direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C42⋊C2, (C2×C42)⋊3C10, (C22×C20)⋊32C4, (C4×C20)⋊48C22, (C22×C4)⋊10C20, C42⋊13(C2×C10), (C23×C4).9C10, C2.3(C23×C20), (C23×C20).24C2, C4.30(C22×C20), C23.34(C2×C20), C24.29(C2×C10), C10.76(C23×C4), (C2×C20).706C23, C20.247(C22×C4), (C2×C10).334C24, C22.7(C23×C10), (C23×C10).89C22, C22.25(C22×C20), C23.67(C22×C10), (C22×C20).609C22, (C22×C10).467C23, (C2×C4×C20)⋊5C2, (C10×C4⋊C4)⋊51C2, (C2×C4⋊C4)⋊24C10, C4⋊C4⋊18(C2×C10), (C2×C20)⋊53(C2×C4), (C2×C4)⋊11(C2×C20), C2.1(C10×C4○D4), (C5×C4⋊C4)⋊75C22, C10.220(C2×C4○D4), C22.26(C5×C4○D4), C22⋊C4.27(C2×C10), (C10×C22⋊C4).35C2, (C2×C22⋊C4).15C10, (C22×C4).97(C2×C10), (C2×C10).226(C4○D4), (C2×C10).265(C22×C4), (C2×C4).133(C22×C10), (C22×C10).188(C2×C4), (C5×C22⋊C4).158C22, SmallGroup(320,1516)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 402 in 330 conjugacy classes, 258 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×10], C22 [×12], C5, C2×C4 [×36], C2×C4 [×8], C23, C23 [×6], C23 [×4], C10, C10 [×6], C10 [×4], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×16], C24, C20 [×8], C20 [×8], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×8], C23×C4, C2×C20 [×36], C2×C20 [×8], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C42⋊C2, C4×C20 [×8], C5×C22⋊C4 [×8], C5×C4⋊C4 [×8], C22×C20 [×2], C22×C20 [×16], C23×C10, C2×C4×C20 [×2], C10×C22⋊C4 [×2], C10×C4⋊C4 [×2], C5×C42⋊C2 [×8], C23×C20, C10×C42⋊C2
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C5, C2×C4 [×28], C23 [×15], C10 [×15], C22×C4 [×14], C4○D4 [×4], C24, C20 [×8], C2×C10 [×35], C42⋊C2 [×4], C23×C4, C2×C4○D4 [×2], C2×C20 [×28], C22×C10 [×15], C2×C42⋊C2, C22×C20 [×14], C5×C4○D4 [×4], C23×C10, C5×C42⋊C2 [×4], C23×C20, C10×C4○D4 [×2], C10×C42⋊C2
Generators and relations
G = < a,b,c,d | a10=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, cd=dc >
►On 160 points
Generators in S160
(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)
(1 131 54 147)(2 132 55 148)(3 133 56 149)(4 134 57 150)(5 135 58 141)(6 136 59 142)(7 137 60 143)(8 138 51 144)(9 139 52 145)(10 140 53 146)(11 78 21 82)(12 79 22 83)(13 80 23 84)(14 71 24 85)(15 72 25 86)(16 73 26 87)(17 74 27 88)(18 75 28 89)(19 76 29 90)(20 77 30 81)(31 103 160 97)(32 104 151 98)(33 105 152 99)(34 106 153 100)(35 107 154 91)(36 108 155 92)(37 109 156 93)(38 110 157 94)(39 101 158 95)(40 102 159 96)(41 126 65 112)(42 127 66 113)(43 128 67 114)(44 129 68 115)(45 130 69 116)(46 121 70 117)(47 122 61 118)(48 123 62 119)(49 124 63 120)(50 125 64 111)
(1 79 48 107)(2 80 49 108)(3 71 50 109)(4 72 41 110)(5 73 42 101)(6 74 43 102)(7 75 44 103)(8 76 45 104)(9 77 46 105)(10 78 47 106)(11 118 34 146)(12 119 35 147)(13 120 36 148)(14 111 37 149)(15 112 38 150)(16 113 39 141)(17 114 40 142)(18 115 31 143)(19 116 32 144)(20 117 33 145)(21 122 153 140)(22 123 154 131)(23 124 155 132)(24 125 156 133)(25 126 157 134)(26 127 158 135)(27 128 159 136)(28 129 160 137)(29 130 151 138)(30 121 152 139)(51 90 69 98)(52 81 70 99)(53 82 61 100)(54 83 62 91)(55 84 63 92)(56 85 64 93)(57 86 65 94)(58 87 66 95)(59 88 67 96)(60 89 68 97)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 51)(9 52)(10 53)(11 153)(12 154)(13 155)(14 156)(15 157)(16 158)(17 159)(18 160)(19 151)(20 152)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)(27 40)(28 31)(29 32)(30 33)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 61)(48 62)(49 63)(50 64)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 81)(78 82)(79 83)(80 84)(91 107)(92 108)(93 109)(94 110)(95 101)(96 102)(97 103)(98 104)(99 105)(100 106)(111 133)(112 134)(113 135)(114 136)(115 137)(116 138)(117 139)(118 140)(119 131)(120 132)(121 145)(122 146)(123 147)(124 148)(125 149)(126 150)(127 141)(128 142)(129 143)(130 144)
