)(10 51 112 125)(11 126 113 52)(12 53 114 127)(13 128 115 54)(14 55 116 129)(15 130 117 56)(16 57 118 131)(17 132 119 58)(18 59 120 133)(19 134 81 60)(20 61 82 135)(21 136 83 62)(22 63 84 137)(23 138 85 64)(24 65 86 139)(25 140 87 66)(26 67 88 141)(27 142 89 68)(28 69 90 143)(29 144 91 70)(30 71 92 145)(31 146 93 72)(32 73 94 147)(33 148 95 74)(34 75 96 149)(35 150 97 76)(36 77 98 151)(37 152 99 78)(38 79 100 153)(39 154 101 80)(40 41 102 155)
(1 11 21 31)(2 12 22 32)(3 13 23 33)(4 14 24 34)(5 15 25 35)(6 16 26 36)(7 17 27 37)(8 18 28 38)(9 19 29 39)(10 20 30 40)(41 51 61 71)(42 52 62 72)(43 53 63 73)(44 54 64 74)(45 55 65 75)(46 56 66 76)(47 57 67 77)(48 58 68 78)(49 59 69 79)(50 60 70 80)(81 91 101 111)(82 92 102 112)(83 93 103 113)(84 94 104 114)(85 95 105 115)(86 96 106 116)(87 97 107 117)(88 98 108 118)(89 99 109 119)(90 100 110 120)(121 131 141 151)(122 132 142 152)(123 133 143 153)(124 134 144 154)(125 135 145 155)(126 136 146 156)(127 137 147 157)(128 138 148 158)(129 139 149 159)(130 140 150 160)
(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 10 113 82 21 30 93 102)(2 81 114 29 22 101 94 9)(3 28 115 100 23 8 95 120)(4 99 116 7 24 119 96 27)(5 6 117 118 25 26 97 98)(11 20 83 92 31 40 103 112)(12 91 84 39 32 111 104 19)(13 38 85 110 33 18 105 90)(14 109 86 17 34 89 106 37)(15 16 87 88 35 36 107 108)(41 136 125 72 61 156 145 52)(42 71 126 155 62 51 146 135)(43 154 127 50 63 134 147 70)(44 49 128 133 64 69 148 153)(45 132 129 68 65 152 149 48)(46 67 130 151 66 47 150 131)(53 124 137 60 73 144 157 80)(54 59 138 143 74 79 158 123)(55 142 139 78 75 122 159 58)(56 77 140 121 76 57 160 141)
G:=sub<Sym(160)| (1,156,103,42)(2,43,104,157)(3,158,105,44)(4,45,106,159)(5,160,107,46)(6,47,108,121)(7,122,109,48)(8,49,110,123)(9,124,111,50)(10,51,112,125)(11,126,113,52)(12,53,114,127)(13,128,115,54)(14,55,116,129)(15,130,117,56)(16,57,118,131)(17,132,119,58)(18,59,120,133)(19,134,81,60)(20,61,82,135)(21,136,83,62)(22,63,84,137)(23,138,85,64)(24,65,86,139)(25,140,87,66)(26,67,88,141)(27,142,89,68)(28,69,90,143)(29,144,91,70)(30,71,92,145)(31,146,93,72)(32,73,94,147)(33,148,95,74)(34,75,96,149)(35,150,97,76)(36,77,98,151)(37,152,99,78)(38,79,100,153)(39,154,101,80)(40,41,102,155), (1,11,21,31)(2,12,22,32)(3,13,23,33)(4,14,24,34)(5,15,25,35)(6,16,26,36)(7,17,27,37)(8,18,28,38)(9,19,29,39)(10,20,30,40)(41,51,61,71)(42,52,62,72)(43,53,63,73)(44,54,64,74)(45,55,65,75)(46,56,66,76)(47,57,67,77)(48,58,68,78)(49,59,69,79)(50,60,70,80)(81,91,101,111)(82,92,102,112)(83,93,103,113)(84,94,104,114)(85,95,105,115)(86,96,106,116)(87,97,107,117)(88,98,108,118)(89,99,109,119)(90,100,110,120)(121,131,141,151)(122,132,142,152)(123,133,143,153)(124,134,144,154)(125,135,145,155)(126,136,146,156)(127,137,147,157)(128,138,148,158)(129,139,149,159)(130,140,150,160