metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.36D10, Dic10.18D4, C4⋊C8⋊6D5, (C2×C4).39D20, C4.132(D4×D5), (C2×Dic20)⋊7C2, C20.341(C2×D4), (C2×C8).131D10, (C2×C20).245D4, C5⋊3(Q8.D4), (C4×Dic10)⋊17C2, D20⋊5C4.2C2, C10.13(C4○D8), (C2×C40).24C22, (C4×C20).71C22, C4.D20.7C2, C20.330(C4○D4), C20.44D4⋊13C2, C2.13(C4⋊D20), C10.40(C4⋊D4), (C2×C20).755C23, C4.46(Q8⋊2D5), (C2×D20).18C22, C22.118(C2×D20), C2.15(D40⋊7C2), C2.18(C8.D10), C10.15(C8.C22), C4⋊Dic5.275C22, (C2×Dic10).220C22, (C5×C4⋊C8)⋊8C2, (C2×C40⋊C2).6C2, (C2×C10).138(C2×D4), (C2×C4).700(C22×D5), SmallGroup(320,472)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.36D10
G = < a,b,c,d | a4=b4=1, c10=b, d2=b2, ab=ba, cac-1=a-1b2, ad=da, bc=cb, dbd-1=b-1, dcd-1=bc9 >
Subgroups: 518 in 112 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C40, Dic10, Dic10, D20, C2×Dic5, C2×C20, C22×D5, Q8.D4, C40⋊C2, Dic20, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×Dic10, C2×D20, C20.44D4, D20⋊5C4, C5×C4⋊C8, C4×Dic10, C4.D20, C2×C40⋊C2, C2×Dic20, C42.36D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C4○D8, C8.C22, D20, C22×D5, Q8.D4, C2×D20, D4×D5, Q8⋊2D5, C4⋊D20, D40⋊7C2, C8.D10, C42.36D10
►On 160 points
Generators in S160
(1 83 59 160)(2 141 60 104)(3 85 61 122)(4 143 62 106)(5 87 63 124)(6 145 64 108)(7 89 65 126)(8 147 66 110)(9 91 67 128)(10 149 68 112)(11 93 69 130)(12 151 70 114)(13 95 71 132)(14 153 72 116)(15 97 73 134)(16 155 74 118)(17 99 75 136)(18 157 76 120)(19 101 77 138)(20 159 78 82)(21 103 79 140)(22 121 80 84)(23 105 41 142)(24 123 42 86)(25 107 43 144)(26 125 44 88)(27 109 45 146)(28 127 46 90)(29 111 47 148)(30 129 48 92)(31 113 49 150)(32 131 50 94)(33 115 51 152)(34 133 52 96)(35 117 53 154)(36 135 54 98)(37 119 55 156)(38 137 56 100)(39 81 57 158)(40 139 58 102)
(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 21 30)(2 29 22 9)(3 8 23 28)(4 27 24 7)(5 6 25 26)(11 40 31 20)(12 19 32 39)(13 38 33 18)(14 17 34 37)(15 36 35 16)(41 46 61 66)(42 65 62 45)(43 44 63 64)(47 80 67 60)(48 59 68 79)(49 78 69 58)(50 57 70 77)(51 76 71 56)(52 55 72 75)(53 74 73 54)(81 151 101 131)(82 130 102 150)(83 149 103 129)(84 128 104 148)(85 147 105 127)(86 126 106 146)(87 145 107 125)(88 124 108 144)(89 143 109 123)(90 122 110 142)(91 141 111 121)(92 160 112 140)(93 139 113 159)(94 158 114 138)(95 137 115 157)(96 156 116 136)(97 135 117 155)(98 154 118 134)(99 133 119 153)(100 152 120 132)
G:=sub<Sym(160)| (1,83,59,160)(2,141,60,104)(3,85,61,122)(4,143,62,106)(5,87,63,124)(6,145,64,108)(7,89,65,126)(8,147,66,110)(9,91,67,128)(10,149,68,112)(11,93,69,130)(12,151,70,114)(13,95,71,132)(14,153,72,116)(15,97,73,134)(16,155,74,118)(17,99,75,136)(18,157,76,120)(19,101,77,138)(20,159,78,82)(21,103,79,140)(22,121,80,84)(23,105,41,142)(24,123,42,86)(25,107,43,144)(26,125,44,88)(27,109,45,146)(28,127,46,90)(29,111,47,148)(30,129,48,92)(31,113,49,150)(32,131,50,94)(33,115,51,152)(34,133,52,96)(35,117,53,154)(36,135,54,98)(37,119,55,156)(38,137,56,100)(39,81,57,158)(40,139,58,102), (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), (1,10,21,30)(2,29,22,9)(3,8,23,28)(4,27,24,7)(5,6,25,26)(11,40,31,20)(12,19,32,39)(13,38,33,18)(14,17,34,37)(15,36,35,16)(41,46,61,66)(42,65,62,45)(43,44,63,64)(47,80,67,60)(48,59,68,79)(49,78,69,58)(50,57,70,77)(51,76,71,56)(52,55,72,75)(53,74,73,54)(81,151,101,131)(82,130,102,150)(83,149,103,129)(84,128,104,148)(85,147,105,127)(86,126,106,146)(87,145,107,125)(88,124,108,144)(89,143,109,123)(90,122,110,142)(91,141,111,121)(92,160,112,140)(93,139,113,159)(94,158,114,138)(95,137,115,157)(96,156,116,136)(97,135,117,155)(98,154,118,134)(99,133,119,153)(100,152,120,132)>;