G:=sub<Sym(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), (1,131,54,147)(2,132,55,148)(3,133,56,149)(4,134,57,150)(5,135,58,141)(6,136,59,142)(7,137,60,143)(8,138,51,144)(9,139,52,145)(10,140,53,146)(11,78,21,82)(12,79,22,83)(13,80,23,84)(14,71,24,85)(15,72,25,86)(16,73,26,87)(17,74,27,88)(18,75,28,89)(19,76,29,90)(20,77,30,81)(31,103,160,97)(32,104,151,98)(33,105,152,99)(34,106,153,100)(35,107,154,91)(36,108,155,92)(37,109,156,93)(38,110,157,94)(39,101,158,95)(40,102,159,96)(41,126,65,112)(42,127,66,113)(43,128,67,114)(44,129,68,115)(45,130,69,116)(46,121,70,117)(47,122,61,118)(48,123,62,119)(49,124,63,120)(50,125,64,111), (1,79,48,107)(2,80,49,108)(3,71,50,109)(4,72,41,110)(5,73,42,101)(6,74,43,102)(7,75,44,103)(8,76,45,104)(9,77,46,105)(10,78,47,106)(11,118,34,146)(12,119,35,147)(13,120,36,148)(14,111,37,149)(15,112,38,150)(16,113,39,141)(17,114,40,142)(18,115,31,143)(19,116,32,144)(20,117,33,145)(21,122,153,140)(22,123,154,131)(23,124,155,132)(24,125,156,133)(25,126,157,134)(26,127,158,135)(27,128,159,136)(28,129,160,137)(29,130,151,138)(30,121,152,139)(51,90,69,98)(52,81,70,99)(53,82,61,100)(54,83,62,91)(55,84,63,92)(56,85,64,93)(57,86,65,94)(58,87,66,95)(59,88,67,96)(60,89,68,97), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,51)(9,52)(10,53)(11,153)(12,154)(13,155)(14,156)(15,157)(16,158)(17,159)(18,160)(19,151)(20,152)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,31)(29,32)(30,33)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,61)(48,62)(49,63)(50,64)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,81)(78,82)(79,83)(80,84)(91,107)(92,108)(93,109)(94,110)(95,101)(96,102)(97,103)(98,104)(99,105)(100,106)(111,133)(112,134)(113,135)(114,136)(115,137)(116,138)(117,139)(118,140)(119,131)(120,132)(121,145)(122,146)(123,147)(124,148)(125,149)(126,150)(127,141)(128,142)(129,143)(130,144)>;
G:=Group( (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), (1,131,54,147)(2,132,55,148)(3,133,56,149)(4,134,57,150)(5,135,58,141)(6,136,59,142)(7,137,60,143)(8,138,51,144)(9,139,52,145)(10,140,53,146)(11,78,21,82)(12,79,22,83)(13,80,23,84)(14,71,24,85)(15,72,25,86)(16,73,26,87)(17,74,27,88)(18,75,28,89)(19,76,29,90)(20,77,30,81)(31,103,160,97)(32,104,151,98)(33,105,152,99)(34,106,153,100)(35,107,154,91)(36,108,155,92)(37,109,156,93)(38,110,157,94)(39,101,158,95)(40,102,159,96)(41,126,65,112)(42,127,66,113)(43,128,67,114)(44,129,68,115)(45,130,69,116)(46,121,70,117)(47,122,61,118)(48,123,62,119)(49,124,63,120)(50,125,64,111), (1,79,48,107)(2,80,49,108)(3,71,50,109)(4,72,41,110)(5,73,42,101)(6,74,43,102)(7,75,44,103)(8,76,45,104)(9,77,46,105)(10,78,47,106)(11,118,34,146)(12,119,35,147)(13,120,36,148)(14,111,37,149)(15,112,38,150)(16,113,39,141)(17,114,40,142)(18,115,31,143)(19,116,32,144)(20,117,33,145)(21,122,153,140)(22,123,154,131)(23,124,155,132)(24,125,156,133)(25,126,157,134)(26,127,158,135)(27,128,159,136)(28,129,160,137)(29,130,151,138)(30,121,152,139)(51,90,69,98)(52,81,70,99)(53,82,61,100)(54,83,62,91)(55,84,63,92)(56,85,64,93)(57,86,65,94)(58,87,66,95)(59,88,67,96)(60,89,68,97), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,51)(9,52)(10,53)(11,153)(12,154)(13,155)(14,156)