), (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), (1,10,113,82,21,30,93,102)(2,81,114,29,22,101,94,9)(3,28,115,100,23,8,95,120)(4,99,116,7,24,119,96,27)(5,6,117,118,25,26,97,98)(11,20,83,92,31,40,103,112)(12,91,84,39,32,111,104,19)(13,38,85,110,33,18,105,90)(14,109,86,17,34,89,106,37)(15,16,87,88,35,36,107,108)(41,136,125,72,61,156,145,52)(42,71,126,155,62,51,146,135)(43,154,127,50,63,134,147,70)(44,49,128,133,64,69,148,153)(45,132,129,68,65,152,149,48)(46,67,130,151,66,47,150,131)(53,124,137,60,73,144,157,80)(54,59,138,143,74,79,158,123)(55,142,139,78,75,122,159,58)(56,77,140,121,76,57,160,141)>;
G:=Group( (1,156,103,42)(2,43,104,157)(3,158,105,44)(4,45,106,159)(5,160,107,46)(6,47,108,121)(7,122,109,48)(8,49,110,123)(9,124,111,50)(10,51,112,125)(11,126,113,52)(12,53,114,127)(13,128,115,54)(14,55,116,129)(15,130,117,56)(16,57,118,131)(17,132,119,58)(18,59,120,133)(19,134,81,60)(20,61,82,135)(21,136,83,62)(22,63,84,137)(23,138,85,64)(24,65,86,139)(25,140,87,66)(26,67,88,141)(27,142,89,68)(28,69,90,143)(29,144,91,70)(30,71,92,145)(31,146,93,72)(32,73,94,147)(33,148,95,74)(34,75,96,149)(35,150,97,76)(36,77,98,151)(37,152,99,78)(38,79,100,153)(39,154,101,80)(40,41,102,155), (1,11,21,31)(2,12,22,32)(3,13,23,33)(4,14,24,34)(5,15,25,35)(6,16,26,36)(7,17,27,37)(8,18,28,38)(9,19,29,39)(10,20,30,40)(41,51,61,71)(42,52,62,72)(43,53,63,73)(44,54,64,74)(45,55,65,75)(46,56,66,76)(47,57,67,77)(48,58,68,78)(49,59,69,79)(50,60,70,80)(81,91,101,111)(82,92,102,112)(83,93,103,113)(84,94,104,114)(85,95,105,115)(86,96,106,116)(87,97,107,117)(88,98,108,118)(89,99,109,119)(90,100,110,120)(121,131,141,151)(122,132,142,152)(123,133,143,153)(124,134,144,154)(125,135,145,155)(126,136,146,156)(127,137,147,157)(128,138,148,158)(129,139,149,159)(130,140,150,160), (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), (1,10,113,82,21,30,93,102)(2,81,114,29,22,101,94,9)(3,28,115,100,23,8,95,120)(4,99,116,7,24,119,96,27)(5,6,117,118,25,26,97,98)(11,20,83,92,31,40,103,112)(12,91,84,39,32,111,104,19)(13,38,85,110,33,18,105,90)(14,109,86,17,34,89,106,37)(15,16,87,88,35,36,107,108)(41,136,125,72,61,156,145,52)(42,71,126,155,62,51,146,135)(43,154,127,50,63,134,147,70)(44,49,128,133,64,69,148,153)(45,132,129,68,65,152,149,48)(46,67,130,151,66,47,150,131)(53,124,137,60,73,144,157,80)(54,59,138,143,74,79,158,123)(55,142,139,78,75,122,159,58)(56,77,140,121,76,57,160,141) );