G:=Group( (1,83,59,160)(2,141,60,104)(3,85,61,122)(4,143,62,106)(5,87,63,124)(6,145,64,108)(7,89,65,126)(8,147,66,110)(9,91,67,128)(10,149,68,112)(11,93,69,130)(12,151,70,114)(13,95,71,132)(14,153,72,116)(15,97,73,134)(16,155,74,118)(17,99,75,136)(18,157,76,120)(19,101,77,138)(20,159,78,82)(21,103,79,140)(22,121,80,84)(23,105,41,142)(24,123,42,86)(25,107,43,144)(26,125,44,88)(27,109,45,146)(28,127,46,90)(29,111,47,148)(30,129,48,92)(31,113,49,150)(32,131,50,94)(33,115,51,152)(34,133,52,96)(35,117,53,154)(36,135,54,98)(37,119,55,156)(38,137,56,100)(39,81,57,158)(40,139,58,102), (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), (1,10,21,30)(2,29,22,9)(3,8,23,28)(4,27,24,7)(5,6,25,26)(11,40,31,20)(12,19,32,39)(13,38,33,18)(14,17,34,37)(15,36,35,16)(41,46,61,66)(42,65,62,45)(43,44,63,64)(47,80,67,60)(48,59,68,79)(49,78,69,58)(50,57,70,77)(51,76,71,56)(52,55,72,75)(53,74,73,54)(81,151,101,131)(82,130,102,150)(83,149,103,129)(84,128,104,148)(85,147,105,127)(86,126,106,146)(87,145,107,125)(88,124,108,144)(89,143,109,123)(90,122,110,142)(91,141,111,121)(92,160,112,140)(93,139,113,159)(94,158,114,138)(95,137,115,157)(96,156,116,136)(97,135,117,155)(98,154,118,134)(99,133,119,153)(100,152,120,132) );
G=PermutationGroup([[(1,83,59,160),(2,141,60,104),(3,85,61,122),(4,143,62,106),(5,87,63,124),(6,145,64,108),(7,89,65,126),(8,147,66,110),(9,91,67,128),(10,149,68,112),(11,93,69,130),(12,151,70,114),(13,95,71,132),(14,153,72,116),(15,97,73,134),(16,155,74,118),(17,99,75,136),(18,157,76,120),(19,101,77,138),(20,159,78,82),(21,103,79,140),(22,121,80,84),(23,105,41,142),(24,123,42,86),(25,107,43,144),(26,125,44,88),(27,109,45,146),(28,127,46,90),(29,111,47,148),(30,129,48,92),(31,113,49,150),(32,131,50,94),(33,115,51,152),(34,133,52,96),(35,117,53,154),(36,135,54,98),(37,119,55,156),(38,137,56,100),(39,81,57,158),(40,139,58,102)], [(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)], [(1,10,21,30),(2,29,22,9),(3,8,23,28),(4,27,24,7),(5,6,25,26),(11,40,31,20),(12,19,32,39),(13,38,33,18),(14,17,34,37),(15,36,35,16),(41,46,61,66),(42,65,62,45),(43,44,63,64),(47,80,67,60),(48,59,68,79),(49,78,69,58),(50,57,70,77),(51,76,71,56),(52,55,72,75),(53,74,73,54),(81,151,101,131),(82,130,102,150),(83,149,103,129),(84,128,104,148),(85,147,105,127),(86,126,106,146),(87,145,107,125),(88,124,108,144),(89,143,109,123),(90,122,110,142),(91,141,111,121),(92,160,112,140),(93,139,113,159),(94,158,114,138),(95,137,115,157),(96,156,116,136),(97,135,117,155),(98,154,118,134),(99,133,119,153),(100,152,120,132)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 40 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C4○D8 | D20 | D40⋊7C2 | C8.C22 | D4×D5 | Q8⋊2D5 | C8.D10 |
| kernel | C42.36D10 | C20.44D4 | D20⋊5C4 | C5×C4⋊C8 | C4×Dic10 | C4.D20 | C2×C40⋊C2 | C2×Dic20 | Dic10 | C2×C20 | C4⋊C8 | C20 | C42 | C2×C8 | C10 | C2×C4 | C2 | C10 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 16 | 1 | 2 | 2 | 4 |
Matrix representation of C42.36D10 ►in GL4(𝔽41) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 33 | 5 |
| 0 | 0 | 28 | 8 |
| 39 | 32 | 0 | 0 |
| 37 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 31 | 23 | 0 | 0 |
| 33 | 39 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 5 | 1 |
| 2 | 23 | 0 | 0 |
| 39 | 39 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,33,28,0,0,5,8],[39,37,0,0,32,2,0,0,0,0,1,0,0,0,0,1],[31,33,0,0,23,39,0,0,0,0,40,5,0,0,0,1],[2,39,0,0,23,39,0,0,0,0,40,0,0,0,0,40] >;
C42.36D10 in GAP, Magma, Sage, TeX
C_4^2._{36}D_{10} % in TeX
G:=Group("C4^2.36D10"); // GroupNames label
G:=SmallGroup(320,472);
// by ID
G=gap.SmallGroup(320,472);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,219,58,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b,d^2=b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b*c^9>;
// generators/relations