metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.154D10, C10.302- 1+4, C20⋊Q8⋊37C2, C4⋊C4.210D10, C42.C2⋊10D5, (C4×Dic10)⋊49C2, D10⋊Q8.3C2, C4.Dic10⋊36C2, C42⋊2D5.1C2, Dic5⋊3Q8⋊37C2, (C2×C20).602C23, (C2×C10).240C24, (C4×C20).199C22, Dic5.19(C4○D4), Dic5.Q8⋊35C2, C4⋊Dic5.316C22, C22.261(C23×D5), D10⋊C4.42C22, C5⋊4(C22.35C24), (C2×Dic5).270C23, (C4×Dic5).236C22, (C22×D5).105C23, C2.59(D4.10D10), C2.31(Q8.10D10), (C2×Dic10).260C22, C10.D4.162C22, C2.91(D5×C4○D4), C4⋊C4⋊D5.2C2, C4⋊C4⋊7D5.13C2, C10.202(C2×C4○D4), (C5×C42.C2)⋊13C2, (C2×C4×D5).139C22, (C5×C4⋊C4).195C22, (C2×C4).205(C22×D5), SmallGroup(320,1368)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.154D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b, dcd-1=c9 >
Subgroups: 590 in 192 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, Dic5, C20, D10, C2×C10, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, Dic10, C4×D5, C2×Dic5, C2×C20, C22×D5, C22.35C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C4×Dic10, C42⋊2D5, Dic5⋊3Q8, C20⋊Q8, Dic5.Q8, C4.Dic10, C4⋊C4⋊7D5, D10⋊Q8, C4⋊C4⋊D5, C5×C42.C2, C42.154D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.35C24, C23×D5, Q8.10D10, D5×C4○D4, D4.10D10, C42.154D10
►On 160 points
Generators in S160
(1 87 11 97)(2 23 12 33)(3 89 13 99)(4 25 14 35)(5 91 15 81)(6 27 16 37)(7 93 17 83)(8 29 18 39)(9 95 19 85)(10 31 20 21)(22 72 32 62)(24 74 34 64)(26 76 36 66)(28 78 38 68)(30 80 40 70)(41 137 51 127)(42 142 52 152)(43 139 53 129)(44 144 54 154)(45 121 55 131)(46 146 56 156)(47 123 57 133)(48 148 58 158)(49 125 59 135)(50 150 60 160)(61 86 71 96)(63 88 73 98)(65 90 75 100)(67 92 77 82)(69 94 79 84)(101 147 111 157)(102 124 112 134)(103 149 113 159)(104 126 114 136)(105 151 115 141)(106 128 116 138)(107 153 117 143)(108 130 118 140)(109 155 119 145)(110 132 120 122)
(1 109 72 45)(2 46 73 110)(3 111 74 47)(4 48 75 112)(5 113 76 49)(6 50 77 114)(7 115 78 51)(8 52 79 116)(9 117 80 53)(10 54 61 118)(11 119 62 55)(12 56 63 120)(13 101 64 57)(14 58 65 102)(15 103 66 59)(16 60 67 104)(17 105 68 41)(18 42 69 106)(19 107 70 43)(20 44 71 108)(21 144 96 130)(22 131 97 145)(23 146 98 132)(24 133 99 147)(25 148 100 134)(26 135 81 149)(27 150 82 136)(28 137 83 151)(29 152 84 138)(30 139 85 153)(31 154 86 140)(32 121 87 155)(33 156 88 122)(34 123 89 157)(35 158 90 124)(36 125 91 159)(37 160 92 126)(38 127 93 141)(39 142 94 128)(40 129 95 143)
(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 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 97 31 87)(22 86 32 96)(23 95 33 85)(24 84 34 94)(25 93 35 83)(26 82 36 92)(27 91 37 81)(28 100 38 90)(29 89 39 99)(30 98 40 88)(41 112 51 102)(42 101 52 111)(43 110 53 120)(44 119 54 109)(45 108 55 118)(46 117 56 107)(47 106 57 116)(48 115 58 105)(49 104 59 114)(50 113 60 103)(61 62 71 72)(63 80 73 70)(64 69 74 79)(65 78 75 68)(66 67 76 77)(121 140 131 130)(122 129 132 139)(123 138 133 128)(124 127 134 137)(125 136 135 126)(141 148 151 158)(142 157 152 147)(143 146 153 156)(144 155 154 145)(149 160 159 150)
