metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.122D10, C10.72- (1+4), (C4×Q8)⋊5D5, (Q8×C20)⋊5C2, C4⋊C4.291D10, D10⋊Q8⋊10C2, (C4×Dic10)⋊36C2, Dic5⋊Q8⋊9C2, C4.18(C4○D20), C42⋊D5⋊33C2, C42⋊2D5⋊17C2, (C2×Q8).176D10, Dic5⋊3Q8⋊17C2, D20⋊8C4.10C2, C20.116(C4○D4), (C2×C20).621C23, (C2×C10).112C24, (C4×C20).238C22, C4.D20.12C2, C20.23D4.9C2, Dic5.37(C4○D4), (C2×D20).146C22, C4⋊Dic5.303C22, (Q8×C10).212C22, (C4×Dic5).89C22, (C22×D5).44C23, C22.137(C23×D5), D10⋊C4.68C22, C5⋊3(C22.50C24), (C2×Dic5).221C23, C10.D4.68C22, C2.10(Q8.10D10), (C2×Dic10).153C22, C2.27(D5×C4○D4), C4⋊C4⋊D5⋊10C2, C2.60(C2×C4○D20), C10.53(C2×C4○D4), (C2×C4×D5).257C22, (C5×C4⋊C4).340C22, (C2×C4).653(C22×D5), SmallGroup(320,1240)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 694 in 212 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×13], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×10], D4 [×2], Q8 [×6], C23 [×2], D5 [×2], C10 [×3], C42, C42 [×2], C42 [×4], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×9], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], Dic5 [×2], Dic5 [×5], C20 [×2], C20 [×6], D10 [×6], C2×C10, C42⋊C2 [×2], C4×D4, C4×Q8, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4 [×2], C42⋊2C2 [×4], C4⋊Q8, Dic10 [×4], C4×D5 [×4], D20 [×2], C2×Dic5 [×4], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5 [×2], C22.50C24, C4×Dic5 [×2], C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×6], C4⋊Dic5, D10⋊C4 [×10], C4×C20, C4×C20 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20, Q8×C10, C4×Dic10, C42⋊D5 [×2], C4.D20, C42⋊2D5 [×2], Dic5⋊3Q8, D20⋊8C4, D10⋊Q8 [×2], C4⋊C4⋊D5 [×2], Dic5⋊Q8, C20.23D4, Q8×C20, C42.122D10
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], C22.50C24, C4○D20 [×2], C23×D5, C2×C4○D20, Q8.10D10, D5×C4○D4, C42.122D10
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c10=b2, ab=ba, ac=ca, dad=ab2, cbc-1=dbd=a2b-1, dcd=c9 >
►On 160 points
Generators in S160
(1 71 50 158)(2 72 51 159)(3 73 52 160)(4 74 53 141)(5 75 54 142)(6 76 55 143)(7 77 56 144)(8 78 57 145)(9 79 58 146)(10 80 59 147)(11 61 60 148)(12 62 41 149)(13 63 42 150)(14 64 43 151)(15 65 44 152)(16 66 45 153)(17 67 46 154)(18 68 47 155)(19 69 48 156)(20 70 49 157)(21 137 112 81)(22 138 113 82)(23 139 114 83)(24 140 115 84)(25 121 116 85)(26 122 117 86)(27 123 118 87)(28 124 119 88)(29 125 120 89)(30 126 101 90)(31 127 102 91)(32 128 103 92)(33 129 104 93)(34 130 105 94)(35 131 106 95)(36 132 107 96)(37 133 108 97)(38 134 109 98)(39 135 110 99)(40 136 111 100)
