metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.187D10, C20⋊3C8⋊7C2, C4⋊C4.6Dic5, C10.61(C8○D4), (C4×C20).19C22, C42⋊C2.8D5, C22⋊C4.3Dic5, C20.250(C4○D4), C4.134(C4○D20), C23.8(C2×Dic5), (C2×C20).847C23, C42.D5⋊22C2, (C22×C4).109D10, C2.5(D4.Dic5), C20.55D4.17C2, C5⋊7(C42.7C22), C10.65(C42⋊C2), (C22×C20).373C22, C22.44(C22×Dic5), C2.10(C23.21D10), (C4×C5⋊2C8)⋊26C2, (C5×C4⋊C4).22C4, (C2×C20).334(C2×C4), (C5×C22⋊C4).12C4, (C2×C4).19(C2×Dic5), (C5×C42⋊C2).9C2, (C2×C4).789(C22×D5), (C22×C10).126(C2×C4), (C2×C10).285(C22×C4), (C2×C5⋊2C8).322C22, SmallGroup(320,627)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.187D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2b, ab=ba, cac-1=dad-1=ab2, bc=cb, bd=db, dcd-1=a2c-1 >
Subgroups: 206 in 96 conjugacy classes, 55 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C20, C20, C2×C10, C2×C10, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C42⋊C2, C5⋊2C8, C2×C20, C2×C20, C2×C20, C22×C10, C42.7C22, C2×C5⋊2C8, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C4×C5⋊2C8, C42.D5, C20⋊3C8, C20.55D4, C5×C42⋊C2, C42.187D10
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C4○D4, Dic5, D10, C42⋊C2, C8○D4, C2×Dic5, C22×D5, C42.7C22, C4○D20, C22×Dic5, C23.21D10, D4.Dic5, C42.187D10
►On 160 points
Generators in S160
(1 53 13 41)(2 59 14 47)(3 55 15 43)(4 51 11 49)(5 57 12 45)(6 56 16 44)(7 52 17 50)(8 58 18 46)(9 54 19 42)(10 60 20 48)(21 71 31 61)(22 77 32 67)(23 73 33 63)(24 79 34 69)(25 75 35 65)(26 76 36 66)(27 72 37 62)(28 78 38 68)(29 74 39 64)(30 80 40 70)(81 143 86 140)(82 136 87 149)(83 145 88 132)(84 138 89 141)(85 147 90 134)(91 127 154 122)(92 120 155 115)(93 129 156 124)(94 112 157 117)(95 121 158 126)(96 114 159 119)(97 123 160 128)(98 116 151 111)(99 125 152 130)(100 118 153 113)(101 137 106 150)(102 146 107 133)(103 139 108 142)(104 148 109 135)(105 131 110 144)
(1 28 8 23)(2 29 9 24)(3 30 10 25)(4 26 6 21)(5 27 7 22)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 68 46 63)(42 69 47 64)(43 70 48 65)(44 61 49 66)(45 62 50 67)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)(81 120 109 128)(82 111 110 129)(83 112 101 130)(84 113 102 121)(85 114 103 122)(86 115 104 123)(87 116 105 124)(88 117 106 125)(89 118 107 126)(90 119 108 127)(91 147 159 139)(92 148 160 140)(93 149 151 131)(94 150 152 132)(95 141 153 133)(96 142 154 134)(97 143 155 135)(98 144 156 136)(99 145 157 137)(100 146 158 138)
(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)
(1 129 38 87 8 111 33 105)(2 123 39 81 9 115 34 109)(3 127 40 85 10 119 35 103)(4 121 36 89 6 113 31 107)(5 125 37 83 7 117 32 101)(11 126 26 84 16 118 21 102)(12 130 27 88 17 112 22 106)(13 124 28 82 18 116 23 110)(14 128 29 86 19 120 24 104)(15 122 30 90 20 114 25 108)(41 151 78 144 46 93 73 136)(42 97 79 140 47 155 74 148)(43 159 80 142 48 91 75 134)(44 95 71 138 49 153 76 146)(45 157 72 150 50 99 77 132)(51 100 66 133 56 158 61 141)(52 152 67 145 57 94 62 137)(53 98 68 131 58 156 63 149)(54 160 69 143 59 92 64 135)(55 96 70 139 60 154 65 147)