G:=sub<Sym(160)| (1,87,11,97)(2,23,12,33)(3,89,13,99)(4,25,14,35)(5,91,15,81)(6,27,16,37)(7,93,17,83)(8,29,18,39)(9,95,19,85)(10,31,20,21)(22,72,32,62)(24,74,34,64)(26,76,36,66)(28,78,38,68)(30,80,40,70)(41,137,51,127)(42,142,52,152)(43,139,53,129)(44,144,54,154)(45,121,55,131)(46,146,56,156)(47,123,57,133)(48,148,58,158)(49,125,59,135)(50,150,60,160)(61,86,71,96)(63,88,73,98)(65,90,75,100)(67,92,77,82)(69,94,79,84)(101,147,111,157)(102,124,112,134)(103,149,113,159)(104,126,114,136)(105,151,115,141)(106,128,116,138)(107,153,117,143)(108,130,118,140)(109,155,119,145)(110,132,120,122), (1,109,72,45)(2,46,73,110)(3,111,74,47)(4,48,75,112)(5,113,76,49)(6,50,77,114)(7,115,78,51)(8,52,79,116)(9,117,80,53)(10,54,61,118)(11,119,62,55)(12,56,63,120)(13,101,64,57)(14,58,65,102)(15,103,66,59)(16,60,67,104)(17,105,68,41)(18,42,69,106)(19,107,70,43)(20,44,71,108)(21,144,96,130)(22,131,97,145)(23,146,98,132)(24,133,99,147)(25,148,100,134)(26,135,81,149)(27,150,82,136)(28,137,83,151)(29,152,84,138)(30,139,85,153)(31,154,86,140)(32,121,87,155)(33,156,88,122)(34,123,89,157)(35,158,90,124)(36,125,91,159)(37,160,92,126)(38,127,93,141)(39,142,94,128)(40,129,95,143), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,97,31,87)(22,86,32,96)(23,95,33,85)(24,84,34,94)(25,93,35,83)(26,82,36,92)(27,91,37,81)(28,100,38,90)(29,89,39,99)(30,98,40,88)(41,112,51,102)(42,101,52,111)(43,110,53,120)(44,119,54,109)(45,108,55,118)(46,117,56,107)(47,106,57,116)(48,115,58,105)(49,104,59,114)(50,113,60,103)(61,62,71,72)(63,80,73,70)(64,69,74,79)(65,78,75,68)(66,67,76,77)(121,140,131,130)(122,129,132,139)(123,138,133,128)(124,127,134,137)(125,136,135,126)(141,148,151,158)(142,157,152,147)(143,146,153,156)(144,155,154,145)(149,160,159,150)>;
G:=Group( (1,87,11,97)(2,23,12,33)(3,89,13,99)(4,25,14,35)(5,91,15,81)(6,27,16,37)(7,93,17,83)(8,29,18,39)(9,95,19,85)(10,31,20,21)(22,72,32,62)(24,74,34,64)(26,76,36,66)(28,78,38,68)(30,80,40,70)(41,137,51,127)(42,142,52,152)(43,139,53,129)(44,144,54,154)(45,121,55,131)(46,146,56,156)(47,123,57,133)(48,148,58,158)(49,125,59,135)(50,150,60,160)(61,86,71,96)(63,88,73,98)(65,90,75,100)(67,92,77,82)(69,94,79,84)(101,147,111,157)(102,124,112,134)(103,149,113,159)(104,126,114,136)(105,151,115,141)(106,128,116,138)(107,153,117,143)(108,130,118,140)(109,155,119,145)(110,132,120,122), (1,109,72,45)(2,46,73,110)(3,111,74,47)(4,48,75,112)(5,113,76,49)(6,50,77,114)(7,115,78,51)(8,52,79,116)(9,117,80,53)(