G=PermutationGroup([[(1,156,103,42),(2,43,104,157),(3,158,105,44),(4,45,106,159),(5,160,107,46),(6,47,108,121),(7,122,109,48),(8,49,110,123),(9,124,111,50),(10,51,112,125),(11,126,113,52),(12,53,114,127),(13,128,115,54),(14,55,116,129),(15,130,117,56),(16,57,118,131),(17,132,119,58),(18,59,120,133),(19,134,81,60),(20,61,82,135),(21,136,83,62),(22,63,84,137),(23,138,85,64),(24,65,86,139),(25,140,87,66),(26,67,88,141),(27,142,89,68),(28,69,90,143),(29,144,91,70),(30,71,92,145),(31,146,93,72),(32,73,94,147),(33,148,95,74),(34,75,96,149),(35,150,97,76),(36,77,98,151),(37,152,99,78),(38,79,100,153),(39,154,101,80),(40,41,102,155)], [(1,11,21,31),(2,12,22,32),(3,13,23,33),(4,14,24,34),(5,15,25,35),(6,16,26,36),(7,17,27,37),(8,18,28,38),(9,19,29,39),(10,20,30,40),(41,51,61,71),(42,52,62,72),(43,53,63,73),(44,54,64,74),(45,55,65,75),(46,56,66,76),(47,57,67,77),(48,58,68,78),(49,59,69,79),(50,60,70,80),(81,91,101,111),(82,92,102,112),(83,93,103,113),(84,94,104,114),(85,95,105,115),(86,96,106,116),(87,97,107,117),(88,98,108,118),(89,99,109,119),(90,100,110,120),(121,131,141,151),(122,132,142,152),(123,133,143,153),(124,134,144,154),(125,135,145,155),(126,136,146,156),(127,137,147,157),(128,138,148,158),(129,139,149,159),(130,140,150,160)], [(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)], [(1,10,113,82,21,30,93,102),(2,81,114,29,22,101,94,9),(3,28,115,100,23,8,95,120),(4,99,116,7,24,119,96,27),(5,6,117,118,25,26,97,98),(11,20,83,92,31,40,103,112),(12,91,84,39,32,111,104,19),(13,38,85,110,33,18,105,90),(14,109,86,17,34,89,106,37),(15,16,87,88,35,36,107,108),(41,136,125,72,61,156,145,52),(42,71,126,155,62,51,146,135),(43,154,127,50,63,134,147,70),(44,49,128,133,64,69,148,153),(45,132,129,68,65,152,149,48),(46,67,130,151,66,47,150,131),(53,124,137,60,73,144,157,80),(54,59,138,143,74,79,158,123),(55,142,139,78,75,122,159,58),(56,77,140,121,76,57,160,141)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D5 | C4○D4 | D10 | D10 | C8○D4 | C4×D5 | D20.3C4 | D4⋊2D5 | Q8⋊2D5 | D20.2C4 |
| kernel | C42.31D10 | C20⋊3C8 | C8×Dic5 | C40⋊8C4 | D10⋊1C8 | C5×C4⋊C8 | C42⋊D5 | C10.D4 | D10⋊C4 | C4⋊C8 | C20 | C42 | C2×C8 | C10 | C2×C4 | C2 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 4 | 4 | 2 | 4 | 2 | 4 | 8 | 8 | 16 | 2 | 2 | 4 |
Matrix representation of C42.31D10 ►in GL4(𝔽41) generated by
| 24 | 40 | 0 | 0 |
| 1 | 17 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 38 | 32 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 27 | 16 | 0 | 0 |
| 25 | 16 | 0 | 0 |
| 0 | 0 | 14 | 2 |
| 0 | 0 | 4 | 27 |
| 16 | 27 | 0 | 0 |
| 16 | 25 | 0 | 0 |
| 0 | 0 | 27 | 39 |
| 0 | 0 | 36 | 14 |
G:=sub<GL(4,GF(41))| [24,1,0,0,40,17,0,0,0,0,9,38,0,0,0,32],[32,0,0,0,0,32,0,0,0,0,40,0,0,0,0,40],[27,25,0,0,16,16,0,0,0,0,14,4,0,0,2,27],[16,16,0,0,27,25,0,0,0,0,27,36,0,0,39,14] >;
C42.31D10 in GAP, Magma, Sage, TeX
C_4^2._{31}D_{10} % in TeX
G:=Group("C4^2.31D10"); // GroupNames label
G:=SmallGroup(320,467);
// by ID
G=gap.SmallGroup(320,467);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,422,219,58,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b,d^2=a^2*b,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^9>;
// generators/relations