metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.108D10, C10.582- (1+4), C10.162+ (1+4), (C4×D4)⋊8D5, D4⋊10(C4×D5), D4⋊2D5⋊5C4, (D4×C20)⋊10C2, (D4×Dic5)⋊9C2, C4⋊C4.315D10, (C4×Dic10)⋊30C2, Dic10⋊22(C2×C4), (C2×D4).246D10, C42⋊D5⋊12C2, C20.68(C22×C4), C10.44(C23×C4), (C2×C10).90C24, (C22×C4).46D10, Dic5⋊4D4⋊46C2, Dic5⋊3Q8⋊15C2, C2.4(D4⋊6D10), (C4×C20).150C22, (C2×C20).491C23, C22⋊C4.131D10, D10.18(C22×C4), C22.33(C23×D5), (D4×C10).254C22, C4⋊Dic5.362C22, Dic5.40(C22×C4), C23.169(C22×D5), C2.3(D4.10D10), C23.11D10⋊28C2, (C22×C20).363C22, (C22×C10).160C23, C5⋊4(C23.33C23), (C4×Dic5).223C22, (C2×Dic5).212C23, (C22×D5).179C23, C23.D5.104C22, D10⋊C4.121C22, (C2×Dic10).295C22, C10.D4.134C22, (C22×Dic5).95C22, C4.33(C2×C4×D5), (D5×C4⋊C4)⋊14C2, (C4×D5)⋊4(C2×C4), C5⋊D4⋊8(C2×C4), (C5×D4)⋊23(C2×C4), (C4×C5⋊D4)⋊41C2, C22.3(C2×C4×D5), C2.25(D5×C22×C4), (C2×C4×D5).72C22, (C2×Dic5)⋊13(C2×C4), (C2×D4⋊2D5).9C2, (C2×C10.D4)⋊38C2, (C5×C4⋊C4).324C22, (C2×C10).10(C22×C4), (C2×C4).283(C22×D5), (C2×C5⋊D4).119C22, (C5×C22⋊C4).143C22, SmallGroup(320,1218)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 862 in 294 conjugacy classes, 151 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×14], C22, C22 [×4], C22 [×8], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×25], D4 [×4], D4 [×8], Q8 [×4], C23 [×2], C23, D5 [×2], C10 [×3], C10 [×4], C42, C42 [×5], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×7], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×8], Dic5 [×6], Dic5 [×4], C20 [×2], C20 [×4], D10 [×2], D10 [×2], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C4⋊C4 [×3], C42⋊C2 [×3], C4×D4, C4×D4 [×5], C4×Q8 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×4], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×12], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×3], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C22×D5, C22×C10 [×2], C23.33C23, C4×Dic5, C4×Dic5 [×4], C10.D4 [×2], C10.D4 [×6], C4⋊Dic5, D10⋊C4 [×2], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5 [×2], D4⋊2D5 [×8], C22×Dic5 [×4], C2×C5⋊D4 [×2], C22×C20 [×2], D4×C10, C4×Dic10, C42⋊D5, C23.11D10 [×2], Dic5⋊4D4 [×2], Dic5⋊3Q8, D5×C4⋊C4, C2×C10.D4 [×2], C4×C5⋊D4 [×2], D4×Dic5, D4×C20, C2×D4⋊2D5, C42.108D10
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, D10 [×7], C23×C4, 2+ (1+4), 2- (1+4), C4×D5 [×4], C22×D5 [×7], C23.33C23, C2×C4×D5 [×6], C23×D5, D5×C22×C4, D4⋊6D10, D4.10D10, C42.108D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=a2c-1 >
►On 160 points
Generators in S160
(1 128 63 143)(2 144 64 129)(3 130 65 145)(4 146 66 121)(5 122 67 147)(6 148 68 123)(7 124 69 149)(8 150 70 125)(9 126 61 141)(10 142 62 127)(11 43 82 107)(12 108 83 44)(13 45 84 109)(14 110 85 46)(15 47 86 101)(16 102 87 48)(17 49 88 103)(18 104 89 50)(19 41 90 105)(20 106 81 42)(21 152 72 137)(22 138 73 153)(23 154 74 139)(24 140 75 155)(25 156 76 131)(26 132 77 157)(27 158 78 133)(28 134 79 159)(29 160 80 135)(30 136 71 151)(31 91 56 116)(32 117 57 92)(33 93 58 118)(34 119 59 94)(35 95 60 120)(36 111 51 96)(37 97 52 112)(38 113 53 98)(39 99 54 114)(40 115 55 100)