(1 37 11 27)(2 119 12 109)(3 39 13 29)(4 101 14 111)(5 21 15 31)(6 103 16 113)(7 23 17 33)(8 105 18 115)(9 25 19 35)(10 107 20 117)(22 55 32 45)(24 57 34 47)(26 59 36 49)(28 41 38 51)(30 43 40 53)(42 120 52 110)(44 102 54 112)(46 104 56 114)(48 106 58 116)(50 108 60 118)(61 123 71 133)(62 98 72 88)(63 125 73 135)(64 100 74 90)(65 127 75 137)(66 82 76 92)(67 129 77 139)(68 84 78 94)(69 131 79 121)(70 86 80 96)(81 152 91 142)(83 154 93 144)(85 156 95 146)(87 158 97 148)(89 160 99 150)(122 147 132 157)(124 149 134 159)(126 151 136 141)(128 153 138 143)(130 155 140 145)
(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 36)(2 25)(3 34)(4 23)(5 32)(6 21)(7 30)(8 39)(9 28)(10 37)(11 26)(12 35)(13 24)(14 33)(15 22)(16 31)(17 40)(18 29)(19 38)(20 27)(41 106)(42 115)(43 104)(44 113)(45 102)(46 111)(47 120)(48 109)(49 118)(50 107)(51 116)(52 105)(53 114)(54 103)(55 112)(56 101)(57 110)(58 119)(59 108)(60 117)(61 132)(62 121)(63 130)(64 139)(65 128)(66 137)(67 126)(68 135)(69 124)(70 133)(71 122)(72 131)(73 140)(74 129)(75 138)(76 127)(77 136)(78 125)(79 134)(80 123)(81 153)(82 142)(83 151)(84 160)(85 149)(86 158)(87 147)(88 156)(89 145)(90 154)(91 143)(92 152)(93 141)(94 150)(95 159)(96 148)(97 157)(98 146)(99 155)(100 144)
G:=sub<Sym(160)| (1,71,50,158)(2,72,51,159)(3,73,52,160)(4,74,53,141)(5,75,54,142)(6,76,55,143)(7,77,56,144)(8,78,57,145)(9,79,58,146)(10,80,59,147)(11,61,60,148)(12,62,41,149)(13,63,42,150)(14,64,43,151)(15,65,44,152)(16,66,45,153)(17,67,46,154)(18,68,47,155)(19,69,48,156)(20,70,49,157)(21,137,112,81)(22,138,113,82)(23,139,114,83)(24,140,115,84)(25,121,116,85)(26,122,117,86)(27,123,118,87)(28,124,119,88)(29,125,120,89)(30,126,101,90)(31,127,102,91)(32,128,103,92)(33,129,104,93)(34,130,105,94)(35,131,106,95)(36,132,107,96)(37,133,108,97)(38,134,109,98)(39,135,110,99)(40,136,111,100), (1,37,11,27)(2,119,12,109)(3,39,13,29)(4,101,14,111)(5,21,15,31)(6,103,16,113)(7,23,17,33)(8,105,18,115)(9,25,19,35)(10,107,20,117)(22,55,32,45)(24,57,34,47)(26,59,36,49)(28,41,38,51)(30,43,40,53)(42,120,52,110)(44,102,54,112)(46,104,56,114)(48,106,58,116)(50,108,60,118)(61,123,71,133)(62,98,72,88)(63,125,73,135)(64,100,74,90)(65,127,75,137)(66,82,76,92)(67,129,77,139)(68,84,78,94)(69,131,79,121)(70,86,80,96)(81,152,91,142)(83,154,93,144)(85,156,95,146)(87,158,97,148)(89,160,99,150)(122,147,132,157)(124,149,134,159)(126,151,136,141)(128,153,138,143)(130,155,140,145), (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), (1,36)(2,25)(3,34)(4,23)(5,32)(6,21)(7,30)(8,39)(9,28)(10,37)(11,26)(12,35)(13,24)(14,33)(15,22)(16,31)(17,40)(18,29)(19,38)(20,27)(41,106)(42,115)(43,104)(44,113)(45,102)(46,111)(47,120)(48,109)(49,118)(50,107)(51,116)(52,105)(53,114)(54,103)(55,112)(56,101)(57,110)(58,119)(59,108)(60,117)(61,132)(62,121)(63,130)(64,139)(65,128)(66,137)(67,126)(68,135)(69,124)(70,133)(71,122)(72,131)(73,140)(74,129)(75,138)(76,127)(77,136)(78,125)(79,134)(80,123)(81,153)(82,142)(83,151)(84,160)(85,149)(86,158)(87,147)(88,156)(89,145)(90,154)(91,143)(92,152)(93,141)(94,150)(95,159)(96,148)(97,157)(98,146)(99,155)(100,144)>;
G:=Group( (1,71,50,158)(2,72,51,159)(3,73,52,160)(4,74,53,141)(5,75,54,142)(6,76,55,143)(7,77,56,144)(8,78,57,145)(9,79,58,146)(10,80,59,147)(11,61,60,148)(12,62,41,149)(13,63,42,150)(14,64,43,151)(15,65,44,152)(16,66,45,153)(17,67,46,154)(18,68,47,155)(19,69,48,156)(20,70,49,157)(21,137,112,81)(22,138,113,82)(23,139,114,83)(24,140,115,84)(25,121,116,85)(26,122,117,86)(27,123,118,87)(28,124,119,88)(29,125,120,89)(30,126,101,90)(31,127,102,91)(32,128,103,92)(33,129,104,93)(34,130,105,94)(35,131,106,95)(36,132,107,96)(37,133,108,97)(38,134,109,98)(39,135,110,99)(40,136,111,100), (1,37,11,27)(2,119,12,109)(3,39,13,29)(4,101,14,111)(5,21,15,31)(6,103,16,113)(7,23,17,33)(8,105,18,115)(9,25,19,35)(10,107,20,117)(22,55,32,45)(24,57,34,47)(26,59,36,49)(28,41,38,51)(30,43,40,53)(42,120,52,110)(44,102,54,112)(46,104,56,114)(48,106,58,116)(50,108,60,118)(61,123,71,133)(62,98,72,88)(63,125,73,135)(64,100,74,90)(65,127,75,137)(66,82,76,92)(67,129,77,139)(68,84,78,94)(69,131,79,121)(70,86,80,96)(81,152,91,142)(83,154,93,144)(85,156,95,146)(87,158,97,148)(89,160,99,150)(122,147,132,157)(124,149,134,159)(126,151,136,141)(128,153,138,143)(130,155,140,145), (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), (1,36)(2,25)(3,34)(4,23)(5,32)(6,21)(7,30)(8,39)(9,28)(10,37)(11,26)(12,35)(13,24)(14,33)(15,22)(16,31)(17,40)(18,29)(19,38)(20,27)(41,106)(42,115)(43,104)(44,113)(45,102)(46,111)(47,120)(48,109)(49,118)(50,107)(51,116)(52,105)(53,114)(54,103)(55,112)(56,101)(57,110)(58,119)(59,108)(60,117)(61,132)(62,121)(63,130)(64,139)(65,128)(66,137)(67,126)(68,135)(69,124)(70,133)(71,122)(72,131)(73,140)(74,129)(75,138)(76,127)(77,136)(78,125)(79,134)(80,123)(81,153)(82,142)(83,151)(84,160)(85,149)(86,158)(87,147)(88,156)(89,145)(90,154)(91,143)(92,152)(93,141)(94,150)(95,159)(96,148)(97,157)(98,146)(99,155)(100,144) );
G=PermutationGroup([(1,71,50,158),(2,72,51,159),(3,73,52,160),(4,74,53,141),(5,75,54,142),(6,76,55,143),(7,77,56,144),(8,78,57,145),(9,79,58,146),(10,80,59,147),(11,61,60,148),(12,62,41,149),(13,63,42,150),(14,64,43,151),(15,65,44,152),(16,66,45,153),(17,67,46,154),(18,68,47,155),(19,69,48,156),(20,70,49,157),(21,137,112,81),(22,138,113,82),(23,139,114,83),(24,140,115,84),(25,121,116,85),(26,122,117,86),(27,123,118,87),(28,124,119,88),(29,125,120,89),(30,126,101,90),(31,127,102,91),(32,128,103,92),(33,129,104,93),(34,130,105,94),(35,131,106,95),(36,132,107,96),(37,133,108,97),(38,134,109,98),(39,135,110,99),(40,136,111,100)], [(1,37,11,27),(2,119,12,109),(3,39,13,29),(4,101,14,111),(5,21,15,31),(6,103,16,