metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.168D10, C10.792+ (1+4), C4⋊1D4⋊8D5, C20⋊2D4⋊38C2, (D4×Dic5)⋊36C2, (C2×D4).179D10, C42⋊D5⋊24C2, C20.6Q8⋊23C2, Dic5⋊D4⋊38C2, C20.134(C4○D4), C4.40(D4⋊2D5), (C2×C10).263C24, (C2×C20).637C23, (C4×C20).205C22, C2.83(D4⋊6D10), C23.69(C22×D5), (D4×C10).215C22, C4⋊Dic5.249C22, (C22×C10).77C23, C22.284(C23×D5), C23.D5.74C22, C23.18D10⋊27C2, C5⋊7(C22.34C24), (C2×Dic5).137C23, (C4×Dic5).164C22, C10.D4.87C22, (C22×D5).117C23, D10⋊C4.150C22, (C22×Dic5).159C22, (C5×C4⋊1D4)⋊10C2, C10.98(C2×C4○D4), C2.62(C2×D4⋊2D5), (C2×C4×D5).149C22, (C2×C4).215(C22×D5), (C2×C5⋊D4).79C22, SmallGroup(320,1391)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 846 in 240 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×15], C5, C2×C4, C2×C4 [×2], C2×C4 [×13], D4 [×12], C23 [×4], C23, D5, C10, C10 [×2], C10 [×4], C42, C42, C22⋊C4 [×10], C4⋊C4 [×8], C22×C4 [×5], C2×D4 [×6], C2×D4 [×4], Dic5 [×7], C20 [×2], C20 [×2], D10 [×3], C2×C10, C2×C10 [×12], C42⋊C2, C4×D4 [×2], C4⋊D4 [×6], C22.D4 [×4], C42.C2, C4⋊1D4, C4×D5 [×2], C2×Dic5, C2×Dic5 [×6], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20, C2×C20 [×2], C5×D4 [×8], C22×D5, C22×C10 [×4], C22.34C24, C4×Dic5, C10.D4 [×6], C4⋊Dic5 [×2], D10⋊C4 [×2], C23.D5 [×8], C4×C20, C2×C4×D5, C22×Dic5 [×4], C2×C5⋊D4 [×4], D4×C10 [×6], C20.6Q8, C42⋊D5, D4×Dic5 [×2], C23.18D10 [×4], C20⋊2D4 [×2], Dic5⋊D4 [×4], C5×C4⋊1D4, C42.168D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ (1+4) [×2], C22×D5 [×7], C22.34C24, D4⋊2D5 [×2], C23×D5, C2×D4⋊2D5, D4⋊6D10 [×2], C42.168D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=b2c-1 >
►On 160 points
Generators in S160
(1 108 33 113)(2 114 34 109)(3 110 35 115)(4 116 36 101)(5 102 37 117)(6 118 38 103)(7 104 39 119)(8 120 40 105)(9 106 31 111)(10 112 32 107)(11 98 58 133)(12 134 59 99)(13 100 60 135)(14 136 51 91)(15 92 52 137)(16 138 53 93)(17 94 54 139)(18 140 55 95)(19 96 56 131)(20 132 57 97)(21 90 143 61)(22 62 144 81)(23 82 145 63)(24 64 146 83)(25 84 147 65)(26 66 148 85)(27 86 149 67)(28 68 150 87)(29 88 141 69)(30 70 142 89)(41 122 75 155)(42 156 76 123)(43 124 77 157)(44 158 78 125)(45 126 79 159)(46 160 80 127)(47 128 71 151)(48 152 72 129)(49 130 73 153)(50 154 74 121)
(1 49 13 66)(2 67 14 50)(3 41 15 68)(4 69 16 42)(5 43 17 70)(6 61 18 44)(7 45 19 62)(8 63 20 46)(9 47 11 64)(10 65 12 48)(21 140 158 118)(22 119 159 131)(23 132 160 120)(24 111 151 133)(25 134 152 112)(26 113 153 135)(27 136 154 114)(28 115 155 137)(29 138 156 116)(30 117 157 139)(31 71 58 83)(32 84 59 72)(33 73 60 85)(34 86 51 74)(35 75 52 87)(36 88 53 76)(37 77 54 89)(38 90 55 78)(39 79 56 81)(40 82 57 80)(91 121 109 149)(92 150 110 122)(93 123 101 141)(94 142 102 124)(95 125 103 143)(96 144 104 126)(97 127 105 145)(98 146 106 128)(99 129 107 147)(100 148 108 130)