(1 19 55 27)(2 20 56 28)(3 11 57 29)(4 12 58 30)(5 13 59 21)(6 14 60 22)(7 15 51 23)(8 16 52 24)(9 17 53 25)(10 18 54 26)(31 79 64 81)(32 80 65 82)(33 71 66 83)(34 72 67 84)(35 73 68 85)(36 74 69 86)(37 75 70 87)(38 76 61 88)(39 77 62 89)(40 78 63 90)(41 100 158 128)(42 91 159 129)(43 92 160 130)(44 93 151 121)(45 94 152 122)(46 95 153 123)(47 96 154 124)(48 97 155 125)(49 98 156 126)(50 99 157 127)(101 111 139 149)(102 112 140 150)(103 113 131 141)(104 114 132 142)(105 115 133 143)(106 116 134 144)(107 117 135 145)(108 118 136 146)(109 119 137 147)(110 120 138 148)
(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 63 62)(2 61 64 9)(3 8 65 70)(4 69 66 7)(5 6 67 68)(11 87 82 16)(12 15 83 86)(13 85 84 14)(17 81 88 20)(18 19 89 90)(21 73 72 22)(23 71 74 30)(24 29 75 80)(25 79 76 28)(26 27 77 78)(31 53 56 38)(32 37 57 52)(33 51 58 36)(34 35 59 60)(39 55 54 40)(41 104 105 50)(42 49 106 103)(43 102 107 48)(44 47 108 101)(45 110 109 46)(91 113 116 98)(92 97 117 112)(93 111 118 96)(94 95 119 120)(99 115 114 100)(121 149 146 124)(122 123 147 148)(125 145 150 130)(126 129 141 144)(127 143 142 128)(131 159 156 134)(132 133 157 158)(135 155 160 140)(136 139 151 154)(137 153 152 138)
G:=sub<Sym(160)| (1,128,63,143)(2,144,64,129)(3,130,65,145)(4,146,66,121)(5,122,67,147)(6,148,68,123)(7,124,69,149)(8,150,70,125)(9,126,61,141)(10,142,62,127)(11,43,82,107)(12,108,83,44)(13,45,84,109)(14,110,85,46)(15,47,86,101)(16,102,87,48)(17,49,88,103)(18,104,89,50)(19,41,90,105)(20,106,81,42)(21,152,72,137)(22,138,73,153)(23,154,74,139)(24,140,75,155)(25,156,76,131)(26,132,77,157)(27,158,78,133)(28,134,79,159)(29,160,80,135)(30,136,71,151)(31,91,56,116)(32,117,57,92)(33,93,58,118)(34,119,59,94)(35,95,60,120)(36,111,51,96)(37,97,52,112)(38,113,53,98)(39,99,54,114)(40,115,55,100), (1,19,55,27)(2,20,56,28)(3,11,57,29)(4,12,58,30)(5,13,59,21)(6,14,60,22)(7,15,51,23)(8,16,52,24)(9,17,53,25)(10,18,54,26)(31,79,64,81)(32,80,65,82)(33,71,66,83)(34,72,67,84)(35,73,68,85)(36,74,69,86)(37,75,70,87)(38,76,61,88)(39,77,62,89)(40,78,63,90)(41,100,158,128)(42,91,159,129)(43,92,160,130)(44,93,151,121)(45,94,152,122)(46,95,153,123)(47,96,154,124)(48,97,155,125)(49,98,156,126)(50,99,157,127)(101,111,139,149)(102,112,140,150)(103,113,131,141)(104,114,132,142)(105,115,133,143)(106,116,134,144)(107,117,135,145)(108,118,136,146)(109,119,137,147)(110,120,138,148), (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), (1,10,63,62)(2,61,64,9)(3,8,65,70)(4,69,66,7)(5,6,67,68)(11,87,82,16)(12,15,83,86)(13,85,84,14)(17,81,88,20)(18,19,89,90)(21,73,72,22)(23,71,74,30)(24,29,75,80)(25,79,76,28)(26,27,77,78)(31,53,56,38)(32,37,57,52)(33,51,58,36)(34,35,59,60)(39,55,54,40)(41,104,105,50)(42,49,106,103)(43,102,107,48)(44,47,108,101)(45,110,109,46)(91,113,116,98)(92,97,117,112)(93,111,118,96)(94,95,119,120)(99,115,114,100)(121,149,146,124)(122,123,147,148)(125,145,150,130)(126,129,141,144)(127,143,142,128)(131,159,156,134)(132,133,157,158)(135,155,160,140)(136,139,151,154)(137,153,152,138)>;
