metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4○D20⋊9C4, D20⋊30(C2×C4), C4.60(C2×D20), (C2×C4).43D20, C4⋊C4.226D10, C20.140(C2×D4), (C2×C20).134D4, D20⋊6C4⋊24C2, Dic10⋊27(C2×C4), C10.Q16⋊24C2, (C22×C4).93D10, C10.83(C8⋊C22), C20.67(C22⋊C4), C20.119(C22×C4), (C2×C20).319C23, (C22×C10).184D4, C23.76(C5⋊D4), C5⋊4(C23.36D4), C4.37(D10⋊C4), C2.2(D4.D10), C2.2(C20.C23), (C2×D20).241C22, C10.83(C8.C22), C22.4(D10⋊C4), (C22×C20).134C22, (C2×Dic10).268C22, (C2×C4⋊C4)⋊2D5, (C10×C4⋊C4)⋊2C2, C4.49(C2×C4×D5), (C2×C4).39(C4×D5), (C2×C4○D20).6C2, (C2×C4.Dic5)⋊8C2, (C2×C20).251(C2×C4), (C2×C10).439(C2×D4), C10.80(C2×C22⋊C4), C22.58(C2×C5⋊D4), C2.12(C2×D10⋊C4), (C2×C4).182(C5⋊D4), (C5×C4⋊C4).257C22, (C2×C5⋊2C8).80C22, (C2×C4).419(C22×D5), (C2×C10).122(C22⋊C4), SmallGroup(320,593)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4○D20⋊9C4
G = < a,b,c,d | a4=c2=d4=1, b10=a2, ab=ba, ac=ca, dad-1=a-1, cbc=a2b9, dbd-1=a2b, dcd-1=a2b5c >
Subgroups: 606 in 162 conjugacy classes, 63 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, C20, D10, C2×C10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C2×C4⋊C4, C2×M4(2), C2×C4○D4, C5⋊2C8, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C22×D5, C22×C10, C23.36D4, C2×C5⋊2C8, C4.Dic5, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C4○D20, C2×C5⋊D4, C22×C20, C22×C20, D20⋊6C4, C10.Q16, C2×C4.Dic5, C10×C4⋊C4, C2×C4○D20, C4○D20⋊9C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C8⋊C22, C8.C22, C4×D5, D20, C5⋊D4, C22×D5, C23.36D4, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4, D4.D10, C20.C23, C4○D20⋊9C4
►On 160 points
Generators in S160
(1 86 11 96)(2 87 12 97)(3 88 13 98)(4 89 14 99)(5 90 15 100)(6 91 16 81)(7 92 17 82)(8 93 18 83)(9 94 19 84)(10 95 20 85)(21 116 31 106)(22 117 32 107)(23 118 33 108)(24 119 34 109)(25 120 35 110)(26 101 36 111)(27 102 37 112)(28 103 38 113)(29 104 39 114)(30 105 40 115)(41 133 51 123)(42 134 52 124)(43 135 53 125)(44 136 54 126)(45 137 55 127)(46 138 56 128)(47 139 57 129)(48 140 58 130)(49 121 59 131)(50 122 60 132)(61 156 71 146)(62 157 72 147)(63 158 73 148)(64 159 74 149)(65 160 75 150)(66 141 76 151)(67 142 77 152)(68 143 78 153)(69 144 79 154)(70 145 80 155)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(33 40)(34 39)(35 38)(36 37)(41 51)(42 50)(43 49)(44 48)(45 47)(52 60)(53 59)(54 58)(55 57)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(81 90)(82 89)(83 88)(84 87)(85 86)(91 100)(92 99)(93 98)(94 97)(95 96)(101 102)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(136 140)(137 139)(141 151)(142 150)(143 149)(144 148)(145 147)(152 160)(153 159)(154 158)(155 157)
(1 69 27 44)(2 80 28 55)(3 71 29 46)(4 62 30 57)(5 73 31 48)(6 64 32 59)(7 75 33 50)(8 66 34 41)(9 77 35 52)(10 68 36 43)(11 79 37 54)(12 70 38 45)(13 61 39 56)(14 72 40 47)(15 63 21 58)(16 74 22 49)(17 65 23 60)(18 76 24 51)(19 67 25 42)(20 78 26 53)(81 159 117 131)(82 150 118 122)(83 141 119 133)(84 152 120 124)(85 143 101 135)(86 154 102 126)(87 145 103 137)(88 156 104 128)(89 147 105 139)(90 158 106 130)(91 149 107 121)(92 160 108 132)(93 151 109 123)(94 142 110 134)(95 153 111 125)(96 144 112 136)(97 155 113 127)(98 146 114 138)(99 157 115 129)(100 148 116 140)
