metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.176D10, C10.382- 1+4, C4⋊Q8⋊14D5, C4⋊C4.125D10, (Q8×Dic5)⋊23C2, (C2×Q8).148D10, C20.6Q8⋊25C2, C42⋊D5.9C2, C20.138(C4○D4), C4.42(D4⋊2D5), (C4×C20).216C22, (C2×C10).275C24, (C2×C20).108C23, D10⋊3Q8.14C2, Dic5.Q8⋊42C2, C4⋊Dic5.254C22, (Q8×C10).142C22, C22.296(C23×D5), C5⋊7(C22.35C24), (C4×Dic5).172C22, (C2×Dic5).283C23, C10.D4.63C22, (C22×D5).120C23, D10⋊C4.154C22, C2.39(Q8.10D10), (C5×C4⋊Q8)⋊17C2, C4⋊C4⋊D5.4C2, C10.101(C2×C4○D4), C2.65(C2×D4⋊2D5), (C2×C4×D5).157C22, (C5×C4⋊C4).218C22, (C2×C4).221(C22×D5), SmallGroup(320,1403)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.176D10
G = < a,b,c,d | a4=b4=1, c10=a2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=b2c9 >
Subgroups: 558 in 192 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C23, D5, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C42⋊2C2, C4⋊Q8, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22.35C24, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C2×C4×D5, Q8×C10, C20.6Q8, C42⋊D5, Dic5.Q8, C4⋊C4⋊D5, Q8×Dic5, D10⋊3Q8, C5×C4⋊Q8, C42.176D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.35C24, D4⋊2D5, C23×D5, C2×D4⋊2D5, Q8.10D10, C42.176D10
►On 160 points
Generators in S160
(1 27 11 37)(2 38 12 28)(3 29 13 39)(4 40 14 30)(5 31 15 21)(6 22 16 32)(7 33 17 23)(8 24 18 34)(9 35 19 25)(10 26 20 36)(41 136 51 126)(42 127 52 137)(43 138 53 128)(44 129 54 139)(45 140 55 130)(46 131 56 121)(47 122 57 132)(48 133 58 123)(49 124 59 134)(50 135 60 125)(61 154 71 144)(62 145 72 155)(63 156 73 146)(64 147 74 157)(65 158 75 148)(66 149 76 159)(67 160 77 150)(68 151 78 141)(69 142 79 152)(70 153 80 143)(81 115 91 105)(82 106 92 116)(83 117 93 107)(84 108 94 118)(85 119 95 109)(86 110 96 120)(87 101 97 111)(88 112 98 102)(89 103 99 113)(90 114 100 104)
(1 89 72 126)(2 127 73 90)(3 91 74 128)(4 129 75 92)(5 93 76 130)(6 131 77 94)(7 95 78 132)(8 133 79 96)(9 97 80 134)(10 135 61 98)(11 99 62 136)(12 137 63 100)(13 81 64 138)(14 139 65 82)(15 83 66 140)(16 121 67 84)(17 85 68 122)(18 123 69 86)(19 87 70 124)(20 125 71 88)(21 117 149 55)(22 56 150 118)(23 119 151 57)(24 58 152 120)(25 101 153 59)(26 60 154 102)(27 103 155 41)(28 42 156 104)(29 105 157 43)(30 44 158 106)(31 107 159 45)(32 46 160 108)(33 109 141 47)(34 48 142 110)(35 111 143 49)(36 50 144 112)(37 113 145 51)(38 52 146 114)(39 115 147 53)(40 54 148 116)
(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 62 71)(2 70 63 9)(3 8 64 69)(4 68 65 7)(5 6 66 67)(11 20 72 61)(12 80 73 19)(13 18 74 79)(14 78 75 17)(15 16 76 77)(21 22 159 160)(23 40 141 158)(24 157 142 39)(25 38 143 156)(26 155 144 37)(27 36 145 154)(28 153 146 35)(29 34 147 152)(30 151 148 33)(31 32 149 150)(41 60 113 112)(42 111 114 59)(43 58 115 110)(44 109 116 57)(45 56 117 108)(46 107 118 55)(47 54 119 106)(48 105 120 53)(49 52 101 104)(50 103 102 51)(81 96 128 123)(82 122 129 95)(83 94 130 121)(84 140 131 93)(85 92 132 139)(86 138 133 91)(87 90 134 137)(88 136 135 89)(97 100 124 127)(98 126 125 99)
