metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.171D10, C10.342- (1+4), C4⋊Q8⋊9D5, C4.37(D4×D5), (C4×D5).13D4, C20.69(C2×D4), C4⋊C4.122D10, D10.82(C2×D4), D10⋊Q8⋊47C2, C4.D20⋊26C2, C42⋊D5⋊25C2, (C2×Q8).143D10, Dic5.93(C2×D4), C10.98(C22×D4), Dic5⋊Q8⋊26C2, C20.23D4⋊25C2, (C4×C20).209C22, (C2×C10).268C24, (C2×C20).101C23, D10.13D4⋊45C2, (C2×D20).177C22, (Q8×C10).135C22, C22.289(C23×D5), D10⋊C4.49C22, C5⋊5(C23.38C23), (C2×Dic5).140C23, (C4×Dic5).167C22, (C22×D5).240C23, C2.35(Q8.10D10), (C2×Dic10).194C22, C10.D4.165C22, (C2×Q8×D5)⋊12C2, C2.71(C2×D4×D5), (C5×C4⋊Q8)⋊10C2, (C2×Q8⋊2D5).7C2, (C2×C4×D5).151C22, (C5×C4⋊C4).211C22, (C2×C4).217(C22×D5), SmallGroup(320,1396)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 990 in 270 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×10], C5, C2×C4, C2×C4 [×6], C2×C4 [×17], D4 [×6], Q8 [×10], C23 [×3], D5 [×4], C10, C10 [×2], C42, C42, C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×6], C22×C4 [×5], C2×D4 [×3], C2×Q8 [×2], C2×Q8 [×7], C4○D4 [×4], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×6], D10 [×2], D10 [×8], C2×C10, C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C4⋊Q8, C4⋊Q8, C22×Q8, C2×C4○D4, Dic10 [×6], C4×D5 [×4], C4×D5 [×8], D20 [×6], C2×Dic5, C2×Dic5 [×4], C2×C20, C2×C20 [×6], C5×Q8 [×4], C22×D5, C22×D5 [×2], C23.38C23, C4×Dic5, C10.D4 [×6], D10⋊C4 [×10], C4×C20, C5×C4⋊C4 [×4], C2×Dic10, C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×4], C2×D20, C2×D20 [×2], Q8×D5 [×4], Q8⋊2D5 [×4], Q8×C10 [×2], C42⋊D5, C4.D20, D10.13D4 [×4], D10⋊Q8 [×4], Dic5⋊Q8, C20.23D4, C5×C4⋊Q8, C2×Q8×D5, C2×Q8⋊2D5, C42.171D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2- (1+4) [×2], C22×D5 [×7], C23.38C23, D4×D5 [×2], C23×D5, C2×D4×D5, Q8.10D10 [×2], C42.171D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c9 >
►On 160 points
Generators in S160
(1 37 11 27)(2 28 12 38)(3 39 13 29)(4 30 14 40)(5 21 15 31)(6 32 16 22)(7 23 17 33)(8 34 18 24)(9 25 19 35)(10 36 20 26)(41 159 51 149)(42 150 52 160)(43 141 53 151)(44 152 54 142)(45 143 55 153)(46 154 56 144)(47 145 57 155)(48 156 58 146)(49 147 59 157)(50 158 60 148)(61 102 71 112)(62 113 72 103)(63 104 73 114)(64 115 74 105)(65 106 75 116)(66 117 76 107)(67 108 77 118)(68 119 78 109)(69 110 79 120)(70 101 80 111)(81 136 91 126)(82 127 92 137)(83 138 93 128)(84 129 94 139)(85 140 95 130)(86 131 96 121)(87 122 97 132)(88 133 98 123)(89 124 99 134)(90 135 100 125)
(1 139 145 113)(2 114 146 140)(3 121 147 115)(4 116 148 122)(5 123 149 117)(6 118 150 124)(7 125 151 119)(8 120 152 126)(9 127 153 101)(10 102 154 128)(11 129 155 103)(12 104 156 130)(13 131 157 105)(14 106 158 132)(15 133 159 107)(16 108 160 134)(17 135 141 109)(18 110 142 136)(19 137 143 111)(20 112 144 138)(21 88 41 76)(22 77 42 89)(23 90 43 78)(24 79 44 91)(25 92 45 80)(26 61 46 93)(27 94 47 62)(28 63 48 95)(29 96 49 64)(30 65 50 97)(31 98 51 66)(32 67 52 99)(33 100 53 68)(34 69 54 81)(35 82 55 70)(36 71 56 83)(37 84 57 72)(38 73 58 85)(39 86 59 74)(40 75 60 87)
