metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.80D10, C4⋊Q8⋊4D5, C4.18(D4×D5), C5⋊2C8.8D4, C20.38(C2×D4), C5⋊3(C8.2D4), (C2×C20).295D4, (C2×Q8).46D10, C10.24(C4⋊1D4), C2.15(C20⋊D4), (C4×C20).133C22, (C2×C20).404C23, C4.D20.10C2, C42.D5⋊14C2, (Q8×C10).64C22, (C2×D20).112C22, C10.96(C8.C22), C2.17(C20.C23), (C2×Dic10).117C22, (C5×C4⋊Q8)⋊4C2, (C2×Q8⋊D5).7C2, (C2×C5⋊Q16)⋊15C2, (C2×C10).535(C2×D4), (C2×C4).73(C5⋊D4), (C2×C4).501(C22×D5), C22.207(C2×C5⋊D4), (C2×C5⋊2C8).137C22, SmallGroup(320,713)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.80D10
G = < a,b,c,d | a4=b4=1, c10=b2, d2=b, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c9 >
Subgroups: 510 in 124 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, D10, C2×C10, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16, C2×Q16, C5⋊2C8, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C2×C20, C5×Q8, C22×D5, C8.2D4, C2×C5⋊2C8, D10⋊C4, Q8⋊D5, C5⋊Q16, C4×C20, C5×C4⋊C4, C2×Dic10, C2×D20, Q8×C10, C42.D5, C4.D20, C2×Q8⋊D5, C2×C5⋊Q16, C5×C4⋊Q8, C42.80D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4⋊1D4, C8.C22, C5⋊D4, C22×D5, C8.2D4, D4×D5, C2×C5⋊D4, C20⋊D4, C20.C23, C42.80D10
►On 160 points
Generators in S160
(1 49 84 107)(2 108 85 50)(3 51 86 109)(4 110 87 52)(5 53 88 111)(6 112 89 54)(7 55 90 113)(8 114 91 56)(9 57 92 115)(10 116 93 58)(11 59 94 117)(12 118 95 60)(13 41 96 119)(14 120 97 42)(15 43 98 101)(16 102 99 44)(17 45 100 103)(18 104 81 46)(19 47 82 105)(20 106 83 48)(21 72 153 138)(22 139 154 73)(23 74 155 140)(24 121 156 75)(25 76 157 122)(26 123 158 77)(27 78 159 124)(28 125 160 79)(29 80 141 126)(30 127 142 61)(31 62 143 128)(32 129 144 63)(33 64 145 130)(34 131 146 65)(35 66 147 132)(36 133 148 67)(37 68 149 134)(38 135 150 69)(39 70 151 136)(40 137 152 71)
(1 23 11 33)(2 34 12 24)(3 25 13 35)(4 36 14 26)(5 27 15 37)(6 38 16 28)(7 29 17 39)(8 40 18 30)(9 31 19 21)(10 22 20 32)(41 66 51 76)(42 77 52 67)(43 68 53 78)(44 79 54 69)(45 70 55 80)(46 61 56 71)(47 72 57 62)(48 63 58 73)(49 74 59 64)(50 65 60 75)(81 142 91 152)(82 153 92 143)(83 144 93 154)(84 155 94 145)(85 146 95 156)(86 157 96 147)(87 148 97 158)(88 159 98 149)(89 150 99 160)(90 141 100 151)(101 134 111 124)(102 125 112 135)(103 136 113 126)(104 127 114 137)(105 138 115 128)(106 129 116 139)(107 140 117 130)(108 131 118 121)(109 122 119 132)(110 133 120 123)
(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 23 22 11 20 33 32)(2 31 34 19 12 21 24 9)(3 8 25 40 13 18 35 30)(4 29 36 17 14 39 26 7)(5 6 27 38 15 16 37 28)(41 114 66 137 51 104 76 127)(42 126 77 103 52 136 67 113)(43 112 68 135 53 102 78 125)(44 124 79 101 54 134 69 111)(45 110 70 133 55 120 80 123)(46 122 61 119 56 132 71 109)(47 108 72 131 57 118 62 121)(48 140 63 117 58 130 73 107)(49 106 74 129 59 116 64 139)(50 138 65 115 60 128 75 105)(81 147 142 86 91 157 152 96)(82 95 153 156 92 85 143 146)(83 145 144 84 93 155 154 94)(87 141 148 100 97 151 158 90)(88 89 159 150 98 99 149 160)
