metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.145D10, C10.932- (1+4), C10.742+ (1+4), C4.4D4⋊17D5, (C2×Q8).84D10, D10⋊D4⋊43C2, D10⋊3Q8⋊34C2, (C2×D4).113D10, C4.D20⋊31C2, C22⋊C4.38D10, C20.6Q8⋊29C2, Dic5⋊D4⋊35C2, (C4×C20).222C22, (C2×C20).633C23, (C2×C10).228C24, (C2×D20).37C22, C4⋊Dic5.52C22, D10.12D4⋊47C2, C2.54(D4⋊8D10), C2.78(D4⋊6D10), C23.50(C22×D5), (D4×C10).213C22, C22.D20⋊28C2, (C22×C10).58C23, (Q8×C10).131C22, C22.249(C23×D5), Dic5.14D4⋊43C2, C23.D5.60C22, D10⋊C4.73C22, C5⋊4(C22.56C24), (C2×Dic5).118C23, (C2×Dic10).41C22, C10.D4.84C22, (C22×D5).100C23, C2.54(D4.10D10), (C22×Dic5).147C22, (C5×C4.4D4)⋊20C2, (C2×C4×D5).132C22, (C2×C4).201(C22×D5), (C2×C5⋊D4).66C22, (C5×C22⋊C4).69C22, SmallGroup(320,1356)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 854 in 220 conjugacy classes, 91 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C4 [×11], C22, C22 [×12], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×10], D4 [×6], Q8 [×2], C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×2], C42, C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×10], C22×C4 [×4], C2×D4, C2×D4 [×5], C2×Q8, C2×Q8, Dic5 [×6], C20 [×5], D10 [×6], C2×C10, C2×C10 [×6], C4⋊D4 [×4], C22⋊Q8 [×4], C22.D4 [×4], C4.4D4, C4.4D4, C42.C2, Dic10, C4×D5 [×2], D20, C2×Dic5 [×6], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×3], C2×C20 [×2], C5×D4, C5×Q8, C22×D5 [×2], C22×C10 [×2], C22.56C24, C10.D4 [×6], C4⋊Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×6], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, C22×Dic5 [×2], C2×C5⋊D4 [×4], D4×C10, Q8×C10, C20.6Q8, C4.D20, Dic5.14D4 [×2], D10.12D4 [×2], D10⋊D4 [×2], C22.D20 [×2], Dic5⋊D4 [×2], D10⋊3Q8 [×2], C5×C4.4D4, C42.145D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ (1+4) [×2], 2- (1+4), C22×D5 [×7], C22.56C24, C23×D5, D4⋊6D10, D4⋊8D10, D4.10D10, C42.145D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c9 >
►On 160 points
Generators in S160
(1 121 108 62)(2 73 109 132)(3 123 110 64)(4 75 111 134)(5 125 112 66)(6 77 113 136)(7 127 114 68)(8 79 115 138)(9 129 116 70)(10 61 117 140)(11 131 118 72)(12 63 119 122)(13 133 120 74)(14 65 101 124)(15 135 102 76)(16 67 103 126)(17 137 104 78)(18 69 105 128)(19 139 106 80)(20 71 107 130)(21 49 153 91)(22 82 154 60)(23 51 155 93)(24 84 156 42)(25 53 157 95)(26 86 158 44)(27 55 159 97)(28 88 160 46)(29 57 141 99)(30 90 142 48)(31 59 143 81)(32 92 144 50)(33 41 145 83)(34 94 146 52)(35 43 147 85)(36 96 148 54)(37 45 149 87)(38 98 150 56)(39 47 151 89)(40 100 152 58)
(1 46 11 56)(2 57 12 47)(3 48 13 58)(4 59 14 49)(5 50 15 60)(6 41 16 51)(7 52 17 42)(8 43 18 53)(9 54 19 44)(10 45 20 55)(21 134 31 124)(22 125 32 135)(23 136 33 126)(24 127 34 137)(25 138 35 128)(26 129 36 139)(27 140 37 130)(28 131 38 121)(29 122 39 132)(30 133 40 123)(61 149 71 159)(62 160 72 150)(63 151 73 141)(64 142 74 152)(65 153 75 143)(66 144 76 154)(67 155 77 145)(68 146 78 156)(69 157 79 147)(70 148 80 158)(81 101 91 111)(82 112 92 102)(83 103 93 113)(84 114 94 104)(85 105 95 115)(86 116 96 106)(87 107 97 117)(88 118 98 108)(89 109 99 119)(90 120 100 110)