(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)
(1 117 11 107)(2 106 12 116)(3 115 13 105)(4 104 14 114)(5 113 15 103)(6 102 16 112)(7 111 17 101)(8 120 18 110)(9 109 19 119)(10 118 20 108)(21 62 31 72)(22 71 32 61)(23 80 33 70)(24 69 34 79)(25 78 35 68)(26 67 36 77)(27 76 37 66)(28 65 38 75)(29 74 39 64)(30 63 40 73)(41 94 51 84)(42 83 52 93)(43 92 53 82)(44 81 54 91)(45 90 55 100)(46 99 56 89)(47 88 57 98)(48 97 58 87)(49 86 59 96)(50 95 60 85)(121 157 131 147)(122 146 132 156)(123 155 133 145)(124 144 134 154)(125 153 135 143)(126 142 136 152)(127 151 137 141)(128 160 138 150)(129 149 139 159)(130 158 140 148)
G:=sub<Sym(160)| (1,37,11,27)(2,28,12,38)(3,39,13,29)(4,30,14,40)(5,21,15,31)(6,32,16,22)(7,23,17,33)(8,34,18,24)(9,25,19,35)(10,36,20,26)(41,159,51,149)(42,150,52,160)(43,141,53,151)(44,152,54,142)(45,143,55,153)(46,154,56,144)(47,145,57,155)(48,156,58,146)(49,147,59,157)(50,158,60,148)(61,102,71,112)(62,113,72,103)(63,104,73,114)(64,115,74,105)(65,106,75,116)(66,117,76,107)(67,108,77,118)(68,119,78,109)(69,110,79,120)(70,101,80,111)(81,136,91,126)(82,127,92,137)(83,138,93,128)(84,129,94,139)(85,140,95,130)(86,131,96,121)(87,122,97,132)(88,133,98,123)(89,124,99,134)(90,135,100,125), (1,139,145,113)(2,114,146,140)(3,121,147,115)(4,116,148,122)(5,123,149,117)(6,118,150,124)(7,125,151,119)(8,120,152,126)(9,127,153,101)(10,102,154,128)(11,129,155,103)(12,104,156,130)(13,131,157,105)(14,106,158,132)(15,133,159,107)(16,108,160,134)(17,135,141,109)(18,110,142,136)(19,137,143,111)(20,112,144,138)(21,88,41,76)(22,77,42,89)(23,90,43,78)(24,79,44,91)(25,92,45,80)(26,61,46,93)(27,94,47,62)(28,63,48,95)(29,96,49,64)(30,65,50,97)(31,98,51,66)(32,67,52,99)(33,100,53,68)(34,69,54,81)(35,82,55,70)(36,71,56,83)(37,84,57,72)(38,73,58,85)(39,86,59,74)(40,75,60,87), (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), (1,117,11,107)(2,106,12,116)(3,115,13,105)(4,104,14,114)(5,113,15,103)(6,102,16,112)(7,111,17,101)(8,120,18,110)(9,109,19,119)(10,118,20,108)(21,62,31,72)(22,71,32,61)(23,80,33,70)(24,69,34,79)(25,78,35,68)(26,67,36,77)(27,76,37,66)(28,65,38,75)(29,74,39,64)(30,63,40,73)(41,94,51,84)(42,83,52,93)(43,92,53,82)(44,81,54,91)(45,90,55,100)(46,99,56,89)(47,88,57,98)(48,97,58,87)(49,86,59,96)(50,95,60,85)(121,157,131,147)(122,146,132,156)(123,155,133,145)(124,144,134,154)(125,153,135,143)(126,142,136,152)(127,151,137,141)(128,160,138,150)(129,149,139,159)(130,158,140,148)>;