G:=sub<Sym(160)| (1,49,84,107)(2,108,85,50)(3,51,86,109)(4,110,87,52)(5,53,88,111)(6,112,89,54)(7,55,90,113)(8,114,91,56)(9,57,92,115)(10,116,93,58)(11,59,94,117)(12,118,95,60)(13,41,96,119)(14,120,97,42)(15,43,98,101)(16,102,99,44)(17,45,100,103)(18,104,81,46)(19,47,82,105)(20,106,83,48)(21,72,153,138)(22,139,154,73)(23,74,155,140)(24,121,156,75)(25,76,157,122)(26,123,158,77)(27,78,159,124)(28,125,160,79)(29,80,141,126)(30,127,142,61)(31,62,143,128)(32,129,144,63)(33,64,145,130)(34,131,146,65)(35,66,147,132)(36,133,148,67)(37,68,149,134)(38,135,150,69)(39,70,151,136)(40,137,152,71), (1,23,11,33)(2,34,12,24)(3,25,13,35)(4,36,14,26)(5,27,15,37)(6,38,16,28)(7,29,17,39)(8,40,18,30)(9,31,19,21)(10,22,20,32)(41,66,51,76)(42,77,52,67)(43,68,53,78)(44,79,54,69)(45,70,55,80)(46,61,56,71)(47,72,57,62)(48,63,58,73)(49,74,59,64)(50,65,60,75)(81,142,91,152)(82,153,92,143)(83,144,93,154)(84,155,94,145)(85,146,95,156)(86,157,96,147)(87,148,97,158)(88,159,98,149)(89,150,99,160)(90,141,100,151)(101,134,111,124)(102,125,112,135)(103,136,113,126)(104,127,114,137)(105,138,115,128)(106,129,116,139)(107,140,117,130)(108,131,118,121)(109,122,119,132)(110,133,120,123), (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), (1,10,23,22,11,20,33,32)(2,31,34,19,12,21,24,9)(3,8,25,40,13,18,35,30)(4,29,36,17,14,39,26,7)(5,6,27,38,15,16,37,28)(41,114,66,137,51,104,76,127)(42,126,77,103,52,136,67,113)(43,112,68,135,53,102,78,125)(44,124,79,101,54,134,69,111)(45,110,70,133,55,120,80,123)(46,122,61,119,56,132,71,109)(47,108,72,131,57,118,62,121)(48,140,63,117,58,130,73,107)(49,106,74,129,59,116,64,139)(50,138,65,115,60,128,75,105)(81,147,142,86,91,157,152,96)(82,95,153,156,92,85,143,146)(83,145,144,84,93,155,154,94)(87,141,148,100,97,151,158,90)(88,89,159,150,98,99,149,160)>;
G:=Group( (1,49,84,107)(2,108,85,50)(3,51,86,109)(4,110,87,52)(5,53,88,111)(6,112,89,54)(7,55,90,113)(8,114,91,56)(9,57,92,115)(10,116,93,58)(11,59,94,117)(12,118,95,60)(13,41,96,119)(14,120,97,42)(15,43,98,101)(16,102,99,44)(17,45,100,103)(18,104,81,46)(19,47,82,105)(20,106,83,48)(21,72,153,138)(22,139,154,73)(23,74,155,140)(24,121,156,75)(25,76,157,122)(26,123,158,77)(27,78,159,124)(28,125,160,79)(29,80,141,126)(30,127,142,61)(31,62,143,128)(32,129,144,63)(33,64,145,130)(34,131,146,65)(35,66,147,132)(36,133,148,67)(37,68,149,134)(38,135,150,69)(39,70,151,136)(40,137,152,71), (1,23,11,33)(2,34,12,24)(3,25,13,35)(4,36,14,26)(5,27,15,37)(6,38,16,28)(7,29,17,39)(8,40,18,30)(9,31,19,21)(10,22,20,32)(41,66,51,76)(42,77,52,67)(43,68,53,78)(44,79,54,69)(45,70,55,80)(46,61,56,71)(47,72,57,62)(48,63,58,73)(49,74,59,64)(50,65,60,75)(81,142,91,152)(82,153,92,143)(83,144,93,154)(84,155,94,145)(85,146,95,156)(86,157,96,147)(87,148,97,158)(88,159,98,149)(89,150,99,160)(90,141,100,151)(101,134,111,124)(102,125,112,135)(103,136,113,126)(104,127,114,137)(105,138,115,128)(106,129,116,139)(107,140,117,130)(108,131,118,121)(109,122,119,132)(110,133,120,123), (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), (1,10,23,22,11,20,33,32)(2,31,34,19,12,21,24,9)(3,8,25,40,13,18,35,30)(4,29,36,17,14,39,26,7)(5,6,27,38,15,16,37,28)(41,114,66,137,51,104,76,127)(42,126,77,103,52,136,67,113)(43,112,68,135,53,102,78,125)(44,124,79,101,54,134,69,111)(45,110,70,133,55,120,80,123)(46,122,61,119,56,132,71,109)(47,108,72,131,57,118,62,121)(48,140,63,117,58,130,73,107)(49,106,74,129,59,116,64,139)(50,138,65,115,60,128,75,105)(81,147,142,86,91,157,152,96)(82,95,153,156,92,85,143,146)(83,145,144,84,93,155,154,94)(87,141,148,100,97,151,158,90)(88,89,159,150,98,99,149,160) );