(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)(21 156 31 146)(22 145 32 155)(23 154 33 144)(24 143 34 153)(25 152 35 142)(26 141 36 151)(27 150 37 160)(28 159 38 149)(29 148 39 158)(30 157 40 147)(41 92 51 82)(42 81 52 91)(43 90 53 100)(44 99 54 89)(45 88 55 98)(46 97 56 87)(47 86 57 96)(48 95 58 85)(49 84 59 94)(50 93 60 83)(61 72 71 62)(63 70 73 80)(64 79 74 69)(65 68 75 78)(66 77 76 67)(101 114 111 104)(102 103 112 113)(105 110 115 120)(106 119 116 109)(107 108 117 118)(121 140 131 130)(122 129 132 139)(123 138 133 128)(124 127 134 137)(125 136 135 126)
G:=sub<Sym(160)| (1,121,108,62)(2,73,109,132)(3,123,110,64)(4,75,111,134)(5,125,112,66)(6,77,113,136)(7,127,114,68)(8,79,115,138)(9,129,116,70)(10,61,117,140)(11,131,118,72)(12,63,119,122)(13,133,120,74)(14,65,101,124)(15,135,102,76)(16,67,103,126)(17,137,104,78)(18,69,105,128)(19,139,106,80)(20,71,107,130)(21,49,153,91)(22,82,154,60)(23,51,155,93)(24,84,156,42)(25,53,157,95)(26,86,158,44)(27,55,159,97)(28,88,160,46)(29,57,141,99)(30,90,142,48)(31,59,143,81)(32,92,144,50)(33,41,145,83)(34,94,146,52)(35,43,147,85)(36,96,148,54)(37,45,149,87)(38,98,150,56)(39,47,151,89)(40,100,152,58), (1,46,11,56)(2,57,12,47)(3,48,13,58)(4,59,14,49)(5,50,15,60)(6,41,16,51)(7,52,17,42)(8,43,18,53)(9,54,19,44)(10,45,20,55)(21,134,31,124)(22,125,32,135)(23,136,33,126)(24,127,34,137)(25,138,35,128)(26,129,36,139)(27,140,37,130)(28,131,38,121)(29,122,39,132)(30,133,40,123)(61,149,71,159)(62,160,72,150)(63,151,73,141)(64,142,74,152)(65,153,75,143)(66,144,76,154)(67,155,77,145)(68,146,78,156)(69,157,79,147)(70,148,80,158)(81,101,91,111)(82,112,92,102)(83,103,93,113)(84,114,94,104)(85,105,95,115)(86,116,96,106)(87,107,97,117)(88,118,98,108)(89,109,99,119)(90,120,100,110), (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)(21,156,31,146)(22,145,32,155)(23,154,33,144)(24,143,34,153)(25,152,35,142)(26,141,36,151)(27,150,37,160)(28,159,38,149)(29,148,39,158)(30,157,40,147)(41,92,51,82)(42,81,52,91)(43,90,53,100)(44,99,54,89)(45,88,55,98)(46,97,56,87)(47,86,57,96)(48,95,58,85)(49,84,59,94)(50,93,60,83)(61,72,71,62)(63,70,73,80)(64,79,74,69)(65,68,75,78)(66,77,76,67)(101,114,111,104)(102,103,112,113)(105,110,115,120)(106,119,116,109)(107,108,117,118)(121,140,131,130)(122,129,132,139)(123,138,133,128)(124,127,134,137)(125,136,135,126)>;
