metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.89D10, C10.472- 1+4, C20⋊2Q8⋊5C2, C4⋊C4.267D10, (C4×Dic10)⋊7C2, C20.6Q8⋊3C2, (C2×C10).62C24, (C4×C20).22C22, C22⋊C4.90D10, C4.119(C4○D20), C20.235(C4○D4), (C2×C20).141C23, Dic5.Q8⋊4C2, C42⋊C2.12D5, (C22×C4).186D10, C4⋊Dic5.31C22, C22.95(C23×D5), C23.83(C22×D5), C20.48D4.18C2, C23.D5.3C22, (C2×Dic5).21C23, C23.D10.1C2, C2.6(D4.10D10), (C22×C10).132C23, (C22×C20).307C22, C5⋊1(C22.35C24), (C4×Dic5).214C22, C10.D4.74C22, (C2×Dic10).237C22, C10.27(C2×C4○D4), C2.29(C2×C4○D20), (C5×C4⋊C4).303C22, (C2×C4).269(C22×D5), (C5×C42⋊C2).13C2, (C5×C22⋊C4).111C22, SmallGroup(320,1190)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.89D10
G = < a,b,c,d | a4=b4=1, c10=a2, d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=b2c9 >
Subgroups: 542 in 192 conjugacy classes, 95 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C23, C10, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C42⋊2C2, C4⋊Q8, Dic10, C2×Dic5, C2×C20, C2×C20, C2×C20, C22×C10, C22.35C24, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×C20, C4×Dic10, C20⋊2Q8, C20.6Q8, C23.D10, Dic5.Q8, C20.48D4, C5×C42⋊C2, C42.89D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.35C24, C4○D20, C23×D5, C2×C4○D20, D4.10D10, C42.89D10
►On 160 points
Generators in S160
(1 16 11 6)(2 17 12 7)(3 18 13 8)(4 19 14 9)(5 20 15 10)(21 103 31 113)(22 104 32 114)(23 105 33 115)(24 106 34 116)(25 107 35 117)(26 108 36 118)(27 109 37 119)(28 110 38 120)(29 111 39 101)(30 112 40 102)(41 46 51 56)(42 47 52 57)(43 48 53 58)(44 49 54 59)(45 50 55 60)(61 66 71 76)(62 67 72 77)(63 68 73 78)(64 69 74 79)(65 70 75 80)(81 142 91 152)(82 143 92 153)(83 144 93 154)(84 145 94 155)(85 146 95 156)(86 147 96 157)(87 148 97 158)(88 149 98 159)(89 150 99 160)(90 151 100 141)(121 136 131 126)(122 137 132 127)(123 138 133 128)(124 139 134 129)(125 140 135 130)
(1 68 130 47)(2 79 131 58)(3 70 132 49)(4 61 133 60)(5 72 134 51)(6 63 135 42)(7 74 136 53)(8 65 137 44)(9 76 138 55)(10 67 139 46)(11 78 140 57)(12 69 121 48)(13 80 122 59)(14 71 123 50)(15 62 124 41)(16 73 125 52)(17 64 126 43)(18 75 127 54)(19 66 128 45)(20 77 129 56)(21 86 118 152)(22 97 119 143)(23 88 120 154)(24 99 101 145)(25 90 102 156)(26 81 103 147)(27 92 104 158)(28 83 105 149)(29 94 106 160)(30 85 107 151)(31 96 108 142)(32 87 109 153)(33 98 110 144)(34 89 111 155)(35 100 112 146)(36 91 113 157)(37 82 114 148)(38 93 115 159)(39 84 116 150)(40 95 117 141)
(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 37 130 114)(2 103 131 26)(3 35 132 112)(4 101 133 24)(5 33 134 110)(6 119 135 22)(7 31 136 108)(8 117 137 40)(9 29 138 106)(10 115 139 38)(11 27 140 104)(12 113 121 36)(13 25 122 102)(14 111 123 34)(15 23 124 120)(16 109 125 32)(17 21 126 118)(18 107 127 30)(19 39 128 116)(20 105 129 28)(41 144 62 98)(42 87 63 153)(43 142 64 96)(44 85 65 151)(45 160 66 94)(46 83 67 149)(47 158 68 92)(48 81 69 147)(49 156 70 90)(50 99 71 145)(51 154 72 88)(52 97 73 143)(53 152 74 86)(54 95 75 141)(55 150 76 84)(56 93 77 159)(57 148 78 82)(58 91 79 157)(59 146 80 100)(60 89 61 155)
G:=sub<Sym(160)| (1,16,11,6)(2,17,12,7)(3,18,13,8)(4,19,14,9)(5,20,15,10)(21,103,31,113)(22,104,32,114)(23,105,33,115)(24,106,34,116)(25,107,35,117)(26,108,36,118)(27,109,37,119)(28,110,38,120)(29,111,39,101)(30,112,40,102)(41,46,51,56)(42,47,52,57)(43,48,53,58)(44,49,54,59)(45,50,55,60)(61,66,71,76)(62,67,72,77)(63,68,73,78)(64,69,74,79)(65,70,75,80)(81,142,91,152)(82,143,92,153)(83,144,93,154)(84,145,94,155)(85,146,95,156)(86,147,96,157)(87,148,97,158)(88,149,98,159)(89,150,99,160)(90,151,100,141)(121,136,131,126)(122,137,132,127)(123,138,133,128)(124,139,134,129)(125,140,135,130), (1,68,130,47)(2,79,131,58)(3,70,132,49)(4,61,133,60)(5,72,134,51)(6,63,135,42)(7,74,136,53)(8,65,137,44)(9,76,138,55)(10,67,139,46)(11,78,140,57)(12,69,121,48)(13,80,122,59)(14,71,123,50)(15,62,124,41)(16,73,125,52)(17,64,126,43)(18,75,127,54)(19,66,128,45)(20,77,129,56)(21,86,118,152)(22,97,119,143)(23,88,120,154)(24,99,101,145)(25,90,102,156)(26,81,103,147)(27,92,104,158)(28,83,105,149)(29,94,106,160)(30,85,107,151)(31,96,108,142)(32,87,109,153)(33,98,110,144)(34,89,111,155)(35,100,112,146)(36,91,113,157)(37,82,114,148)(38,93,115,159)(39,84,116,150)(40,95,117,141), (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), (1,37,130,114)(2,103,131,26)(3,35,132,112)(4,101,133,24)(5,33,134,110)(6,119,135,22)(7,31,136,108)(8,117,137,40)(9,29,138,106)(10,115,139,38)(11,27,140,104)(12,113,121,36)(13,25,122,102)(14,111,123,34)(15,23,124,120)(16,109,125,32)(17,21,126,118)(18,107,127,30)(19,39,128,116)(20,105,129,28)(41,144,62,98)(42,87,63,153)(43,142,64,96)(44,85,65,151)(45,160,66,94)(46,83,67,149)(47,158,68,92)(48,81,69,147)(49,156,70,90)(50,99,71,145)(51,154,72,88)(52,97,73,143)(53,152,74,86)(54,95,75,141)(55,150,76,84)(56,93,77,159)(57,148,78,82)(58,91,79,157)(59,146,80,100)(60,89,61,155)>;
G:=Group( (1,16,11,6)(2,17,12,7)(3,18,13,8)(4,19,14,9)(5,20,15,10)(21,103,31,113)(22,104,32,114)(23,105,33,115)(24,106,34,116)(25,107,35,117)(26,108,36,118)(27,109,37,119)(28,110,38,120)(29,111,39,101)(30,112,40,102)(41,46,51,56)(42,47,52,57)(43,48,53,58)(44,49,54,59)(45,50,55,60)(61,66,71,76)(62,67,72,77)(63,68,73,78)(64,69,74,79)(65,70,75,80)(81,142,91,152)(82,143,92,153)(83,144,93,154)(84,145,94,155)(85,146,95,156)(86,147,96,157)(87,148,97,158)(88,149,98,159)(89,150,99,160)(90,151,100,141)(121,136,131,126)(122,137,132,127)(123,138,133,128)(124,139,134,129)(125,140,135,130), (1,68,130,47)(2,79,131,58)(3,70,132,49)(4,61,133,60)(5,72,134,51)(6,63,135,42)(7,74,136,53)(8,65,137,44)(9,76,138,55)(10,67,139,46)(11,78,140,57)(12,69,121,48)(13,80,122,59)(14,71,123,50)(15,62,124,41)(16,73,125,52)(17,64,126,43)(18,75,127,54)(19,66,128,45)(20,77,129,56)(21,86,118,152)(22,97,119,143)(23,88,120,154)(24,99,101,145)(25,90,102,156)(26,81,103,147)(27,92,104,158)(28,83,105,149)(29,94,106,160)(30,85,107,151)(31,96,108,142)(32,87,109,153)(33,98,110,144)(34,89,111,155)(35,100,112,146)(36,91,113,157)(37,82,114,148)(38,93,115,159)(39,84,116,150)(40,95,117,141), (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), (1,37,130,114)(2,103,131,26)(3,35,132,112)(4,101,133,24)(5,33,134,110)(6,119,135,22)(7,31,136,108)(8,117,137,40)(9,29,138,106)(10,115,139,38)(11,27,140,104)(12,113,121,36)(13,25,122,102)(14,111,123,34)(15,23,124,120)(16,109,125,32)(17,21,126,118)(18,107,127,30)(19,39,128,116)(20,105,129,28)(41,144,62,98)(42,87,63,153)(43,142,64,96)(44,85,65,151)(45,160,66,94)(46,83,67,149)(47,158,68,92)(48,81,69,147)(49,156,70,90)(50,99,71,145)(51,154,72,88)(52,97,73,143)(53,152,74,86)(54,95,75,141)(55,150,76,84)(56,93,77,159)(57,148,78,82)(58,91,79,157)(59,146,80,100)(60,89,61,155) );