G:=Group( (1,128,63,143)(2,144,64,129)(3,130,65,145)(4,146,66,121)(5,122,67,147)(6,148,68,123)(7,124,69,149)(8,150,70,125)(9,126,61,141)(10,142,62,127)(11,43,82,107)(12,108,83,44)(13,45,84,109)(14,110,85,46)(15,47,86,101)(16,102,87,48)(17,49,88,103)(18,104,89,50)(19,41,90,105)(20,106,81,42)(21,152,72,137)(22,138,73,153)(23,154,74,139)(24,140,75,155)(25,156,76,131)(26,132,77,157)(27,158,78,133)(28,134,79,159)(29,160,80,135)(30,136,71,151)(31,91,56,116)(32,117,57,92)(33,93,58,118)(34,119,59,94)(35,95,60,120)(36,111,51,96)(37,97,52,112)(38,113,53,98)(39,99,54,114)(40,115,55,100), (1,19,55,27)(2,20,56,28)(3,11,57,29)(4,12,58,30)(5,13,59,21)(6,14,60,22)(7,15,51,23)(8,16,52,24)(9,17,53,25)(10,18,54,26)(31,79,64,81)(32,80,65,82)(33,71,66,83)(34,72,67,84)(35,73,68,85)(36,74,69,86)(37,75,70,87)(38,76,61,88)(39,77,62,89)(40,78,63,90)(41,100,158,128)(42,91,159,129)(43,92,160,130)(44,93,151,121)(45,94,152,122)(46,95,153,123)(47,96,154,124)(48,97,155,125)(49,98,156,126)(50,99,157,127)(101,111,139,149)(102,112,140,150)(103,113,131,141)(104,114,132,142)(105,115,133,143)(106,116,134,144)(107,117,135,145)(108,118,136,146)(109,119,137,147)(110,120,138,148), (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), (1,10,63,62)(2,61,64,9)(3,8,65,70)(4,69,66,7)(5,6,67,68)(11,87,82,16)(12,15,83,86)(13,85,84,14)(17,81,88,20)(18,19,89,90)(21,73,72,22)(23,71,74,30)(24,29,75,80)(25,79,76,28)(26,27,77,78)(31,53,56,38)(32,37,57,52)(33,51,58,36)(34,35,59,60)(39,55,54,40)(41,104,105,50)(42,49,106,103)(43,102,107,48)(44,47,108,101)(45,110,109,46)(91,113,116,98)(92,97,117,112)(93,111,118,96)(94,95,119,120)(99,115,114,100)(121,149,146,124)(122,123,147,148)(125,145,150,130)(126,129,141,144)(127,143,142,128)(131,159,156,134)(132,133,157,158)(135,155,160,140)(136,139,151,154)(137,153,152,138) );
G=PermutationGroup([(1,128,63,143),(2,144,64,129),(3,130,65,145),(4,146,66,121),(5,122,67,147),(6,148,68,123),(7,124,69,149),(8,150,70,125),(9,126,61,141),(10,142,62,127),(11,43,82,107),(12,108,83,44),(13,45,84,109),(14,110,85,46),(15,47,86,101),(16,102,87,48),(17,49,88,103),(18,104,89,50),(19,41,90,105),(20,106,81,42),(21,152,72,137),(22,138,73,153),(23,154,74,139),(24,140,75,155),(25,156,76,131),(26,132,77,157),(27,158,78,133),(28,134,79,159),(29,160,80,135),(30,136,71,151),(31,91,56,116),(32,117,57,92),(33,93,58,118),(34,119,59,94),(35,95,60,120),(36,111,51,96),(37,97,52,112),(38,113,53,98),(39,99,54,114),(40,115,55,100)], [(1,19,55,27),(2,20,56,28),(3,11,57,29),(4,12,58,30),(5,13,59,21),(6,14,60,22),(7,15,51,23),(8,16,52,24),(9,17,53,25),(10,18,54,26),(31,79,64,81),(32,80,65,82),(33,71,66,83),(34,72,67,84),(35,73,68,85),(36,74,