G:=sub<Sym(160)| (1,27,11,37)(2,38,12,28)(3,29,13,39)(4,40,14,30)(5,31,15,21)(6,22,16,32)(7,33,17,23)(8,24,18,34)(9,35,19,25)(10,26,20,36)(41,136,51,126)(42,127,52,137)(43,138,53,128)(44,129,54,139)(45,140,55,130)(46,131,56,121)(47,122,57,132)(48,133,58,123)(49,124,59,134)(50,135,60,125)(61,154,71,144)(62,145,72,155)(63,156,73,146)(64,147,74,157)(65,158,75,148)(66,149,76,159)(67,160,77,150)(68,151,78,141)(69,142,79,152)(70,153,80,143)(81,115,91,105)(82,106,92,116)(83,117,93,107)(84,108,94,118)(85,119,95,109)(86,110,96,120)(87,101,97,111)(88,112,98,102)(89,103,99,113)(90,114,100,104), (1,89,72,126)(2,127,73,90)(3,91,74,128)(4,129,75,92)(5,93,76,130)(6,131,77,94)(7,95,78,132)(8,133,79,96)(9,97,80,134)(10,135,61,98)(11,99,62,136)(12,137,63,100)(13,81,64,138)(14,139,65,82)(15,83,66,140)(16,121,67,84)(17,85,68,122)(18,123,69,86)(19,87,70,124)(20,125,71,88)(21,117,149,55)(22,56,150,118)(23,119,151,57)(24,58,152,120)(25,101,153,59)(26,60,154,102)(27,103,155,41)(28,42,156,104)(29,105,157,43)(30,44,158,106)(31,107,159,45)(32,46,160,108)(33,109,141,47)(34,48,142,110)(35,111,143,49)(36,50,144,112)(37,113,145,51)(38,52,146,114)(39,115,147,53)(40,54,148,116), (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), (1,10,62,71)(2,70,63,9)(3,8,64,69)(4,68,65,7)(5,6,66,67)(11,20,72,61)(12,80,73,19)(13,18,74,79)(14,78,75,17)(15,16,76,77)(21,22,159,160)(23,40,141,158)(24,157,142,39)(25,38,143,156)(26,155,144,37)(27,36,145,154)(28,153,146,35)(29,34,147,152)(30,151,148,33)(31,32,149,150)(41,60,113,112)(42,111,114,59)(43,58,115,110)(44,109,116,57)(45,56,117,108)(46,107,118,55)(47,54,119,106)(48,105,120,53)(49,52,101,104)(50,103,102,51)(81,96,128,123)(82,122,129,95)(83,94,130,121)(84,140,131,93)(85,92,132,139)(86,138,133,91)(87,90,134,137)(88,136,135,89)(97,100,124,127)(98,126,125,99)>;
G:=Group( (1,27,11,37)(2,38,12,28)(3,29,13,39)(4,40,14,30)(5,31,15,21)(6,22,16,32)(7,33,17,23)(8,24,18,34)(9,35,19,25)(10,26,20,36)(41,136,51,126)(42,127,52,137)(43,138,53,128)(44,129,54,139)(45,140,55,130)(46,131,56,121)(47,122,57,132)(48,133,58,123)(49,124,59,134)(50,135,60,125)(61,154,71,144)(62,145,72,155)(63,156,73,146)(64,147,74,157)(65,158,75,148)(66,149,76,159)(67,160,77,150)(68,151,78,141)(69,142,79,152)(70,153,80,143)(81,115,91,105)(82,106,92,116)(83,117,93,107)(84,108,94,118)(85,119,95,109)(86,110,96,120)(87,101,97,111)(88,112,98,102)(89,103,99,113)(90,114,100,104), (1,89,72,126)(2,127,73,90)(3,91,74,128