G=PermutationGroup([[(1,49,84,107),(2,108,85,50),(3,51,86,109),(4,110,87,52),(5,53,88,111),(6,112,89,54),(7,55,90,113),(8,114,91,56),(9,57,92,115),(10,116,93,58),(11,59,94,117),(12,118,95,60),(13,41,96,119),(14,120,97,42),(15,43,98,101),(16,102,99,44),(17,45,100,103),(18,104,81,46),(19,47,82,105),(20,106,83,48),(21,72,153,138),(22,139,154,73),(23,74,155,140),(24,121,156,75),(25,76,157,122),(26,123,158,77),(27,78,159,124),(28,125,160,79),(29,80,141,126),(30,127,142,61),(31,62,143,128),(32,129,144,63),(33,64,145,130),(34,131,146,65),(35,66,147,132),(36,133,148,67),(37,68,149,134),(38,135,150,69),(39,70,151,136),(40,137,152,71)], [(1,23,11,33),(2,34,12,24),(3,25,13,35),(4,36,14,26),(5,27,15,37),(6,38,16,28),(7,29,17,39),(8,40,18,30),(9,31,19,21),(10,22,20,32),(41,66,51,76),(42,77,52,67),(43,68,53,78),(44,79,54,69),(45,70,55,80),(46,61,56,71),(47,72,57,62),(48,63,58,73),(49,74,59,64),(50,65,60,75),(81,142,91,152),(82,153,92,143),(83,144,93,154),(84,155,94,145),(85,146,95,156),(86,157,96,147),(87,148,97,158),(88,159,98,149),(89,150,99,160),(90,141,100,151),(101,134,111,124),(102,125,112,135),(103,136,113,126),(104,127,114,137),(105,138,115,128),(106,129,116,139),(107,140,117,130),(108,131,118,121),(109,122,119,132),(110,133,120,123)], [(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)], [(1,10,23,22,11,20,33,32),(2,31,34,19,12,21,24,9),(3,8,25,40,13,18,35,30),(4,29,36,17,14,39,26,7),(5,6,27,38,15,16,37,28),(41,114,66,137,51,104,76,127),(42,126,77,103,52,136,67,113),(43,112,68,135,53,102,78,125),(44,124,79,101,54,134,69,111),(45,110,70,133,55,120,80,123),(46,122,61,119,56,132,71,109),(47,108,72,131,57,118,62,121),(48,140,63,117,58,130,73,107),(49,106,74,129,59,116,64,139),(50,138,65,115,60,128,75,105),(81,147,142,86,91,157,152,96),(82,95,153,156,92,85,143,146),(83,145,144,84,93,155,154,94),(87,141,148,100,97,151,158,90),(88,89,159,150,98,99,149,160)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 40 | 2 | 2 | 4 | 4 | 8 | 8 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | C5⋊D4 | C8.C22 | D4×D5 | C20.C23 |
| kernel | C42.80D10 | C42.D5 | C4.D20 | C2×Q8⋊D5 | C2×C5⋊Q16 | C5×C4⋊Q8 | C5⋊2C8 | C2×C20 | C4⋊Q8 | C42 | C2×Q8 | C2×C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 8 | 2 | 4 | 8 |
Matrix representation of C42.80D10 ►in GL6(𝔽41)
| 30 | 33 | 0 | 0 | 0 | 0 |
| 5 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 1 |
| 0 | 0 | 0 | 0 | 40 | 17 |
| 0 | 0 | 17 | 40 | 0 | 0 |
| 0 | 0 | 1 | 24 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 28 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 16 | 16 | 27 |
| 0 | 0 | 25 | 38 | 14 | 32 |
| 0 | 0 | 16 | 27 | 14 | 25 |
| 0 | 0 | 14 | 32 | 16 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 27 | 14 | 25 |
| 0 | 0 | 38 | 25 | 9 | 27 |
| 0 | 0 | 27 | 16 | 16 | 27 |
| 0 | 0 | 32 | 14 | 38 | 25 |
G:=sub<GL(6,GF(41))| [30,5,0,0,0,0,33,11,0,0,0,0,0,0,0,0,17,1,0,0,0,0,40,24,0,0,24,40,0,0,0,0,1,17,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0],[1,28,0,0,0,0,0,40,0,0,0,0,0,0,27,25,16,14,0,0,16,38,27,32,0,0,16,14,14,16,0,0,27,32,25,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,38,27,32,0,0,27,25,16,14,0,0,14,9,16,38,0,0,25,27,27,25] >;
C42.80D10 in GAP, Magma, Sage, TeX
C_4^2._{80}D_{10} % in TeX
G:=Group("C4^2.80D10"); // GroupNames label
G:=SmallGroup(320,713);
// by ID
G=gap.SmallGroup(320,713);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,555,184,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b^2,d^2=b,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^9>;
// generators/relations