113),(7,23,17,33),(8,105,18,115),(9,25,19,35),(10,107,20,117),(22,55,32,45),(24,57,34,47),(26,59,36,49),(28,41,38,51),(30,43,40,53),(42,120,52,110),(44,102,54,112),(46,104,56,114),(48,106,58,116),(50,108,60,118),(61,123,71,133),(62,98,72,88),(63,125,73,135),(64,100,74,90),(65,127,75,137),(66,82,76,92),(67,129,77,139),(68,84,78,94),(69,131,79,121),(70,86,80,96),(81,152,91,142),(83,154,93,144),(85,156,95,146),(87,158,97,148),(89,160,99,150),(122,147,132,157),(124,149,134,159),(126,151,136,141),(128,153,138,143),(130,155,140,145)], [(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)], [(1,36),(2,25),(3,34),(4,23),(5,32),(6,21),(7,30),(8,39),(9,28),(10,37),(11,26),(12,35),(13,24),(14,33),(15,22),(16,31),(17,40),(18,29),(19,38),(20,27),(41,106),(42,115),(43,104),(44,113),(45,102),(46,111),(47,120),(48,109),(49,118),(50,107),(51,116),(52,105),(53,114),(54,103),(55,112),(56,101),(57,110),(58,119),(59,108),(60,117),(61,132),(62,121),(63,130),(64,139),(65,128),(66,137),(67,126),(68,135),(69,124),(70,133),(71,122),(72,131),(73,140),(74,129),(75,138),(76,127),(77,136),(78,125),(79,134),(80,123),(81,153),(82,142),(83,151),(84,160),(85,149),(86,158),(87,147),(88,156),(89,145),(90,154),(91,143),(92,152),(93,141),(94,150),(95,159),(96,148),(97,157),(98,146),(99,155),(100,144)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 18 | 1 |
| 1 | 6 | 0 | 0 | 0 | 0 |
| 35 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 5 |
| 0 | 0 | 0 | 0 | 38 | 37 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 35 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 36 |
| 0 | 0 | 0 | 0 | 3 | 4 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,18,0,0,0,0,0,1],[1,35,0,0,0,0,6,6,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,4,38,0,0,0,0,5,37],[1,35,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,37,3,0,0,0,0,36,4] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 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 | 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 | C4○D20 | 2- (1+4) | Q8.10D10 | D5×C4○D4 |
| kernel | C42.122D10 | C4×Dic10 | C42⋊D5 | C4.D20 | C42⋊2D5 | Dic5⋊3Q8 | D20⋊8C4 | D10⋊Q8 | C4⋊C4⋊D5 | Dic5⋊Q8 | C20.23D4 | Q8×C20 | C4×Q8 | Dic5 | C20 | C42 | C4⋊C4 | C2×Q8 | C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 6 | 6 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{122}D_{10} % in TeX
G:=Group("C4^2.122D10"); // GroupNames label
G:=SmallGroup(320,1240);
// by ID
G=gap.SmallGroup(320,1240);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,232,758,100,794,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^10=b^2,a*b=b*a,a*c=c*a,d*a*d=a*b^2,c*b*c^-1=d*b*d=a^2*b^-1,d*c*d=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