69,86),(37,75,70,87),(38,76,61,88),(39,77,62,89),(40,78,63,90),(41,100,158,128),(42,91,159,129),(43,92,160,130),(44,93,151,121),(45,94,152,122),(46,95,153,123),(47,96,154,124),(48,97,155,125),(49,98,156,126),(50,99,157,127),(101,111,139,149),(102,112,140,150),(103,113,131,141),(104,114,132,142),(105,115,133,143),(106,116,134,144),(107,117,135,145),(108,118,136,146),(109,119,137,147),(110,120,138,148)], [(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)], [(1,10,63,62),(2,61,64,9),(3,8,65,70),(4,69,66,7),(5,6,67,68),(11,87,82,16),(12,15,83,86),(13,85,84,14),(17,81,88,20),(18,19,89,90),(21,73,72,22),(23,71,74,30),(24,29,75,80),(25,79,76,28),(26,27,77,78),(31,53,56,38),(32,37,57,52),(33,51,58,36),(34,35,59,60),(39,55,54,40),(41,104,105,50),(42,49,106,103),(43,102,107,48),(44,47,108,101),(45,110,109,46),(91,113,116,98),(92,97,117,112),(93,111,118,96),(94,95,119,120),(99,115,114,100),(121,149,146,124),(122,123,147,148),(125,145,150,130),(126,129,141,144),(127,143,142,128),(131,159,156,134),(132,133,157,158),(135,155,160,140),(136,139,151,154),(137,153,152,138)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 22 | 0 |
| 0 | 0 | 0 | 40 | 0 | 22 |
| 0 | 0 | 26 | 0 | 1 | 0 |
| 0 | 0 | 0 | 26 | 0 | 1 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 1 | 0 | 0 |
| 0 | 0 | 40 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 1 |
| 0 | 0 | 0 | 0 | 40 | 24 |
| 0 | 34 | 0 | 0 | 0 | 0 |
| 6 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 25 | 26 | 7 |
| 0 | 0 | 16 | 16 | 34 | 34 |
| 0 | 0 | 7 | 35 | 36 | 16 |
| 0 | 0 | 6 | 6 | 25 | 25 |
| 6 | 34 | 0 | 0 | 0 | 0 |
| 5 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 5 | 7 | 26 |
| 0 | 0 | 16 | 16 | 34 | 34 |
| 0 | 0 | 35 | 7 | 16 | 36 |
| 0 | 0 | 6 | 6 | 25 | 25 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,26,0,0,0,0,40,0,26,0,0,22,0,1,0,0,0,0,22,0,1],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,17,40,0,0,0,0,1,24,0,0,0,0,0,0,17,40,0,0,0,0,1,24],[0,6,0,0,0,0,34,35,0,0,0,0,0,0,5,16,7,6,0,0,25,16,35,6,0,0,26,34,36,25,0,0,7,34,16,25],[6,5,0,0,0,0,34,35,0,0,0,0,0,0,25,16,35,6,0,0,5,16,7,6,0,0,7,34,16,25,0,0,26,34,36,25] >;
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4J | 4K | ··· | 4X | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | 10 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D5 | D10 | D10 | D10 | D10 | D10 | C4×D5 | 2+ (1+4) | 2- (1+4) | D4⋊6D10 | D4.10D10 |
| kernel | C42.108D10 | C4×Dic10 | C42⋊D5 | C23.11D10 | Dic5⋊4D4 | Dic5⋊3Q8 | D5×C4⋊C4 | C2×C10.D4 | C4×C5⋊D4 | D4×Dic5 | D4×C20 | C2×D4⋊2D5 | D4⋊2D5 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 16 | 2 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{108}D_{10} % in TeX
G:=Group("C4^2.108D10"); // GroupNames label
G:=SmallGroup(320,1218);
// by ID
G=gap.SmallGroup(320,1218);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,570,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