G:=sub<Sym(160)| (1,86,11,96)(2,87,12,97)(3,88,13,98)(4,89,14,99)(5,90,15,100)(6,91,16,81)(7,92,17,82)(8,93,18,83)(9,94,19,84)(10,95,20,85)(21,116,31,106)(22,117,32,107)(23,118,33,108)(24,119,34,109)(25,120,35,110)(26,101,36,111)(27,102,37,112)(28,103,38,113)(29,104,39,114)(30,105,40,115)(41,133,51,123)(42,134,52,124)(43,135,53,125)(44,136,54,126)(45,137,55,127)(46,138,56,128)(47,139,57,129)(48,140,58,130)(49,121,59,131)(50,122,60,132)(61,156,71,146)(62,157,72,147)(63,158,73,148)(64,159,74,149)(65,160,75,150)(66,141,76,151)(67,142,77,152)(68,143,78,153)(69,144,79,154)(70,145,80,155), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(33,40)(34,39)(35,38)(36,37)(41,51)(42,50)(43,49)(44,48)(45,47)(52,60)(53,59)(54,58)(55,57)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(81,90)(82,89)(83,88)(84,87)(85,86)(91,100)(92,99)(93,98)(94,97)(95,96)(101,102)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,140)(137,139)(141,151)(142,150)(143,149)(144,148)(145,147)(152,160)(153,159)(154,158)(155,157), (1,69,27,44)(2,80,28,55)(3,71,29,46)(4,62,30,57)(5,73,31,48)(6,64,32,59)(7,75,33,50)(8,66,34,41)(9,77,35,52)(10,68,36,43)(11,79,37,54)(12,70,38,45)(13,61,39,56)(14,72,40,47)(15,63,21,58)(16,74,22,49)(17,65,23,60)(18,76,24,51)(19,67,25,42)(20,78,26,53)(81,159,117,131)(82,150,118,122)(83,141,119,133)(84,152,120,124)(85,143,101,135)(86,154,102,126)(87,145,103,137)(88,156,104,128)(89,147,105,139)(90,158,106,130)(91,149,107,121)(92,160,108,132)(93,151,109,123)(94,142,110,134)(95,153,111,125)(96,144,112,136)(97,155,113,127)(98,146,114,138)(99,157,115,129)(100,148,116,140)>;
G:=Group( (1,86,11,96)(2,87,12,97)(3,88,13,98)(4,89,14,99)(5,90,15,100)(6,91,16,81)(7,92,17,82)(8,93,18,83)(9,94,19,84)(10,95,20,85)(21,116,31,106)(22,117,32,107)(23,118,33,108)(24,119,34,109)(25,120,35,110)(26,101,36,111)(27,102,37,112)(28,103,38,113)(29,104,39,114)(30,105,40,115)(41,133,51,123)(42,134,52,124)(43,135,53,125)(44,136,54,126)(45,137,55,127)(46,138,56,128)(47,139,57,129)(48,140,58,130)(49,121,59,131)(50,122,60,132)(61,156,71,146)(62,157,72,147)(63,158,73,148)(64,159,74,149)(65,160,75,150)(66,141,76,151)(67,142,77,152)(68,143,78,153)(69,144,79,154)(70,145,80,155), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(33,40)(34,39)(35,38)(36,37)(41,51)(42,50)(43,49)(44,48)(45,47)(52,60)(53,59)(54,58)(55,57)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(81,90)(82,89)(83,88)(84,87)(85,86)(91,100)(92,99)(93,98)(94,97)(95,96)(101,102)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,140)(137,139)(141,151)(142,150)(143,149)(144,148)(145,147)(152,160)(153,159)(154,158)(155,157), (1,69,27,44)(2,80,28,55)(3,71,29,46)(4,62,30,57)(5,73,31,48)(6,64,32,59)(7,75,33,50)(8,66,34,41)(9,77,35,52)(10,68,36,43)(11,79,37,54)(12,70,38,45)(13,61,39,56)(14,72,40,47)(15,63,21,58)(16,74,22,49)(17,65,23,60)(18,76,24,51)(19,67,25,42)(20,78,26,53)(81,159,117,131)(82,150,118,122)(83,141,119,133)(84,152,120,124)(85,143,101,135)(86,154,102,126)(87,145,103,137)(88,156,104,128)(89,147,105,139)(90,158,106,130)(91,149,107,121)(92,160,108,132)(93,151,109,123)(94,142,110,134)(95,153,111,125)(96,144,112,136)(97,155,113,127)(98,146,114,138)(99,157,115,129)(100,148,116,140) );