)(4,129,75,92)(5,93,76,130)(6,131,77,94)(7,95,78,132)(8,133,79,96)(9,97,80,134)(10,135,61,98)(11,99,62,136)(12,137,63,100)(13,81,64,138)(14,139,65,82)(15,83,66,140)(16,121,67,84)(17,85,68,122)(18,123,69,86)(19,87,70,124)(20,125,71,88)(21,117,149,55)(22,56,150,118)(23,119,151,57)(24,58,152,120)(25,101,153,59)(26,60,154,102)(27,103,155,41)(28,42,156,104)(29,105,157,43)(30,44,158,106)(31,107,159,45)(32,46,160,108)(33,109,141,47)(34,48,142,110)(35,111,143,49)(36,50,144,112)(37,113,145,51)(38,52,146,114)(39,115,147,53)(40,54,148,116), (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), (1,10,62,71)(2,70,63,9)(3,8,64,69)(4,68,65,7)(5,6,66,67)(11,20,72,61)(12,80,73,19)(13,18,74,79)(14,78,75,17)(15,16,76,77)(21,22,159,160)(23,40,141,158)(24,157,142,39)(25,38,143,156)(26,155,144,37)(27,36,145,154)(28,153,146,35)(29,34,147,152)(30,151,148,33)(31,32,149,150)(41,60,113,112)(42,111,114,59)(43,58,115,110)(44,109,116,57)(45,56,117,108)(46,107,118,55)(47,54,119,106)(48,105,120,53)(49,52,101,104)(50,103,102,51)(81,96,128,123)(82,122,129,95)(83,94,130,121)(84,140,131,93)(85,92,132,139)(86,138,133,91)(87,90,134,137)(88,136,135,89)(97,100,124,127)(98,126,125,99) );
G=PermutationGroup([[(1,27,11,37),(2,38,12,28),(3,29,13,39),(4,40,14,30),(5,31,15,21),(6,22,16,32),(7,33,17,23),(8,24,18,34),(9,35,19,25),(10,26,20,36),(41,136,51,126),(42,127,52,137),(43,138,53,128),(44,129,54,139),(45,140,55,130),(46,131,56,121),(47,122,57,132),(48,133,58,123),(49,124,59,134),(50,135,60,125),(61,154,71,144),(62,145,72,155),(63,156,73,146),(64,147,74,157),(65,158,75,148),(66,149,76,159),(67,160,77,150),(68,151,78,141),(69,142,79,152),(70,153,80,143),(81,115,91,105),(82,106,92,116),(83,117,93,107),(84,108,94,118),(85,119,95,109),(86,110,96,120),(87,101,97,111),(88,112,98,102),(89,103,99,113),(90,114,100,104)], [(1,89,72,126),(2,127,73,90),(3,91,74,128),(4,129,75,92),(5,93,76,130),(6,131,77,94),(7,95,78,132),(8,133,79,96),(9,97,80,134),(10,135,61,98),(11,99,62,136),(12,137,63,100),(13,81,64,138),(14,139,65,82),(15,83,66,140),(16,121,67,84),(17,85,68,122),(18,123,69,86),(19,87,70,124),(20,125,71,88),(21,117,149,55),(22,56,150,118),(23,119,151,57),(24,58,152,120),(25,101,153,59),(26,60,154,102),(27,103,155,41),(28,42,156,104),(29,105,157,43),(30,44,158,106),(31,107,159,45),(32,46,160,108),(33,109,141,47),(34,48,142,110),(35,111,143,49),(36,50,144,112),(37,113,145,51),(38,52,146,114),(39,115,147,53),(40,54,148,116)], [(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)], [(1,10,62,71),(2,70,63,9),(3,8,64,69),(4,68,65,7),(5,6,66,67),(11,20,72,61),(12,80,73,19),(13,18,74,79),(14,78,75,17),(15,16,76,77),(21,22,159,160),(23,40,141,158),(24,157,142,39),(25,38,143,156),(26,155,144,37),(27,36,145,154),(28,153,146,35),(29,34,147,152),(30,151,148,33),(31,32,149,150),(41,60,113,112),(42,111,114,59),(43,58,115,110),(44,109,116,57),(45,56,117,108),(46,107,118,55),(47,54,119,106),(48,105,120,53),(49,52,101,104),(50,103,102,51),(81,96,128,123),(82,122,129,95),(83,94,130,121),(84,140,131,93),(85,92,132,139),(86,138,133,91),(87,90,134,137),(88,136,135,89),(97,100,124,127),(98,126,125,99)]])