G:=Group( (1,37,11,27)(2,28,12,38)(3,39,13,29)(4,30,14,40)(5,21,15,31)(6,32,16,22)(7,23,17,33)(8,34,18,24)(9,25,19,35)(10,36,20,26)(41,159,51,149)(42,150,52,160)(43,141,53,151)(44,152,54,142)(45,143,55,153)(46,154,56,144)(47,145,57,155)(48,156,58,146)(49,147,59,157)(50,158,60,148)(61,102,71,112)(62,113,72,103)(63,104,73,114)(64,115,74,105)(65,106,75,116)(66,117,76,107)(67,108,77,118)(68,119,78,109)(69,110,79,120)(70,101,80,111)(81,136,91,126)(82,127,92,137)(83,138,93,128)(84,129,94,139)(85,140,95,130)(86,131,96,121)(87,122,97,132)(88,133,98,123)(89,124,99,134)(90,135,100,125), (1,139,145,113)(2,114,146,140)(3,121,147,115)(4,116,148,122)(5,123,149,117)(6,118,150,124)(7,125,151,119)(8,120,152,126)(9,127,153,101)(10,102,154,128)(11,129,155,103)(12,104,156,130)(13,131,157,105)(14,106,158,132)(15,133,159,107)(16,108,160,134)(17,135,141,109)(18,110,142,136)(19,137,143,111)(20,112,144,138)(21,88,41,76)(22,77,42,89)(23,90,43,78)(24,79,44,91)(25,92,45,80)(26,61,46,93)(27,94,47,62)(28,63,48,95)(29,96,49,64)(30,65,50,97)(31,98,51,66)(32,67,52,99)(33,100,53,68)(34,69,54,81)(35,82,55,70)(36,71,56,83)(37,84,57,72)(38,73,58,85)(39,86,59,74)(40,75,60,87), (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), (1,117,11,107)(2,106,12,116)(3,115,13,105)(4,104,14,114)(5,113,15,103)(6,102,16,112)(7,111,17,101)(8,120,18,110)(9,109,19,119)(10,118,20,108)(21,62,31,72)(22,71,32,61)(23,80,33,70)(24,69,34,79)(25,78,35,68)(26,67,36,77)(27,76,37,66)(28,65,38,75)(29,74,39,64)(30,63,40,73)(41,94,51,84)(42,83,52,93)(43,92,53,82)(44,81,54,91)(45,90,55,100)(46,99,56,89)(47,88,57,98)(48,97,58,87)(49,86,59,96)(50,95,60,85)(121,157,131,147)(122,146,132,156)(123,155,133,145)(124,144,134,154)(125,153,135,143)(126,142,136,152)(127,151,137,141)(128,160,138,150)(129,149,139,159)(130,158,140,148) );
G=PermutationGroup([(1,37,11,27),(2,28,12,38),(3,39,13,29),(4,30,14,40),(5,21,15,31),(6,32,16,22),(7,23,17,33),(8,34,18,24),(9,25,19,35),(10,36,20,26),(41,159,51,149),(42,150,52,160),(43,141,53,151),(44,152,54,142),(45,143,55,153),(46,154,56,144),(47,145,57,155),(48,156,58,146),(49,147,59,157),(50,158,60,148),(61,102,71,112),(62,113,72,103),(63,104,73,114),(64,115,74,105),(65,106,75,116),(66,117,76,107),(67,108,77,118),(68,119,78,109),(69,110,79,120),(70,101,80,111),(81,136,91,126),(82,127,92,137),(83,138,93,128),(84,129,94,139),(85,140,95,130),(86,131,96,121),(87,122,97,132),(88,133,98,123),(89,124,99,134),(90,135,100,125)], [(1,139,145,113),(2,114,146,140),(3,121,147,115),(4,116,148,122),(5,123,149,117),(6,118,150,124),(7,125,151,119),(8,120,152,126),(9,127,153,101),(10,102,154,128),(11,129,155,103),(12,104,156,130),(13,131,157,105),(14,106,158,132),(15,133,159,107),(16,108,160,134),(17,135,141,109),(18,110,142,136),(19,137,143,111),