10,54,61,118)(11,119,62,55)(12,56,63,120)(13,101,64,57)(14,58,65,102)(15,103,66,59)(16,60,67,104)(17,105,68,41)(18,42,69,106)(19,107,70,43)(20,44,71,108)(21,144,96,130)(22,131,97,145)(23,146,98,132)(24,133,99,147)(25,148,100,134)(26,135,81,149)(27,150,82,136)(28,137,83,151)(29,152,84,138)(30,139,85,153)(31,154,86,140)(32,121,87,155)(33,156,88,122)(34,123,89,157)(35,158,90,124)(36,125,91,159)(37,160,92,126)(38,127,93,141)(39,142,94,128)(40,129,95,143), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,97,31,87)(22,86,32,96)(23,95,33,85)(24,84,34,94)(25,93,35,83)(26,82,36,92)(27,91,37,81)(28,100,38,90)(29,89,39,99)(30,98,40,88)(41,112,51,102)(42,101,52,111)(43,110,53,120)(44,119,54,109)(45,108,55,118)(46,117,56,107)(47,106,57,116)(48,115,58,105)(49,104,59,114)(50,113,60,103)(61,62,71,72)(63,80,73,70)(64,69,74,79)(65,78,75,68)(66,67,76,77)(121,140,131,130)(122,129,132,139)(123,138,133,128)(124,127,134,137)(125,136,135,126)(141,148,151,158)(142,157,152,147)(143,146,153,156)(144,155,154,145)(149,160,159,150) );
G=PermutationGroup([[(1,87,11,97),(2,23,12,33),(3,89,13,99),(4,25,14,35),(5,91,15,81),(6,27,16,37),(7,93,17,83),(8,29,18,39),(9,95,19,85),(10,31,20,21),(22,72,32,62),(24,74,34,64),(26,76,36,66),(28,78,38,68),(30,80,40,70),(41,137,51,127),(42,142,52,152),(43,139,53,129),(44,144,54,154),(45,121,55,131),(46,146,56,156),(47,123,57,133),(48,148,58,158),(49,125,59,135),(50,150,60,160),(61,86,71,96),(63,88,73,98),(65,90,75,100),(67,92,77,82),(69,94,79,84),(101,147,111,157),(102,124,112,134),(103,149,113,159),(104,126,114,136),(105,151,115,141),(106,128,116,138),(107,153,117,143),(108,130,118,140),(109,155,119,145),(110,132,120,122)], [(1,109,72,45),(2,46,73,110),(3,111,74,47),(4,48,75,112),(5,113,76,49),(6,50,77,114),(7,115,78,51),(8,52,79,116),(9,117,80,53),(10,54,61,118),(11,119,62,55),(12,56,63,120),(13,101,64,57),(14,58,65,102),(15,103,66,59),(16,60,67,104),(17,105,68,41),(18,42,69,106),(19,107,70,43),(20,44,71,108),(21,144,96,130),(22,131,97,145),(23,146,98,132),(24,133,99,147),(25,148,100,134),(26,135,81,149),(27,150,82,136),(28,137,83,151),(29,152,84,138),(30,139,85,153),(31,154,86,140),(32,121,87,155),(33,156,88,122),(34,123,89,157),(35,158,90,124),(36,125,91,159),(37,160,92,126),(38,127,93,141),(39,142,94,128),(40,129,95,143)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,97,31,87),(22,86,32,96),(23,95,33,85),(24,84,34,94),(25,93,35,83),(26,82,36,92),(27,91,37,81),(28,100,38,90),(29,89,39,99),(30,98,40,88),(41,112,51,102),(42,101,52,111),(43,110,53,120),(44,119,54,109),(45,108,55,118),(46,117,56,107),(47,106,57,116),(48,115,58,105),(49,104,59,114),(50,113,60,103),(61,62,71,72),(63,80,73,70),(64,69,74,79),(65,78,75,68),(66,67,76,77),(121,140,131,130),(122,129,132,139),(123,138,133,128),(124,127,134,137),(125,136,135,126),(141,148,151,158),(142,157,152,147),(143,146,153,156),(144,155,154,145),(149,160,159,150)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4Q | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | 2- 1+4 | Q8.10D10 | D5×C4○D4 | D4.10D10 |