G=PermutationGroup([[(1,16,11,6),(2,17,12,7),(3,18,13,8),(4,19,14,9),(5,20,15,10),(21,103,31,113),(22,104,32,114),(23,105,33,115),(24,106,34,116),(25,107,35,117),(26,108,36,118),(27,109,37,119),(28,110,38,120),(29,111,39,101),(30,112,40,102),(41,46,51,56),(42,47,52,57),(43,48,53,58),(44,49,54,59),(45,50,55,60),(61,66,71,76),(62,67,72,77),(63,68,73,78),(64,69,74,79),(65,70,75,80),(81,142,91,152),(82,143,92,153),(83,144,93,154),(84,145,94,155),(85,146,95,156),(86,147,96,157),(87,148,97,158),(88,149,98,159),(89,150,99,160),(90,151,100,141),(121,136,131,126),(122,137,132,127),(123,138,133,128),(124,139,134,129),(125,140,135,130)], [(1,68,130,47),(2,79,131,58),(3,70,132,49),(4,61,133,60),(5,72,134,51),(6,63,135,42),(7,74,136,53),(8,65,137,44),(9,76,138,55),(10,67,139,46),(11,78,140,57),(12,69,121,48),(13,80,122,59),(14,71,123,50),(15,62,124,41),(16,73,125,52),(17,64,126,43),(18,75,127,54),(19,66,128,45),(20,77,129,56),(21,86,118,152),(22,97,119,143),(23,88,120,154),(24,99,101,145),(25,90,102,156),(26,81,103,147),(27,92,104,158),(28,83,105,149),(29,94,106,160),(30,85,107,151),(31,96,108,142),(32,87,109,153),(33,98,110,144),(34,89,111,155),(35,100,112,146),(36,91,113,157),(37,82,114,148),(38,93,115,159),(39,84,116,150),(40,95,117,141)], [(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)], [(1,37,130,114),(2,103,131,26),(3,35,132,112),(4,101,133,24),(5,33,134,110),(6,119,135,22),(7,31,136,108),(8,117,137,40),(9,29,138,106),(10,115,139,38),(11,27,140,104),(12,113,121,36),(13,25,122,102),(14,111,123,34),(15,23,124,120),(16,109,125,32),(17,21,126,118),(18,107,127,30),(19,39,128,116),(20,105,129,28),(41,144,62,98),(42,87,63,153),(43,142,64,96),(44,85,65,151),(45,160,66,94),(46,83,67,149),(47,158,68,92),(48,81,69,147),(49,156,70,90),(50,99,71,145),(51,154,72,88),(52,97,73,143),(53,152,74,86),(54,95,75,141),(55,150,76,84),(56,93,77,159),(57,148,78,82),(58,91,79,157),(59,146,80,100),(60,89,61,155)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4F | 4G | 4H | 4I | 4J | ··· | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | C4○D20 | 2- 1+4 | D4.10D10 |
| kernel | C42.89D10 | C4×Dic10 | C20⋊2Q8 | C20.6Q8 | C23.D10 | Dic5.Q8 | C20.48D4 | C5×C42⋊C2 | C42⋊C2 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C4 | C10 | C2 |
| # reps | 1 | 2 | 1 | 1 | 4 | 4 | 2 | 1 | 2 | 4 | 4 | 4 | 4 | 2 | 16 | 2 | 8 |
Matrix representation of C42.89D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 28 | 0 | 0 |
| 0 | 0 | 38 | 40 | 0 | 0 |
| 0 | 0 | 35 | 6 | 30 | 28 |
| 0 | 0 | 19 | 32 | 22 | 11 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 23 | 10 | 7 |
| 0 | 0 | 6 | 30 | 10 | 33 |
| 0 | 0 | 2 | 21 | 36 | 40 |
| 0 | 0 | 36 | 38 | 8 | 15 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 16 | 0 | 0 |
| 0 | 0 | 10 | 8 | 0 | 0 |
| 0 | 0 | 35 | 34 | 27 | 25 |
| 0 | 0 | 28 | 10 | 5 | 32 |
| 0 | 18 | 0 | 0 | 0 | 0 |
| 25 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 26 | 37 | 19 |
| 0 | 0 | 34 | 20 | 10 | 31 |
| 0 | 0 | 17 | 13 | 27 | 9 |
| 0 | 0 | 7 | 6 | 17 | 29 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,38,35,19,0,0,28,40,6,32,0,0,0,0,30,22,0,0,0,0,28,11],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,6,2,36,0,0,23,30,21,38,0,0,10,10,36,8,0,0,7,33,40,15],[16,0,0,0,0,0,0,23,0,0,0,0,0,0,15,10,35,28,0,0,16,8,34,10,0,0,0,0,27,5,0,0,0,0,25,32],[0,25,0,0,0,0,18,0,0,0,0,0,0,0,6,34,17,7,0,0,26,20,13,6,0,0,37,10,27,17,0,0,19,31,9,29] >;
C42.89D10 in GAP, Magma, Sage, TeX
C_4^2._{89}D_{10} % in TeX
G:=Group("C4^2.89D10"); // GroupNames label
G:=SmallGroup(320,1190);
// by ID
G=gap.SmallGroup(320,1190);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,477,232,100,675,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^9>;
// generators/relations