G:=Group( (1,121,108,62)(2,73,109,132)(3,123,110,64)(4,75,111,134)(5,125,112,66)(6,77,113,136)(7,127,114,68)(8,79,115,138)(9,129,116,70)(10,61,117,140)(11,131,118,72)(12,63,119,122)(13,133,120,74)(14,65,101,124)(15,135,102,76)(16,67,103,126)(17,137,104,78)(18,69,105,128)(19,139,106,80)(20,71,107,130)(21,49,153,91)(22,82,154,60)(23,51,155,93)(24,84,156,42)(25,53,157,95)(26,86,158,44)(27,55,159,97)(28,88,160,46)(29,57,141,99)(30,90,142,48)(31,59,143,81)(32,92,144,50)(33,41,145,83)(34,94,146,52)(35,43,147,85)(36,96,148,54)(37,45,149,87)(38,98,150,56)(39,47,151,89)(40,100,152,58), (1,46,11,56)(2,57,12,47)(3,48,13,58)(4,59,14,49)(5,50,15,60)(6,41,16,51)(7,52,17,42)(8,43,18,53)(9,54,19,44)(10,45,20,55)(21,134,31,124)(22,125,32,135)(23,136,33,126)(24,127,34,137)(25,138,35,128)(26,129,36,139)(27,140,37,130)(28,131,38,121)(29,122,39,132)(30,133,40,123)(61,149,71,159)(62,160,72,150)(63,151,73,141)(64,142,74,152)(65,153,75,143)(66,144,76,154)(67,155,77,145)(68,146,78,156)(69,157,79,147)(70,148,80,158)(81,101,91,111)(82,112,92,102)(83,103,93,113)(84,114,94,104)(85,105,95,115)(86,116,96,106)(87,107,97,117)(88,118,98,108)(89,109,99,119)(90,120,100,110), (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)(21,156,31,146)(22,145,32,155)(23,154,33,144)(24,143,34,153)(25,152,35,142)(26,141,36,151)(27,150,37,160)(28,159,38,149)(29,148,39,158)(30,157,40,147)(41,92,51,82)(42,81,52,91)(43,90,53,100)(44,99,54,89)(45,88,55,98)(46,97,56,87)(47,86,57,96)(48,95,58,85)(49,84,59,94)(50,93,60,83)(61,72,71,62)(63,70,73,80)(64,79,74,69)(65,68,75,78)(66,77,76,67)(101,114,111,104)(102,103,112,113)(105,110,115,120)(106,119,116,109)(107,108,117,118)(121,140,131,130)(122,129,132,139)(123,138,133,128)(124,127,134,137)(125,136,135,126) );
G=PermutationGroup([(1,121,108,62),(2,73,109,132),(3,123,110,64),(4,75,111,134),(5,125,112,66),(6,77,113,136),(7,127,114,68),(8,79,115,138),(9,129,116,70),(10,61,117,140),(11,131,118,72),(12,63,119,122),(13,133,120,74),(14,65,101,124),(15,135,102,76),(16,67,103,126),(17,137,104,78),(18,69,105,128),(19,139,106,80),(20,71,107,130),(21,49,153,91),(22,82,154,60),(23,51,155,93),(24,84,156,42),(25,53,157,95),(26,86,158,44),(27,55,159,97),(28,88,160,46),(29,57,141,99),(30,90,142,48),(31,59,143,81),(32,92,144,50),(33,41,145,83),(34,94,146,52),(35,43,147,85),(36,96,148,54),(37,45,149,87),(38,98,150,56),(39,47,151,89),(40,100,152,58)], [(1,46,11,56),(2,57,12,47),(3,48,13,58),(4,59,14,49),(5,50,15,60),(6,41,16,51),(7,52,17,42),(8,43,18,53),(9,54,19,44),(10,45,20,55),(21,134,31,124),(22,125,32,135),(23,136,33,126),(24,127,34,137),(25,138,35,128),(26,129,36,139),(27,140,37,130),(28,131,38,121),(29,122,39,132),(30,133,40,123),(61,149,71,159),(62,160,72,150),(63,151,73,141),(64,142,74,152),(65,153,75,143),(66,144,76,154),(67,155,77,145),(68,146,78,156),(69,157,79,147),(70,148,80,158),(81,101,91,111),(82,112,92,102),(83,103,93,113),(84,114,94,104),(85,105,95,115),(86,116,96,106),(87,107,97,117),(88,118,98,108),(89,109,99,119),(90,120,100,110)], [(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),(21,156,31,146),(22,145,32,155),(23,154,33,144),(24,143,34,153),(25,152,35,142),(26,141,36,151),(27,150,37,160),(28,159,38,149),(29,148,39,158),(30,157,40,147),(41,92,51,82),(42,81,52,91),(43,90,53,100),(44,99,54,89),(45,88,55,98),(46,97,56,87),(47,86,57,96),(48,95,58,85),(49,84,59,94),(50,93,60,83),(61,72,71,62),(63,70,73,80),(64,79,74,69),(65,68,75,78),(66,77,76,67),(101,114,111,104),(102,103,112,113),(105,110,115,120),(106,119,116,109),(107,108,117,118),(121,140,131,130),(122,129,132,139),(123,138,133,128),(124,127,134,137),(125,136,135,126)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 39 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 10 | 23 | 7 |