G:=sub<Sym(160)| (1,53,13,41)(2,59,14,47)(3,55,15,43)(4,51,11,49)(5,57,12,45)(6,56,16,44)(7,52,17,50)(8,58,18,46)(9,54,19,42)(10,60,20,48)(21,71,31,61)(22,77,32,67)(23,73,33,63)(24,79,34,69)(25,75,35,65)(26,76,36,66)(27,72,37,62)(28,78,38,68)(29,74,39,64)(30,80,40,70)(81,143,86,140)(82,136,87,149)(83,145,88,132)(84,138,89,141)(85,147,90,134)(91,127,154,122)(92,120,155,115)(93,129,156,124)(94,112,157,117)(95,121,158,126)(96,114,159,119)(97,123,160,128)(98,116,151,111)(99,125,152,130)(100,118,153,113)(101,137,106,150)(102,146,107,133)(103,139,108,142)(104,148,109,135)(105,131,110,144), (1,28,8,23)(2,29,9,24)(3,30,10,25)(4,26,6,21)(5,27,7,22)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,68,46,63)(42,69,47,64)(43,70,48,65)(44,61,49,66)(45,62,50,67)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75)(81,120,109,128)(82,111,110,129)(83,112,101,130)(84,113,102,121)(85,114,103,122)(86,115,104,123)(87,116,105,124)(88,117,106,125)(89,118,107,126)(90,119,108,127)(91,147,159,139)(92,148,160,140)(93,149,151,131)(94,150,152,132)(95,141,153,133)(96,142,154,134)(97,143,155,135)(98,144,156,136)(99,145,157,137)(100,146,158,138), (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), (1,129,38,87,8,111,33,105)(2,123,39,81,9,115,34,109)(3,127,40,85,10,119,35,103)(4,121,36,89,6,113,31,107)(5,125,37,83,7,117,32,101)(11,126,26,84,16,118,21,102)(12,130,27,88,17,112,22,106)(13,124,28,82,18,116,23,110)(14,128,29,86,19,120,24,104)(15,122,30,90,20,114,25,108)(41,151,78,144,46,93,73,136)(42,97,79,140,47,155,74,148)(43,159,80,142,48,91,75,134)(44,95,71,138,49,153,76,146)(45,157,72,150,50,99,77,132)(51,100,66,133,56,158,61,141)(52,152,67,145,57,94,62,137)(53,98,68,131,58,156,63,149)(54,160,69,143,59,92,64,135)(55,96,70,139,60,154,65,147)>;
G:=Group( (1,53,13,41)(2,59,14,47)(3,55,15,43)(4,51,11,49)(5,57,12,45)(6,56,16,44)(7,52,17,50)(8,58,18,46)(9,54,19,42)(10,60,20,48)(21,71,31,61)(22,77,32,67)(23,73,33,63)(24,79,34,69)(25,75,35,65)(26,76,36,66)(27,72,37,62)(28,78,38,68)(29,74,39,64)(30,80,40,70)(81,143,86,140)(82,136,87,149)(83,145,88,132)(84,138,89,141)(85,147,90,134)(91,127,154,122)(92,120,155,115)(93,129,156,124)(94,112,157,117)(95,121,158,126)(96,114,159,119)(97,123,160,128)(98,116,151,111)(99,125,152,130)(100,118,153,113)(101,137,106,150)(102,146,107,133)(103,139,108,142)(104,148,109,135)(105,131,110,144), (1,28,8,23)(2,29,9,24)(3,30,10,25)(4,26,6,21)(5,27,7,22)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,68,46,63)(42,69,47,64)(43,70,48,65)(44,61,49,66)(45,62,50,67)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75)(81,120,109,128)(82,111,110,129)(83,112,101,130)(84,113,102,121)(85,114,103,122)(86,115,104,123)(87,116,105,124)(88,117,106,125)(89,118,107,126)(90,119,108,127)(91,147,159,139)(92,148,160,140)(93,149,151,131)(94,150,152,132)(95,141,153,133)(96,142,154,134)(97,143,155,135)(98,144,156,136)(99,145,157,137)(100,146,158,138), (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), (1,129,38,87,8,111,33,105)(2,123,39,81,9,115,34,109)(3,127,40,85,10,119,35,103)(4,121,36,89,6,113,31,107)(5,125,37,83,7,117,32,101)(11,126,26,84,16,118,21,102)(12,130,27,88,17,112,22,106)(13,124,28,82,18,116,23,110)(14,128,29,86,19,120,24,104)(15,122,30,90,20,114,25,108)(41,151,78,144,46,93,73,136)(42,97,79,140,47,155,74,148)(43,159,80,142,48,91,75,134)(44,95,71,138,49,153,76,146)(45,157,72,150,50,99,77,132)(51,100,66,133,56,158,61,141)(52,152,67,145,57,94,62,137)(53,98,68,131,58,156,63,149)(54,160,69,143,59,92,64,135)(55,96,70,139,60,154,65,147) );