G=PermutationGroup([[(1,86,11,96),(2,87,12,97),(3,88,13,98),(4,89,14,99),(5,90,15,100),(6,91,16,81),(7,92,17,82),(8,93,18,83),(9,94,19,84),(10,95,20,85),(21,116,31,106),(22,117,32,107),(23,118,33,108),(24,119,34,109),(25,120,35,110),(26,101,36,111),(27,102,37,112),(28,103,38,113),(29,104,39,114),(30,105,40,115),(41,133,51,123),(42,134,52,124),(43,135,53,125),(44,136,54,126),(45,137,55,127),(46,138,56,128),(47,139,57,129),(48,140,58,130),(49,121,59,131),(50,122,60,132),(61,156,71,146),(62,157,72,147),(63,158,73,148),(64,159,74,149),(65,160,75,150),(66,141,76,151),(67,142,77,152),(68,143,78,153),(69,144,79,154),(70,145,80,155)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27),(33,40),(34,39),(35,38),(36,37),(41,51),(42,50),(43,49),(44,48),(45,47),(52,60),(53,59),(54,58),(55,57),(62,80),(63,79),(64,78),(65,77),(66,76),(67,75),(68,74),(69,73),(70,72),(81,90),(82,89),(83,88),(84,87),(85,86),(91,100),(92,99),(93,98),(94,97),(95,96),(101,102),(103,120),(104,119),(105,118),(106,117),(107,116),(108,115),(109,114),(110,113),(111,112),(121,135),(122,134),(123,133),(124,132),(125,131),(126,130),(127,129),(136,140),(137,139),(141,151),(142,150),(143,149),(144,148),(145,147),(152,160),(153,159),(154,158),(155,157)], [(1,69,27,44),(2,80,28,55),(3,71,29,46),(4,62,30,57),(5,73,31,48),(6,64,32,59),(7,75,33,50),(8,66,34,41),(9,77,35,52),(10,68,36,43),(11,79,37,54),(12,70,38,45),(13,61,39,56),(14,72,40,47),(15,63,21,58),(16,74,22,49),(17,65,23,60),(18,76,24,51),(19,67,25,42),(20,78,26,53),(81,159,117,131),(82,150,118,122),(83,141,119,133),(84,152,120,124),(85,143,101,135),(86,154,102,126),(87,145,103,137),(88,156,104,128),(89,147,105,139),(90,158,106,130),(91,149,107,121),(92,160,108,132),(93,151,109,123),(94,142,110,134),(95,153,111,125),(96,144,112,136),(97,155,113,127),(98,146,114,138),(99,157,115,129),(100,148,116,140)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | D10 | D10 | C4×D5 | D20 | C5⋊D4 | C5⋊D4 | C8⋊C22 | C8.C22 | D4.D10 | C20.C23 |
| kernel | C4○D20⋊9C4 | D20⋊6C4 | C10.Q16 | C2×C4.Dic5 | C10×C4⋊C4 | C2×C4○D20 | C4○D20 | C2×C20 | C22×C10 | C2×C4⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 3 | 1 | 2 | 4 | 2 | 8 | 8 | 4 | 4 | 1 | 1 | 4 | 4 |
Matrix representation of C4○D20⋊9C4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 6 | 5 | 12 |
| 0 | 0 | 35 | 18 | 29 | 36 |
| 0 | 0 | 18 | 35 | 18 | 35 |
| 0 | 0 | 6 | 23 | 6 | 23 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 39 |
| 0 | 0 | 1 | 35 | 2 | 29 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 40 | 6 | 40 | 6 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 9 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 39 |
| 0 | 0 | 40 | 0 | 39 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 32 | 39 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 13 | 34 | 13 |
| 0 | 0 | 28 | 39 | 28 | 30 |
| 0 | 0 | 15 | 14 | 39 | 28 |
| 0 | 0 | 27 | 17 | 13 | 2 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,23,35,18,6,0,0,6,18,35,23,0,0,5,29,18,6,0,0,12,36,35,23],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,40,0,0,40,35,1,6,0,0,0,2,0,40,0,0,39,29,1,6],[40,9,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,39,0,1,0,0,39,0,1,0],[32,0,0,0,0,0,39,9,0,0,0,0,0,0,2,28,15,27,0,0,13,39,14,17,0,0,34,28,39,13,0,0,13,30,28,2] >;
C4○D20⋊9C4 in GAP, Magma, Sage, TeX
C_4\circ D_{20}\rtimes_9C_4 % in TeX
G:=Group("C4oD20:9C4"); // GroupNames label
G:=SmallGroup(320,593);
// by ID
G=gap.SmallGroup(320,593);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,422,387,58,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^2=d^4=1,b^10=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c=a^2*b^9,d*b*d^-1=a^2*b,d*c*d^-1=a^2*b^5*c>;
// generators/relations