(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 13 12)(2 11 14 9)(3 8 15 20)(4 19 16 7)(5 6 17 18)(21 124 158 142)(22 141 159 123)(23 122 160 150)(24 149 151 121)(25 130 152 148)(26 147 153 129)(27 128 154 146)(28 145 155 127)(29 126 156 144)(30 143 157 125)(31 34 58 51)(32 60 59 33)(35 40 52 57)(36 56 53 39)(37 38 54 55)(41 80 68 82)(42 81 69 79)(43 78 70 90)(44 89 61 77)(45 76 62 88)(46 87 63 75)(47 74 64 86)(48 85 65 73)(49 72 66 84)(50 83 67 71)(91 106 109 98)(92 97 110 105)(93 104 101 96)(94 95 102 103)(99 108 107 100)(111 114 133 136)(112 135 134 113)(115 120 137 132)(116 131 138 119)(117 118 139 140)
G:=sub<Sym(160)| (1,108,33,113)(2,114,34,109)(3,110,35,115)(4,116,36,101)(5,102,37,117)(6,118,38,103)(7,104,39,119)(8,120,40,105)(9,106,31,111)(10,112,32,107)(11,98,58,133)(12,134,59,99)(13,100,60,135)(14,136,51,91)(15,92,52,137)(16,138,53,93)(17,94,54,139)(18,140,55,95)(19,96,56,131)(20,132,57,97)(21,90,143,61)(22,62,144,81)(23,82,145,63)(24,64,146,83)(25,84,147,65)(26,66,148,85)(27,86,149,67)(28,68,150,87)(29,88,141,69)(30,70,142,89)(41,122,75,155)(42,156,76,123)(43,124,77,157)(44,158,78,125)(45,126,79,159)(46,160,80,127)(47,128,71,151)(48,152,72,129)(49,130,73,153)(50,154,74,121), (1,49,13,66)(2,67,14,50)(3,41,15,68)(4,69,16,42)(5,43,17,70)(6,61,18,44)(7,45,19,62)(8,63,20,46)(9,47,11,64)(10,65,12,48)(21,140,158,118)(22,119,159,131)(23,132,160,120)(24,111,151,133)(25,134,152,112)(26,113,153,135)(27,136,154,114)(28,115,155,137)(29,138,156,116)(30,117,157,139)(31,71,58,83)(32,84,59,72)(33,73,60,85)(34,86,51,74)(35,75,52,87)(36,88,53,76)(37,77,54,89)(38,90,55,78)(39,79,56,81)(40,82,57,80)(91,121,109,149)(92,150,110,122)(93,123,101,141)(94,142,102,124)(95,125,103,143)(96,144,104,126)(97,127,105,145)(98,146,106,128)(99,129,107,147)(100,148,108,130), (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), (1,10,13,12)(2,11,14,9)(3,8,15,20)(4,19,16,7)(5,6,17,18)(21,124,158,142)(22,141,159,123)(23,122,160,150)(24,149,151,121)(25,130,152,148)(26,147,153,129)(27,128,154,146)(28,145,155,127)(29,126,156,144)(30,143,157,125)(31,34,58,51)(32,60,59,33)(35,40,52,57)(36,56,53,39)(37,38,54,55)(41,80,68,82)(42,81,69,79)(43,78,70,90)(44,89,61,77)(45,76,62,88)(46,87,63,75)(47,74,64,86)(48,85,65,73)(49,72,66,84)(50,83,67,71)(91,106,109,98)(92,97,110,105)(93,104,101,96)(94,95,102,103)(99,108,107,100)(111,114,133,136)(112,135,134,113)(115,120,137,132)(116,131,138,119)(117,118,139,140)>;
