metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.114D10, C10.202+ (1+4), (C4×D4)⋊21D5, (D4×C20)⋊23C2, C4⋊C4.319D10, C20⋊D4.9C2, D20⋊8C4⋊16C2, (C4×Dic10)⋊34C2, (C2×D4).220D10, C4.16(C4○D20), C4.D20⋊19C2, C20.17D4⋊9C2, (C22×C4).48D10, Dic5⋊3Q8⋊16C2, D10.12D4⋊8C2, C20.111(C4○D4), (C2×C10).103C24, (C2×C20).701C23, (C4×C20).158C22, C22⋊C4.116D10, Dic5.5D4⋊8C2, C2.21(D4⋊6D10), Dic5.61(C4○D4), (D4×C10).263C22, (C2×D20).145C22, C4⋊Dic5.301C22, (C4×Dic5).84C22, (C2×Dic5).44C23, (C22×D5).37C23, C22.128(C23×D5), C23.100(C22×D5), D10⋊C4.87C22, C23.23D10⋊18C2, (C22×C20).365C22, (C22×C10).173C23, C5⋊1(C22.53C24), C10.D4.66C22, C23.D5.107C22, (C2×Dic10).151C22, (C4×C5⋊D4)⋊45C2, C2.26(D5×C4○D4), C2.52(C2×C4○D20), C10.45(C2×C4○D4), (C2×C4×D5).253C22, (C5×C4⋊C4).332C22, (C2×C4).286(C22×D5), (C2×C5⋊D4).124C22, (C5×C22⋊C4).127C22, SmallGroup(320,1231)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 838 in 236 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×12], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×10], D4 [×10], Q8 [×4], C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×2], C42, C42 [×4], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×5], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×5], C2×Q8 [×2], Dic5 [×2], Dic5 [×5], C20 [×2], C20 [×4], D10 [×6], C2×C10, C2×C10 [×6], C4×D4, C4×D4 [×3], C4×Q8 [×2], C22.D4 [×4], C4.4D4 [×4], C4⋊1D4, Dic10 [×4], C4×D5 [×2], D20 [×2], C2×Dic5 [×4], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×3], C2×C20 [×2], C2×C20 [×2], C5×D4 [×2], C22×D5 [×2], C22×C10 [×2], C22.53C24, C4×Dic5 [×2], C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×6], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20, C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C4×Dic10, C4.D20, D10.12D4 [×2], Dic5.5D4 [×2], Dic5⋊3Q8, D20⋊8C4, C4×C5⋊D4 [×2], C23.23D10 [×2], C20.17D4, C20⋊D4, D4×C20, C42.114D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D5 [×7], C22.53C24, C4○D20 [×2], C23×D5, C2×C4○D20, D4⋊6D10, D5×C4○D4, C42.114D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=c9 >
►On 160 points
Generators in S160
(1 55 40 155)(2 146 21 46)(3 57 22 157)(4 148 23 48)(5 59 24 159)(6 150 25 50)(7 41 26 141)(8 152 27 52)(9 43 28 143)(10 154 29 54)(11 45 30 145)(12 156 31 56)(13 47 32 147)(14 158 33 58)(15 49 34 149)(16 160 35 60)(17 51 36 151)(18 142 37 42)(19 53 38 153)(20 144 39 44)(61 94 129 119)(62 110 130 85)(63 96 131 101)(64 112 132 87)(65 98 133 103)(66 114 134 89)(67 100 135 105)(68 116 136 91)(69 82 137 107)(70 118 138 93)(71 84 139 109)(72 120 140 95)(73 86 121 111)(74 102 122 97)(75 88 123 113)(76 104 124 99)(77 90 125 115)(78 106 126 81)(79 92 127 117)(80 108 128 83)
(1 102 11 112)(2 103 12 113)(3 104 13 114)(4 105 14 115)(5 106 15 116)(6 107 16 117)(7 108 17 118)(8 109 18 119)(9 110 19 120)(10 111 20 101)(21 98 31 88)(22 99 32 89)(23 100 33 90)(24 81 34 91)(25 82 35 92)(26 83 36 93)(27 84 37 94)(28 85 38 95)(29 86 39 96)(30 87 40 97)(41 128 51 138)(42 129 52 139)(43 130 53 140)(44 131 54 121)(45 132 55 122)(46 133 56 123)(47 134 57 124)(48 135 58 125)(49 136 59 126)(50 137 60 127)(61 152 71 142)(62 153 72 143)(63 154 73 144)(64 155 74 145)(65 156 75 146)(66 157 76 147)(67 158 77 148)(68 159 78 149)(69 160 79 150)(70 141 80 151)