G=PermutationGroup([[(1,53,13,41),(2,59,14,47),(3,55,15,43),(4,51,11,49),(5,57,12,45),(6,56,16,44),(7,52,17,50),(8,58,18,46),(9,54,19,42),(10,60,20,48),(21,71,31,61),(22,77,32,67),(23,73,33,63),(24,79,34,69),(25,75,35,65),(26,76,36,66),(27,72,37,62),(28,78,38,68),(29,74,39,64),(30,80,40,70),(81,143,86,140),(82,136,87,149),(83,145,88,132),(84,138,89,141),(85,147,90,134),(91,127,154,122),(92,120,155,115),(93,129,156,124),(94,112,157,117),(95,121,158,126),(96,114,159,119),(97,123,160,128),(98,116,151,111),(99,125,152,130),(100,118,153,113),(101,137,106,150),(102,146,107,133),(103,139,108,142),(104,148,109,135),(105,131,110,144)], [(1,28,8,23),(2,29,9,24),(3,30,10,25),(4,26,6,21),(5,27,7,22),(11,36,16,31),(12,37,17,32),(13,38,18,33),(14,39,19,34),(15,40,20,35),(41,68,46,63),(42,69,47,64),(43,70,48,65),(44,61,49,66),(45,62,50,67),(51,76,56,71),(52,77,57,72),(53,78,58,73),(54,79,59,74),(55,80,60,75),(81,120,109,128),(82,111,110,129),(83,112,101,130),(84,113,102,121),(85,114,103,122),(86,115,104,123),(87,116,105,124),(88,117,106,125),(89,118,107,126),(90,119,108,127),(91,147,159,139),(92,148,160,140),(93,149,151,131),(94,150,152,132),(95,141,153,133),(96,142,154,134),(97,143,155,135),(98,144,156,136),(99,145,157,137),(100,146,158,138)], [(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)], [(1,129,38,87,8,111,33,105),(2,123,39,81,9,115,34,109),(3,127,40,85,10,119,35,103),(4,121,36,89,6,113,31,107),(5,125,37,83,7,117,32,101),(11,126,26,84,16,118,21,102),(12,130,27,88,17,112,22,106),(13,124,28,82,18,116,23,110),(14,128,29,86,19,120,24,104),(15,122,30,90,20,114,25,108),(41,151,78,144,46,93,73,136),(42,97,79,140,47,155,74,148),(43,159,80,142,48,91,75,134),(44,95,71,138,49,153,76,146),(45,157,72,150,50,99,77,132),(51,100,66,133,56,158,61,141),(52,152,67,145,57,94,62,137),(53,98,68,131,58,156,63,149),(54,160,69,143,59,92,64,135),(55,96,70,139,60,154,65,147)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 5A | 5B | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| 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 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D5 | C4○D4 | D10 | Dic5 | Dic5 | D10 | C8○D4 | C4○D20 | D4.Dic5 |
| kernel | C42.187D10 | C4×C5⋊2C8 | C42.D5 | C20⋊3C8 | C20.55D4 | C5×C42⋊C2 | C5×C22⋊C4 | C5×C4⋊C4 | C42⋊C2 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 4 | 4 | 4 | 4 | 2 | 8 | 16 | 8 |
Matrix representation of C42.187D10 ►in GL6(𝔽41)
| 9 | 0 | 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 | 40 | 23 |
| 0 | 0 | 0 | 0 | 32 | 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 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 23 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 0 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 9 | 40 |
| 23 | 39 | 0 | 0 | 0 | 0 |
| 18 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 38 | 0 | 0 |
| 0 | 0 | 25 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 38 | 28 |
| 0 | 0 | 0 | 0 | 0 | 3 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,32,0,0,0,0,23,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,23,0,0,0,0,0,40,0,0,0,0,0,0,37,7,0,0,0,0,0,10,0,0,0,0,0,0,1,9,0,0,0,0,0,40],[23,18,0,0,0,0,39,18,0,0,0,0,0,0,19,25,0,0,0,0,38,22,0,0,0,0,0,0,38,0,0,0,0,0,28,3] >;
C42.187D10 in GAP, Magma, Sage, TeX
C_4^2._{187}D_{10} % in TeX
G:=Group("C4^2.187D10"); // GroupNames label
G:=SmallGroup(320,627);
// by ID
G=gap.SmallGroup(320,627);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,422,387,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2*b,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^-1>;
// generators/relations