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 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | 2- 1+4 | D4⋊2D5 | Q8.10D10 |
| kernel | C42.176D10 | C20.6Q8 | C42⋊D5 | Dic5.Q8 | C4⋊C4⋊D5 | Q8×Dic5 | D10⋊3Q8 | C5×C4⋊Q8 | C4⋊Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 2 | 4 | 2 | 8 | 4 | 2 | 4 | 8 |
Matrix representation of C42.176D10 ►in GL8(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 0 | 35 | 0 |
| 0 | 0 | 0 | 0 | 0 | 39 | 0 | 35 |
| 0 | 0 | 0 | 0 | 35 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 35 | 0 | 2 |
| 17 | 40 | 14 | 11 | 0 | 0 | 0 | 0 |
| 1 | 24 | 14 | 14 | 0 | 0 | 0 | 0 |
| 13 | 22 | 17 | 1 | 0 | 0 | 0 | 0 |
| 28 | 13 | 40 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 17 | 35 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 24 |
| 10 | 23 | 0 | 11 | 0 | 0 | 0 | 0 |
| 18 | 13 | 30 | 0 | 0 | 0 | 0 | 0 |
| 34 | 35 | 28 | 18 | 0 | 0 | 0 | 0 |
| 39 | 34 | 23 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 34 | 35 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 7 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 0 | 0 | 0 |
| 31 | 18 | 0 | 30 | 0 | 0 | 0 | 0 |
| 6 | 10 | 30 | 36 | 0 | 0 | 0 | 0 |
| 10 | 19 | 28 | 32 | 0 | 0 | 0 | 0 |
| 19 | 0 | 23 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 34 | 34 |
| 0 | 0 | 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 7 | 0 | 0 |
G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,39,0,35,0,0,0,0,0,0,39,0,35,0,0,0,0,35,0,2,0,0,0,0,0,0,35,0,2],[17,1,13,28,0,0,0,0,40,24,22,13,0,0,0,0,14,14,17,40,0,0,0,0,11,14,1,24,0,0,0,0,0,0,0,0,17,7,0,0,0,0,0,0,35,24,0,0,0,0,0,0,0,0,17,7,0,0,0,0,0,0,35,24],[10,18,34,39,0,0,0,0,23,13,35,34,0,0,0,0,0,30,28,23,0,0,0,0,11,0,18,31,0,0,0,0,0,0,0,0,0,0,7,34,0,0,0,0,0,0,6,0,0,0,0,0,34,7,0,0,0,0,0,0,35,0,0,0],[31,6,10,19,0,0,0,0,18,10,19,0,0,0,0,0,0,30,28,23,0,0,0,0,30,36,32,13,0,0,0,0,0,0,0,0,0,0,34,7,0,0,0,0,0,0,40,7,0,0,0,0,7,34,0,0,0,0,0,0,1,34,0,0] >;
C42.176D10 in GAP, Magma, Sage, TeX
C_4^2._{176}D_{10} % in TeX
G:=Group("C4^2.176D10"); // GroupNames label
G:=SmallGroup(320,1403);
// by ID
G=gap.SmallGroup(320,1403);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,100,675,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2,d^2=a^2*b^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=b^2*c^9>;
// generators/relations