(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 38 31 28)(22 27 32 37)(23 36 33 26)(24 25 34 35)(29 30 39 40)(41 48 51 58)(42 57 52 47)(43 46 53 56)(44 55 54 45)(49 60 59 50)(61 124 71 134)(62 133 72 123)(63 122 73 132)(64 131 74 121)(65 140 75 130)(66 129 76 139)(67 138 77 128)(68 127 78 137)(69 136 79 126)(70 125 80 135)(81 107 91 117)(82 116 92 106)(83 105 93 115)(84 114 94 104)(85 103 95 113)(86 112 96 102)(87 101 97 111)(88 110 98 120)(89 119 99 109)(90 108 100 118)(141 148 151 158)(142 157 152 147)(143 146 153 156)(144 155 154 145)(149 160 159 150)
G:=sub<Sym(160)| (1,55,40,155)(2,146,21,46)(3,57,22,157)(4,148,23,48)(5,59,24,159)(6,150,25,50)(7,41,26,141)(8,152,27,52)(9,43,28,143)(10,154,29,54)(11,45,30,145)(12,156,31,56)(13,47,32,147)(14,158,33,58)(15,49,34,149)(16,160,35,60)(17,51,36,151)(18,142,37,42)(19,53,38,153)(20,144,39,44)(61,94,129,119)(62,110,130,85)(63,96,131,101)(64,112,132,87)(65,98,133,103)(66,114,134,89)(67,100,135,105)(68,116,136,91)(69,82,137,107)(70,118,138,93)(71,84,139,109)(72,120,140,95)(73,86,121,111)(74,102,122,97)(75,88,123,113)(76,104,124,99)(77,90,125,115)(78,106,126,81)(79,92,127,117)(80,108,128,83), (1,102,11,112)(2,103,12,113)(3,104,13,114)(4,105,14,115)(5,106,15,116)(6,107,16,117)(7,108,17,118)(8,109,18,119)(9,110,19,120)(10,111,20,101)(21,98,31,88)(22,99,32,89)(23,100,33,90)(24,81,34,91)(25,82,35,92)(26,83,36,93)(27,84,37,94)(28,85,38,95)(29,86,39,96)(30,87,40,97)(41,128,51,138)(42,129,52,139)(43,130,53,140)(44,131,54,121)(45,132,55,122)(46,133,56,123)(47,134,57,124)(48,135,58,125)(49,136,59,126)(50,137,60,127)(61,152,71,142)(62,153,72,143)(63,154,73,144)(64,155,74,145)(65,156,75,146)(66,157,76,147)(67,158,77,148)(68,159,78,149)(69,160,79,150)(70,141,80,151), (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)(21,38,31,28)(22,27,32,37)(23,36,33,26)(24,25,34,35)(29,30,39,40)(41,48,51,58)(42,57,52,47)(43,46,53,56)(44,55,54,45)(49,60,59,50)(61,124,71,134)(62,133,72,123)(63,122,73,132)(64,131,74,121)(65,140,75,130)(66,129,76,139)(67,138,77,128)(68,127,78,137)(69,136,79,126)(70,125,80,135)(81,107,91,117)(82,116,92,106)(83,105,93,115)(84,114,94,104)(85,103,95,113)(86,112,96,102)(87,101,97,111)(88,110,98,120)(89,119,99,109)(90,108,100,118)(141,148,151,158)(142,157,152,147)(143,146,153,156)(144,155,154,145)(149,160,159,150)>;
G:=Group( (1,55,40,155)(2,146,21,46)(3,57,22,157)(4,148,23,48)(5,59,24,159)(6,150,25,50)(7,41,26,141)(8,152,27,52)(9,43,28,143)(10,154,29,54)(11,45,30,145)(12,156,31,56)(13,47,32,147)(14,158,33,58)(15,49,34,149)(16,160,35,60)(17,51,36,151)(18,142,37,42)(19,53,38,153)(20,144,39,44)(61,94,129,119)(62,110,130,85)(63,96,131,101)(64,112,132,87)(65,98,133,103)(66,114,134,89)(67,100,135,105)(68,116,136,91)(69,82,137,107)(70,118,138,93)(71,84,139,109)(72,120,140,95)(73,86,121,111)(74,102,122,97)(75,88,123,113)(76,104,124,99)(77,90,125,115)(78,106,126,81)(79,92,127,117)(80,108,128,83), (1,102,11,112)(2,103,12,113)(3,104,13,114)(4,105,14,115)(5,106,15,116)(6,107,16,117)(7,108,17,118)(8,109,18,119)(9,110,19,120)(10,111,20,101)(21,98,31,88)(22,99,32,89)(23,100,33,90)(24,81,34,91)(25,82,35,92)(26,83,36,93)(27,84,37,94)(28,85,38,95)(29,86,39,96)(30,87,40,97)(41,128,51,138)(42,129,52,139)(43,130,53,140)(44,131,54,121)(45,132,55,122)(46,133,56,123)(47,134,57,124)(48,135,58,125)(49,136,59,126)(50,137,60,127)(61,152,71,142)(62,153,72,143)(63,154,73,144)(64,155,74,145)(65,156,75,146)(66,157,76,147)(67,158,77,148)(68,159,78,149)(69,160,79,150)(70,141,80,151), (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)(21,38,31,28)(22,27,32,37)(23,36,33,26)(24,25,34,35)(29,30,39,40)(41,48,51,58)(42,57,52,47)(43,46,53,56)(44,55,54,45)(49,60,59,50)(61,124,71,134)(62,133,72,123)(63,122,73,132)(64,131,74,121)(65,140,75,130)(66,129,76,139)(67,138,77,128)(68,127,78,137)(69,136,79,126)(70,125,80,135)(81,107,91,117)(82,116,92,106)(83,105,93,115)(84,114,94,104)(85,103,95,113)(86,112,96,102)(87,101,97,111)(88,110,98,120)(89,119,99,109)(90,108,100,118)(141,148,151,158)(142,157,152,147)(143,146,153,156)(144,155,154,145)(149,160,159,150) );