| kernel | C42.154D10 | C4×Dic10 | C42⋊2D5 | Dic5⋊3Q8 | C20⋊Q8 | Dic5.Q8 | C4.Dic10 | C4⋊C4⋊7D5 | D10⋊Q8 | C4⋊C4⋊D5 | C5×C42.C2 | C42.C2 | Dic5 | C42 | C4⋊C4 | C10 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 2 | 3 | 1 | 2 | 4 | 2 | 12 | 2 | 4 | 4 | 4 |
Matrix representation of C42.154D10 ►in GL8(𝔽41)
| 9 | 0 | 23 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 23 | 0 | 0 | 0 | 0 |
| 9 | 0 | 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 7 | 25 | 1 |
| 0 | 0 | 0 | 0 | 3 | 19 | 23 | 22 |
| 0 | 0 | 0 | 0 | 7 | 2 | 1 | 34 |
| 0 | 0 | 0 | 0 | 40 | 1 | 26 | 18 |
| 1 | 0 | 39 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 39 | 0 | 0 | 0 | 0 |
| 1 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 40 | 22 | 21 |
| 0 | 0 | 0 | 0 | 1 | 24 | 12 | 7 |
| 0 | 0 | 0 | 0 | 0 | 0 | 18 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 23 |
| 1 | 29 | 28 | 13 | 0 | 0 | 0 | 0 |
| 12 | 11 | 28 | 24 | 0 | 0 | 0 | 0 |
| 15 | 15 | 40 | 12 | 0 | 0 | 0 | 0 |
| 26 | 23 | 29 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 9 | 6 | 24 |
| 0 | 0 | 0 | 0 | 16 | 32 | 33 | 11 |
| 0 | 0 | 0 | 0 | 35 | 6 | 33 | 1 |
| 0 | 0 | 0 | 0 | 27 | 33 | 21 | 9 |
| 1 | 29 | 28 | 13 | 0 | 0 | 0 | 0 |
| 35 | 40 | 17 | 13 | 0 | 0 | 0 | 0 |
| 27 | 14 | 29 | 1 | 0 | 0 | 0 | 0 |
| 12 | 14 | 11 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 32 | 21 | 5 |
| 0 | 0 | 0 | 0 | 38 | 9 | 24 | 30 |
| 0 | 0 | 0 | 0 | 34 | 35 | 11 | 2 |
| 0 | 0 | 0 | 0 | 1 | 8 | 23 | 35 |
G:=sub<GL(8,GF(41))| [9,0,9,0,0,0,0,0,0,9,0,9,0,0,0,0,23,0,32,0,0,0,0,0,0,23,0,32,0,0,0,0,0,0,0,0,3,3,7,40,0,0,0,0,7,19,2,1,0,0,0,0,25,23,1,26,0,0,0,0,1,22,34,18],[1,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,39,0,40,0,0,0,0,0,0,39,0,40,0,0,0,0,0,0,0,0,17,1,0,0,0,0,0,0,40,24,0,0,0,0,0,0,22,12,18,1,0,0,0,0,21,7,5,23],[1,12,15,26,0,0,0,0,29,11,15,23,0,0,0,0,28,28,40,29,0,0,0,0,13,24,12,30,0,0,0,0,0,0,0,0,8,16,35,27,0,0,0,0,9,32,6,33,0,0,0,0,6,33,33,21,0,0,0,0,24,11,1,9],[1,35,27,12,0,0,0,0,29,40,14,14,0,0,0,0,28,17,29,11,0,0,0,0,13,13,1,12,0,0,0,0,0,0,0,0,27,38,34,1,0,0,0,0,32,9,35,8,0,0,0,0,21,24,11,23,0,0,0,0,5,30,2,35] >;
C42.154D10 in GAP, Magma, Sage, TeX
C_4^2._{154}D_{10} % in TeX
G:=Group("C4^2.154D10"); // GroupNames label
G:=SmallGroup(320,1368);
// by ID
G=gap.SmallGroup(320,1368);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,555,100,1571,570,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations