metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.164D10, C10.1412+ (1+4), C10.1022- (1+4), C20⋊Q8⋊41C2, C4⋊C4.119D10, C4.D20⋊9C2, C42⋊2C2⋊7D5, D10⋊Q8⋊45C2, D10⋊2Q8⋊42C2, (C4×Dic10)⋊16C2, D10⋊D4.5C2, (C4×C20).36C22, C22⋊C4.82D10, Dic5⋊4D4⋊38C2, (C2×C10).254C24, (C2×C20).605C23, (C2×D20).39C22, D10.13D4⋊43C2, C2.66(D4⋊8D10), C23.60(C22×D5), Dic5.22(C4○D4), Dic5.5D4⋊47C2, C22.D20⋊31C2, C4⋊Dic5.319C22, (C22×C10).68C23, C22.275(C23×D5), Dic5.14D4⋊47C2, C23.D5.69C22, D10⋊C4.47C22, (C4×Dic5).239C22, (C2×Dic5).277C23, (C22×D5).113C23, C2.66(D4.10D10), C5⋊10(C22.36C24), (C2×Dic10).264C22, C10.D4.127C22, (C22×Dic5).154C22, C4⋊C4⋊D5⋊44C2, C4⋊C4⋊7D5⋊42C2, C2.101(D5×C4○D4), (C5×C42⋊2C2)⋊9C2, C10.212(C2×C4○D4), (C2×C4×D5).145C22, (C5×C4⋊C4).206C22, (C2×C4).210(C22×D5), (C2×C5⋊D4).74C22, (C5×C22⋊C4).79C22, SmallGroup(320,1382)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 798 in 216 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×3], C4 [×13], C22, C22 [×9], C5, C2×C4 [×6], C2×C4 [×10], D4 [×4], Q8 [×4], C23, C23 [×2], D5 [×2], C10 [×3], C10, C42, C42 [×3], C22⋊C4 [×3], C22⋊C4 [×9], C4⋊C4 [×3], C4⋊C4 [×7], C22×C4 [×3], C2×D4 [×3], C2×Q8 [×3], Dic5 [×2], Dic5 [×5], C20 [×6], D10 [×6], C2×C10, C2×C10 [×3], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4 [×3], C42⋊2C2, C42⋊2C2, C4⋊Q8, Dic10 [×4], C4×D5 [×3], D20, C2×Dic5 [×6], C2×Dic5, C5⋊D4 [×3], C2×C20 [×6], C22×D5 [×2], C22×C10, C22.36C24, C4×Dic5 [×3], C10.D4 [×4], C4⋊Dic5 [×3], D10⋊C4 [×8], C23.D5, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×Dic10 [×3], C2×C4×D5 [×2], C2×D20, C22×Dic5, C2×C5⋊D4 [×2], C4×Dic10, C4.D20, Dic5.14D4, Dic5⋊4D4, D10⋊D4, Dic5.5D4 [×2], C22.D20, C20⋊Q8, C4⋊C4⋊7D5, D10.13D4, D10⋊Q8, D10⋊2Q8, C4⋊C4⋊D5, C5×C42⋊2C2, C42.164D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ (1+4), 2- (1+4), C22×D5 [×7], C22.36C24, C23×D5, D5×C4○D4, D4⋊8D10, D4.10D10, C42.164D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c9 >
►On 160 points
Generators in S160
(1 134 109 92)(2 125 110 83)(3 136 111 94)(4 127 112 85)(5 138 113 96)(6 129 114 87)(7 140 115 98)(8 131 116 89)(9 122 117 100)(10 133 118 91)(11 124 119 82)(12 135 120 93)(13 126 101 84)(14 137 102 95)(15 128 103 86)(16 139 104 97)(17 130 105 88)(18 121 106 99)(19 132 107 90)(20 123 108 81)(21 54 154 71)(22 45 155 62)(23 56 156 73)(24 47 157 64)(25 58 158 75)(26 49 159 66)(27 60 160 77)(28 51 141 68)(29 42 142 79)(30 53 143 70)(31 44 144 61)(32 55 145 72)(33 46 146 63)(34 57 147 74)(35 48 148 65)(36 59 149 76)(37 50 150 67)(38 41 151 78)(39 52 152 69)(40 43 153 80)
(1 72 11 62)(2 46 12 56)(3 74 13 64)(4 48 14 58)(5 76 15 66)(6 50 16 60)(7 78 17 68)(8 52 18 42)(9 80 19 70)(10 54 20 44)(21 81 31 91)(22 134 32 124)(23 83 33 93)(24 136 34 126)(25 85 35 95)(26 138 36 128)(27 87 37 97)(28 140 38 130)(29 89 39 99)(30 122 40 132)(41 105 51 115)(43 107 53 117)(45 109 55 119)(47 111 57 101)(49 113 59 103)(61 118 71 108)(63 120 73 110)(65 102 75 112)(67 104 77 114)(69 106 79 116)(82 155 92 145)(84 157 94 147)(86 159 96 149)(88 141 98 151)(90 143 100 153)(121 142 131 152)(123 144 133 154)(125 146 135 156)(127 148 137 158)(129 150 139 160)
(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)(21 155 31 145)(22 144 32 154)(23 153 33 143)(24 142 34 152)(25 151 35 141)(26 160 36 150)(27 149 37 159)(28 158 38 148)(29 147 39 157)(30 156 40 146)(41 48 51 58)(42 57 52 47)(43 46 53 56)(44 55 54 45)(49 60 59 50)(61 72 71 62)(63 70 73 80)(64 79 74 69)(65 68 75 78)(66 77 76 67)(81 134 91 124)(82 123 92 133)(83 132 93 122)(84 121 94 131)(85 130 95 140)(86 139 96 129)(87 128 97 138)(88 137 98 127)(89 126 99 136)(90 135 100 125)(101 106 111 116)(102 115 112 105)(103 104 113 114)(107 120 117 110)(108 109 118 119)
G:=sub<Sym(160)| (1,134,109,92)(2,125,110,83)(3,136,111,94)(4,127,112,85)(5,138,113,96)(6,129,114,87)(7,140,115,98)(8,131,116,89)(9,122,117,100)(10,133,118,91)(11,124,119,82)(12,135,120,93)(13,126,101,84)(14,137,102,95)(15,128,103,86)(16,139,104,97)(17,130,105,88)(18,121,106,99)(19,132,107,90)(20,123,108,81)(21,54,154,71)(22,45,155,62)(23,56,156,73)(24,47,157,64)(25,58,158,75)(26,49,159,66)(27,60,160,77)(28,51,141,68)(29,42,142,79)(30,53,143,70)(31,44,144,61)(32,55,145,72)(33,46,146,63)(34,57,147,74)(35,48,148,65)(36,59,149,76)(37,50,150,67)(38,41,151,78)(39,52,152,69)(40,43,153,80), (1,72,11,62)(2,46,12,56)(3,74,13,64)(4,48,14,58)(5,76,15,66)(6,50,16,60)(7,78,17,68)(8,52,18,42)(9,80,19,70)(10,54,20,44)(21,81,31,91)(22,134,32,124)(23,83,33,93)(24,136,34,126)(25,85,35,95)(26,138,36,128)(27,87,37,97)(28,140,38,130)(29,89,39,99)(30,122,40,132)(41,105,51,115)(43,107,53,117)(45,109,55,119)(47,111,57,101)(49,113,59,103)(61,118,71,108)(63,120,73,110)(65,102,75,112)(67,104,77,114)(69,106,79,116)(82,155,92,145)(84,157,94,147)(86,159,96,149)(88,141,98,151)(90,143,100,153)(121,142,131,152)(123,144,133,154)(125,146,135,156)(127,148,137,158)(129,150,139,160), (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)(21,155,31,145)(22,144,32,154)(23,153,33,143)(24,142,34,152)(25,151,35,141)(26,160,36,150)(27,149,37,159)(28,158,38,148)(29,147,39,157)(30,156,40,146)(41,48,51,58)(42,57,52,47)(43,46,53,56)(44,55,54,45)(49,60,59,50)(61,72,71,62)(63,70,73,80)(64,79,74,69)(65,68,75,78)(66,77,76,67)(81,134,91,124)(82,123,92,133)(83,132,93,122)(84,121,94,131)(85,130,95,140)(86,139,96,129)(87,128,97,138)(88,137,98,127)(89,126,99,136)(90,135,100,125)(101,106,111,116)(102,115,112,105)(103,104,113,114)(107,120,117,110)(108,109,118,119)>;
G:=Group( (1,134,109,92)(2,125,110,83)(3,136,111,94)(4,127,112,85)(5,138,113,96)(6,129,114,87)(7,140,115,98)(8,131,116,89)(9,122,117,100)(10,133,118,91)(11,124,119,82)(12,135,120,93)(13,126,101,84)(14,137,102,95)(15,128,103,86)(16,139,104,97)(17,130,105,88)(18,121,106,99)(19,132,107,90)(20,123,108,81)(21,54,154,71)(22,45,155,62)(23,56,156,73)(24,47,157,64)(25,58,158,75)(26,49,159,66)(27,60,160,77)(28,51,141,68)(29,42,142,79)(30,53,143,70)(31,44,144,61)(32,55,145,72)(33,46,146,63)(34,57,147,74)(35,48,148,65)(36,59,149,76)(37,50,150,67)(38,41,151,78)(39,52,152,69)(40,43,153,80), (1,72,11,62)(2,46,12,56)(3,74,13,64)(4,48,14,58)(5,76,15,66)(6,50,16,60)(7,78,17,68)(8,52,18,42)(9,80,19,70)(10,54,20,44)(21,81,31,91)(22,134,32,124)(23,83,33,93)(24,136,34,126)(25,85,35,95)(26,138,36,128)(27,87,37,97)(28,140,38,130)(29,89,39,99)(30,122,40,132)(41,105,51,115)(43,107,53,117)(45,109,55,119)(47,111,57,101)(49,113,59,103)(61,118,71,108)(63,120,73,110)(65,102,75,112)(67,104,77,114)(69,106,79,116)(82,155,92,145)(84,157,94,147)(86,159,96,149)(88,141,98,151)(90,143,100,153)(121,142,131,152)(123,144,133,154)(125,146,135,156)(127,148,137,158)(129,150,139,160), (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)(21,155,31,145)(22,144,32,154)(23,153,33,143)(24,142,34,152)(25,151,35,141)(26,160,36,150)(27,149,37,159)(28,158,38,148)(29,147,39,157)(30,156,40,146)(41,48,51,58)(42,57,52,47)(43,46,53,56)(44,55,54,45)(49,60,59,50)(61,72,71,62)(63,70,73,80)(64,79,74,69)(65,68,75,78)(66,77,76,67)(81,134,91,124)(82,123,92,133)(83,132,93,122)(84,121,94,131)(85,130,95,140)(86,139,96,129)(87,128,97,138)(88,137,98,127)(89,126,99,136)(90,135,100,125)(101,106,111,116)(102,115,112,105)(103,104,113,114)(107,120,117,110)(108,109,118,119) );
G=PermutationGroup([(1,134,109,92),(2,125,110,83),(3,136,111,94),(4,127,112,85),(5,138,113,96),(6,129,114,87),(7,140,115,98),(8,131,116,89),(9,122,117,100),(10,133,118,91),(11,124,119,82),(12,135,120,93),(13,126,101,84),(14,137,102,95),(15,128,103,86),(16,139,104,97),(17,130,105,88),(18,121,106,99),(19,132,107,90),(20,123,108,81),(21,54,154,71),(22,45,155,62),(23,56,156,73),(24,47,157,64),(25,58,158,75),(26,49,159,66),(27,60,160,77),(28,51,141,68),(29,42,142,79),(30,53,143,70),(31,44,144,61),(32,55,145,72),(33,46,146,63),(34,57,147,74),(35,48,148,65),(36,59,149,76),(37,50,150,67),(38,41,151,78),(39,52,152,69),(40,43,153,80)], [(1,72,11,62),(2,46,12,56),(3,74,13,64),(4,48,14,58),(5,76,15,66),(6,50,16,60),(7,78,17,68),(8,52,18,42),(9,80,19,70),(10,54,20,44),(21,81,31,91),(22,134,32,124),(23,83,33,93),(24,136,34,126),(25,85,35,95),(26,138,36,128),(27,87,37,97),(28,140,38,130),(29,89,39,99),(30,122,40,132),(41,105,51,115),(43,107,53,117),(45,109,55,119),(47,111,57,101),(49,113,59,103),(61,118,71,108),(63,120,73,110),(65,102,75,112),(67,104,77,114),(69,106,79,116),(82,155,92,145),(84,157,94,147),(86,159,96,149),(88,141,98,151),(90,143,100,153),(121,142,131,152),(123,144,133,154),(125,146,135,156),(127,148,137,158),(129,150,139,160)], [(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),(21,155,31,145),(22,144,32,154),(23,153,33,143),(24,142,34,152),(25,151,35,141),(26,160,36,150),(27,149,37,159),(28,158,38,148),(29,147,39,157),(30,156,40,146),(41,48,51,58),(42,57,52,47),(43,46,53,56),(44,55,54,45),(49,60,59,50),(61,72,71,62),(63,70,73,80),(64,79,74,69),(65,68,75,78),(66,77,76,67),(81,134,91,124),(82,123,92,133),(83,132,93,122),(84,121,94,131),(85,130,95,140),(86,139,96,129),(87,128,97,138),(88,137,98,127),(89,126,99,136),(90,135,100,125),(101,106,111,116),(102,115,112,105),(103,104,113,114),(107,120,117,110),(108,109,118,119)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 32 | 39 | 0 | 0 | 0 | 0 |
| 40 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 13 | 0 | 0 |
| 0 | 0 | 28 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 13 |
| 0 | 0 | 0 | 0 | 28 | 39 |
| 40 | 18 | 0 | 0 | 0 | 0 |
| 9 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 1 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 32 | 39 | 39 |
| 0 | 0 | 9 | 19 | 2 | 27 |
| 0 | 0 | 2 | 2 | 9 | 9 |
| 0 | 0 | 39 | 14 | 32 | 22 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 32 | 39 | 39 |
| 0 | 0 | 19 | 9 | 27 | 2 |
| 0 | 0 | 2 | 2 | 9 | 9 |
| 0 | 0 | 14 | 39 | 22 | 32 |
G:=sub<GL(6,GF(41))| [32,40,0,0,0,0,39,9,0,0,0,0,0,0,2,28,0,0,0,0,13,39,0,0,0,0,0,0,2,28,0,0,0,0,13,39],[40,9,0,0,0,0,18,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[9,1,0,0,0,0,0,32,0,0,0,0,0,0,32,9,2,39,0,0,32,19,2,14,0,0,39,2,9,32,0,0,39,27,9,22],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,32,19,2,14,0,0,32,9,2,39,0,0,39,27,9,22,0,0,39,2,9,32] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 20A | ··· | 20L | 20M | ··· | 20R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | 2+ (1+4) | 2- (1+4) | D5×C4○D4 | D4⋊8D10 | D4.10D10 |
| kernel | C42.164D10 | C4×Dic10 | C4.D20 | Dic5.14D4 | Dic5⋊4D4 | D10⋊D4 | Dic5.5D4 | C22.D20 | C20⋊Q8 | C4⋊C4⋊7D5 | D10.13D4 | D10⋊Q8 | D10⋊2Q8 | C4⋊C4⋊D5 | C5×C42⋊2C2 | C42⋊2C2 | Dic5 | C42 | C22⋊C4 | C4⋊C4 | C10 | C10 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 6 | 6 | 1 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{164}D_{10} % in TeX
G:=Group("C4^2.164D10"); // GroupNames label
G:=SmallGroup(320,1382);
// by ID
G=gap.SmallGroup(320,1382);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,268,675,570,80,12550]);
// 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*b^2,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