G=PermutationGroup([(1,55,40,155),(2,146,21,46),(3,57,22,157),(4,148,23,48),(5,59,24,159),(6,150,25,50),(7,41,26,141),(8,152,27,52),(9,43,28,143),(10,154,29,54),(11,45,30,145),(12,156,31,56),(13,47,32,147),(14,158,33,58),(15,49,34,149),(16,160,35,60),(17,51,36,151),(18,142,37,42),(19,53,38,153),(20,144,39,44),(61,94,129,119),(62,110,130,85),(63,96,131,101),(64,112,132,87),(65,98,133,103),(66,114,134,89),(67,100,135,105),(68,116,136,91),(69,82,137,107),(70,118,138,93),(71,84,139,109),(72,120,140,95),(73,86,121,111),(74,102,122,97),(75,88,123,113),(76,104,124,99),(77,90,125,115),(78,106,126,81),(79,92,127,117),(80,108,128,83)], [(1,102,11,112),(2,103,12,113),(3,104,13,114),(4,105,14,115),(5,106,15,116),(6,107,16,117),(7,108,17,118),(8,109,18,119),(9,110,19,120),(10,111,20,101),(21,98,31,88),(22,99,32,89),(23,100,33,90),(24,81,34,91),(25,82,35,92),(26,83,36,93),(27,84,37,94),(28,85,38,95),(29,86,39,96),(30,87,40,97),(41,128,51,138),(42,129,52,139),(43,130,53,140),(44,131,54,121),(45,132,55,122),(46,133,56,123),(47,134,57,124),(48,135,58,125),(49,136,59,126),(50,137,60,127),(61,152,71,142),(62,153,72,143),(63,154,73,144),(64,155,74,145),(65,156,75,146),(66,157,76,147),(67,158,77,148),(68,159,78,149),(69,160,79,150),(70,141,80,151)], [(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),(21,38,31,28),(22,27,32,37),(23,36,33,26),(24,25,34,35),(29,30,39,40),(41,48,51,58),(42,57,52,47),(43,46,53,56),(44,55,54,45),(49,60,59,50),(61,124,71,134),(62,133,72,123),(63,122,73,132),(64,131,74,121),(65,140,75,130),(66,129,76,139),(67,138,77,128),(68,127,78,137),(69,136,79,126),(70,125,80,135),(81,107,91,117),(82,116,92,106),(83,105,93,115),(84,114,94,104),(85,103,95,113),(86,112,96,102),(87,101,97,111),(88,110,98,120),(89,119,99,109),(90,108,100,118),(141,148,151,158),(142,157,152,147),(143,146,153,156),(144,155,154,145),(149,160,159,150)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 32 | 4 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 29 |
| 0 | 0 | 0 | 0 | 5 | 15 |
| 9 | 37 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 13 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 21 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,4,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,26,5,0,0,0,0,29,15],[9,0,0,0,0,0,37,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,0,0,0,0,9,0,0,0,0,0,13,32],[32,21,0,0,0,0,0,9,0,0,0,0,0,0,34,1,0,0,0,0,34,7,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | 2+ (1+4) | D4⋊6D10 | D5×C4○D4 |
| kernel | C42.114D10 | C4×Dic10 | C4.D20 | D10.12D4 | Dic5.5D4 | Dic5⋊3Q8 | D20⋊8C4 | C4×C5⋊D4 | C23.23D10 | C20.17D4 | C20⋊D4 | D4×C20 | C4×D4 | Dic5 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{114}D_{10} % in TeX
G:=Group("C4^2.114D10"); // GroupNames label
G:=SmallGroup(320,1231);
// by ID
G=gap.SmallGroup(320,1231);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,136,12550]);
// 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,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