G:=Group( (1,108,33,113)(2,114,34,109)(3,110,35,115)(4,116,36,101)(5,102,37,117)(6,118,38,103)(7,104,39,119)(8,120,40,105)(9,106,31,111)(10,112,32,107)(11,98,58,133)(12,134,59,99)(13,100,60,135)(14,136,51,91)(15,92,52,137)(16,138,53,93)(17,94,54,139)(18,140,55,95)(19,96,56,131)(20,132,57,97)(21,90,143,61)(22,62,144,81)(23,82,145,63)(24,64,146,83)(25,84,147,65)(26,66,148,85)(27,86,149,67)(28,68,150,87)(29,88,141,69)(30,70,142,89)(41,122,75,155)(42,156,76,123)(43,124,77,157)(44,158,78,125)(45,126,79,159)(46,160,80,127)(47,128,71,151)(48,152,72,129)(49,130,73,153)(50,154,74,121), (1,49,13,66)(2,67,14,50)(3,41,15,68)(4,69,16,42)(5,43,17,70)(6,61,18,44)(7,45,19,62)(8,63,20,46)(9,47,11,64)(10,65,12,48)(21,140,158,118)(22,119,159,131)(23,132,160,120)(24,111,151,133)(25,134,152,112)(26,113,153,135)(27,136,154,114)(28,115,155,137)(29,138,156,116)(30,117,157,139)(31,71,58,83)(32,84,59,72)(33,73,60,85)(34,86,51,74)(35,75,52,87)(36,88,53,76)(37,77,54,89)(38,90,55,78)(39,79,56,81)(40,82,57,80)(91,121,109,149)(92,150,110,122)(93,123,101,141)(94,142,102,124)(95,125,103,143)(96,144,104,126)(97,127,105,145)(98,146,106,128)(99,129,107,147)(100,148,108,130), (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), (1,10,13,12)(2,11,14,9)(3,8,15,20)(4,19,16,7)(5,6,17,18)(21,124,158,142)(22,141,159,123)(23,122,160,150)(24,149,151,121)(25,130,152,148)(26,147,153,129)(27,128,154,146)(28,145,155,127)(29,126,156,144)(30,143,157,125)(31,34,58,51)(32,60,59,33)(35,40,52,57)(36,56,53,39)(37,38,54,55)(41,80,68,82)(42,81,69,79)(43,78,70,90)(44,89,61,77)(45,76,62,88)(46,87,63,75)(47,74,64,86)(48,85,65,73)(49,72,66,84)(50,83,67,71)(91,106,109,98)(92,97,110,105)(93,104,101,96)(94,95,102,103)(99,108,107,100)(111,114,133,136)(112,135,134,113)(115,120,137,132)(116,131,138,119)(117,118,139,140) );
G=PermutationGroup([(1,108,33,113),(2,114,34,109),(3,110,35,115),(4,116,36,101),(5,102,37,117),(6,118,38,103),(7,104,39,119),(8,120,40,105),(9,106,31,111),(10,112,32,107),(11,98,58,133),(12,134,59,99),(13,100,60,135),(14,136,51,91),(15,92,52,137),(16,138,53,93),(17,94,54,139),(18,140,55,95),(19,96,56,131),(20,132,57,97),(21,90,143,61),(22,62,144,81),(23,82,145,63),(24,64,146,83),(25,84,147,65),(26,66,148,85),(27,86,149,67),(28,68,150,87),(29,88,141,69),(30,70,142,89),(41,122,75,155),(42,156,76,123),(43,124,77,157),(44,158,78,125),(45,126,79,159),(46,160,80,127),(47,128,71,151),(48,152,72,129),(49,130,73,153),(50,154,74,121)], [(1,49,13,66),(2,67,14,50),(3,41,15,68),(4,69,16,42),(5,43,17,70),(6,61,18,44),(7,45,19,62),(8,63,20,46),(9,47,11,64),(10,65,12,48),(21,140,158,118),(22,119,159,131),(23,132,160,120),(24,111,151,133),(25,134,152,112),(26,113,153,135),(27,136,154,114),(28,115,155,137),(29,138,156,116),(30,117,157,139),(31,71,58,83),(32,84,59,72),(33,73,60,85),(34,86,51,74),(35,75,52,87),(36,88,53,76),(37,77,54,89),(38,90,55,78),(39,79,56,81),(40,82,57,80),(91,121,109,149),(92,150,110,122),(93,123,101,141),(94,142,102,124),(95,125,103,143),(96,144,104,126),(97,127,105,145),(98,146,106,128),(99,129,107,147),(100,148,108,130)], [(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)], [(1,10,13,12),(2,11,14,9),(3,8,15,20),(4,19,16,7),(5,6,17,18),(21,124,158,142),(22,141,159,123),(23,122,160,150),(24,149,151,121