(20,112,144,138),(21,88,41,76),(22,77,42,89),(23,90,43,78),(24,79,44,91),(25,92,45,80),(26,61,46,93),(27,94,47,62),(28,63,48,95),(29,96,49,64),(30,65,50,97),(31,98,51,66),(32,67,52,99),(33,100,53,68),(34,69,54,81),(35,82,55,70),(36,71,56,83),(37,84,57,72),(38,73,58,85),(39,86,59,74),(40,75,60,87)], [(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)], [(1,117,11,107),(2,106,12,116),(3,115,13,105),(4,104,14,114),(5,113,15,103),(6,102,16,112),(7,111,17,101),(8,120,18,110),(9,109,19,119),(10,118,20,108),(21,62,31,72),(22,71,32,61),(23,80,33,70),(24,69,34,79),(25,78,35,68),(26,67,36,77),(27,76,37,66),(28,65,38,75),(29,74,39,64),(30,63,40,73),(41,94,51,84),(42,83,52,93),(43,92,53,82),(44,81,54,91),(45,90,55,100),(46,99,56,89),(47,88,57,98),(48,97,58,87),(49,86,59,96),(50,95,60,85),(121,157,131,147),(122,146,132,156),(123,155,133,145),(124,144,134,154),(125,153,135,143),(126,142,136,152),(127,151,137,141),(128,160,138,150),(129,149,139,159),(130,158,140,148)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 9 | 0 | 0 |
| 0 | 0 | 4 | 39 | 0 | 0 |
| 0 | 0 | 6 | 1 | 30 | 32 |
| 0 | 0 | 40 | 0 | 9 | 11 |
| 32 | 9 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 3 | 18 | 9 | 0 |
| 0 | 0 | 3 | 18 | 0 | 9 |
| 9 | 32 | 0 | 0 | 0 | 0 |
| 18 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 19 | 27 | 33 |
| 0 | 0 | 32 | 28 | 35 | 34 |
| 0 | 0 | 34 | 0 | 22 | 22 |
| 0 | 0 | 34 | 34 | 22 | 22 |
| 32 | 9 | 0 | 0 | 0 | 0 |
| 23 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 22 | 0 | 0 |
| 0 | 0 | 37 | 13 | 0 | 0 |
| 0 | 0 | 34 | 0 | 22 | 22 |
| 0 | 0 | 1 | 7 | 32 | 19 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,2,4,6,40,0,0,9,39,1,0,0,0,0,0,30,9,0,0,0,0,32,11],[32,0,0,0,0,0,9,9,0,0,0,0,0,0,32,0,3,3,0,0,0,32,18,18,0,0,0,0,9,0,0,0,0,0,0,9],[9,18,0,0,0,0,32,32,0,0,0,0,0,0,10,32,34,34,0,0,19,28,0,34,0,0,27,35,22,22,0,0,33,34,22,22],[32,23,0,0,0,0,9,9,0,0,0,0,0,0,28,37,34,1,0,0,22,13,0,7,0,0,0,0,22,32,0,0,0,0,22,19] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 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 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | D10 | D10 | D10 | 2- (1+4) | D4×D5 | Q8.10D10 |
| kernel | C42.171D10 | C42⋊D5 | C4.D20 | D10.13D4 | D10⋊Q8 | Dic5⋊Q8 | C20.23D4 | C5×C4⋊Q8 | C2×Q8×D5 | C2×Q8⋊2D5 | C4×D5 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 8 | 4 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{171}D_{10} % in TeX
G:=Group("C4^2.171D10"); // GroupNames label
G:=SmallGroup(320,1396);
// by ID
G=gap.SmallGroup(320,1396);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,100,675,297,136,12550]);
// 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=d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