| 0 | 0 | 0 | 0 | 9 | 22 | 9 | 14 |
| 0 | 0 | 0 | 0 | 0 | 16 | 10 | 9 |
| 0 | 0 | 0 | 0 | 25 | 16 | 32 | 38 |
| 40 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 40 | 13 | 13 | 0 | 0 | 0 | 0 |
| 3 | 3 | 1 | 0 | 0 | 0 | 0 | 0 |
| 38 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 36 | 15 | 26 |
| 0 | 0 | 0 | 0 | 40 | 18 | 0 | 26 |
| 0 | 0 | 0 | 0 | 0 | 0 | 17 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 24 |
| 5 | 22 | 30 | 35 | 0 | 0 | 0 | 0 |
| 28 | 33 | 22 | 36 | 0 | 0 | 0 | 0 |
| 0 | 28 | 27 | 19 | 0 | 0 | 0 | 0 |
| 17 | 13 | 22 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 25 | 25 | 38 |
| 0 | 0 | 0 | 0 | 5 | 33 | 18 | 7 |
| 0 | 0 | 0 | 0 | 32 | 37 | 3 | 34 |
| 0 | 0 | 0 | 0 | 32 | 0 | 36 | 26 |
| 21 | 35 | 6 | 11 | 0 | 0 | 0 | 0 |
| 12 | 20 | 5 | 19 | 0 | 0 | 0 | 0 |
| 24 | 12 | 22 | 14 | 0 | 0 | 0 | 0 |
| 39 | 27 | 24 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 25 | 7 | 30 |
| 0 | 0 | 0 | 0 | 3 | 33 | 14 | 2 |
| 0 | 0 | 0 | 0 | 13 | 37 | 38 | 16 |
| 0 | 0 | 0 | 0 | 9 | 0 | 5 | 15 |
G:=sub<GL(8,GF(41))| [39,4,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,11,32,0,0,0,0,0,0,9,30,0,0,0,0,0,0,0,0,12,9,0,25,0,0,0,0,10,22,16,16,0,0,0,0,23,9,10,32,0,0,0,0,7,14,9,38],[40,0,3,38,0,0,0,0,0,40,3,0,0,0,0,0,0,13,1,0,0,0,0,0,28,13,0,1,0,0,0,0,0,0,0,0,23,40,0,0,0,0,0,0,36,18,0,0,0,0,0,0,15,0,17,40,0,0,0,0,26,26,1,24],[5,28,0,17,0,0,0,0,22,33,28,13,0,0,0,0,30,22,27,22,0,0,0,0,35,36,19,17,0,0,0,0,0,0,0,0,20,5,32,32,0,0,0,0,25,33,37,0,0,0,0,0,25,18,3,36,0,0,0,0,38,7,34,26],[21,12,24,39,0,0,0,0,35,20,12,27,0,0,0,0,6,5,22,24,0,0,0,0,11,19,14,19,0,0,0,0,0,0,0,0,37,3,13,9,0,0,0,0,25,33,37,0,0,0,0,0,7,14,38,5,0,0,0,0,30,2,16,15] >;
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4E | 4F | ··· | 4K | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 20 | 20 | 4 | ··· | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | D10 | D10 | 2+ (1+4) | 2- (1+4) | D4⋊6D10 | D4⋊8D10 | D4.10D10 |
| kernel | C42.145D10 | C20.6Q8 | C4.D20 | Dic5.14D4 | D10.12D4 | D10⋊D4 | C22.D20 | Dic5⋊D4 | D10⋊3Q8 | C5×C4.4D4 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C10 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{145}D_{10} % in TeX
G:=Group("C4^2.145D10"); // GroupNames label
G:=SmallGroup(320,1356);
// by ID
G=gap.SmallGroup(320,1356);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=b^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^-1,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