),(25,130,152,148),(26,147,153,129),(27,128,154,146),(28,145,155,127),(29,126,156,144),(30,143,157,125),(31,34,58,51),(32,60,59,33),(35,40,52,57),(36,56,53,39),(37,38,54,55),(41,80,68,82),(42,81,69,79),(43,78,70,90),(44,89,61,77),(45,76,62,88),(46,87,63,75),(47,74,64,86),(48,85,65,73),(49,72,66,84),(50,83,67,71),(91,106,109,98),(92,97,110,105),(93,104,101,96),(94,95,102,103),(99,108,107,100),(111,114,133,136),(112,135,134,113),(115,120,137,132),(116,131,138,119),(117,118,139,140)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 | 0 | 9 |
| 0 | 0 | 0 | 0 | 32 | 0 | 30 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 | 0 | 30 |
| 27 | 0 | 36 | 22 | 0 | 0 | 0 | 0 |
| 0 | 27 | 19 | 5 | 0 | 0 | 0 | 0 |
| 5 | 19 | 14 | 0 | 0 | 0 | 0 | 0 |
| 22 | 36 | 0 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 23 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 18 | 6 |
| 0 | 0 | 0 | 0 | 0 | 0 | 35 | 23 |
| 0 | 0 | 7 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 | 0 | 0 |
| 7 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 34 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 35 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 35 |
| 0 | 0 | 0 | 0 | 40 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 34 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 | 0 | 0 |
| 7 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 35 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 35 | 6 |
| 0 | 0 | 0 | 0 | 35 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 6 | 0 | 0 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,11,0,32,0,0,0,0,0,0,11,0,32,0,0,0,0,9,0,30,0,0,0,0,0,0,9,0,30],[27,0,5,22,0,0,0,0,0,27,19,36,0,0,0,0,36,19,14,0,0,0,0,0,22,5,0,14,0,0,0,0,0,0,0,0,18,35,0,0,0,0,0,0,6,23,0,0,0,0,0,0,0,0,18,35,0,0,0,0,0,0,6,23],[0,0,7,34,0,0,0,0,0,0,7,40,0,0,0,0,7,34,0,0,0,0,0,0,7,40,0,0,0,0,0,0,0,0,0,0,0,0,40,6,0,0,0,0,0,0,35,35,0,0,0,0,40,6,0,0,0,0,0,0,35,35,0,0],[0,0,7,40,0,0,0,0,0,0,7,34,0,0,0,0,34,1,0,0,0,0,0,0,34,7,0,0,0,0,0,0,0,0,0,0,0,0,35,35,0,0,0,0,0,0,40,6,0,0,0,0,35,35,0,0,0,0,0,0,40,6,0,0] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4M | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | 2+ (1+4) | D4⋊2D5 | D4⋊6D10 |
| kernel | C42.168D10 | C20.6Q8 | C42⋊D5 | D4×Dic5 | C23.18D10 | C20⋊2D4 | Dic5⋊D4 | C5×C4⋊1D4 | C4⋊1D4 | C20 | C42 | C2×D4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 1 | 2 | 4 | 2 | 12 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{168}D_{10} % in TeX
G:=Group("C4^2.168D10"); // GroupNames label
G:=SmallGroup(320,1391);
// by ID
G=gap.SmallGroup(320,1391);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,100,675,570,185,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^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^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