(15,157)(16,158)(17,159)(18,160)(19,151)(20,152)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,31)(29,32)(30,33)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,61)(48,62)(49,63)(50,64)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,81)(78,82)(79,83)(80,84)(91,107)(92,108)(93,109)(94,110)(95,101)(96,102)(97,103)(98,104)(99,105)(100,106)(111,133)(112,134)(113,135)(114,136)(115,137)(116,138)(117,139)(118,140)(119,131)(120,132)(121,145)(122,146)(123,147)(124,148)(125,149)(126,150)(127,141)(128,142)(129,143)(130,144) );
G=PermutationGroup([(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)], [(1,131,54,147),(2,132,55,148),(3,133,56,149),(4,134,57,150),(5,135,58,141),(6,136,59,142),(7,137,60,143),(8,138,51,144),(9,139,52,145),(10,140,53,146),(11,78,21,82),(12,79,22,83),(13,80,23,84),(14,71,24,85),(15,72,25,86),(16,73,26,87),(17,74,27,88),(18,75,28,89),(19,76,29,90),(20,77,30,81),(31,103,160,97),(32,104,151,98),(33,105,152,99),(34,106,153,100),(35,107,154,91),(36,108,155,92),(37,109,156,93),(38,110,157,94),(39,101,158,95),(40,102,159,96),(41,126,65,112),(42,127,66,113),(43,128,67,114),(44,129,68,115),(45,130,69,116),(46,121,70,117),(47,122,61,118),(48,123,62,119),(49,124,63,120),(50,125,64,111)], [(1,79,48,107),(2,80,49,108),(3,71,50,109),(4,72,41,110),(5,73,42,101),(6,74,43,102),(7,75,44,103),(8,76,45,104),(9,77,46,105),(10,78,47,106),(11,118,34,146),(12,119,35,147),(13,120,36,148),(14,111,37,149),(15,112,38,150),(16,113,39,141),(17,114,40,142),(18,115,31,143),(19,116,32,144),(20,117,33,145),(21,122,153,140),(22,123,154,131),(23,124,155,132),(24,125,156,133),(25,126,157,134),(26,127,158,135),(27,128,159,136),(28,129,160,137),(29,130,151,138),(30,121,152,139),(51,90,69,98),(52,81,70,99),(53,82,61,100),(54,83,62,91),(55,84,63,92),(56,85,64,93),(57,86,65,94),(58,87,66,95),(59,88,67,96),(60,89,68,97)], [(1,54),(2,55),(3,56),(4,57),(5,58),(6,59),(7,60),(8,51),(9,52),(10,53),(11,153),(12,154),(13,155),(14,156),(15,157),(16,158),(17,159),(18,160),(19,151),(20,152),(21,34),(22,35),(23,36),(24,37),(25,38),(26,39),(27,40),(28,31),(29,32),(30,33),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,61),(48,62),(49,63),(50,64),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,81),(78,82),(79,83),(80,84),(91,107),(92,108),(93,109),(94,110),(95,101),(96,102),(97,103),(98,104),(99,105),(100,106),(111,133),(112,134),(113,135),(114,136),(115,137),(116,138),(117,139),(118,140),(119,131),(120,132),(121,145),(122,146),(123,147),(124,148),(125,149),(126,150),(127,141),(128,142),(129,143),(130,144)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 32 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,16,0,0,0,0,16],[32,0,0,0,0,1,0,0,0,0,0,40,0,0,40,0],[40,0,0,0,0,40,0,0,0,0,32,0,0,0,0,32],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;
200 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4AB | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10AR | 20A | ··· | 20AF | 20AG | ··· | 20DH |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C20 | C4○D4 | C5×C4○D4 |
| kernel | C10×C42⋊C2 | C2×C4×C20 | C10×C22⋊C4 | C10×C4⋊C4 | C5×C42⋊C2 | C23×C20 | C22×C20 | C2×C42⋊C2 | C2×C42 | C2×C22⋊C4 | C2×C4⋊C4 | C42⋊C2 | C23×C4 | C22×C4 | C2×C10 | C22 |
| # reps | 1 | 2 | 2 | 2 | 8 | 1 | 16 | 4 | 8 | 8 | 8 | 32 | 4 | 64 | 8 | 32 |
In GAP, Magma, Sage, TeX
C_{10}\times C_4^2\rtimes C_2 % in TeX
G:=Group("C10xC4^2:C2"); // GroupNames label
G:=SmallGroup(320,1516);
// by ID
G=gap.SmallGroup(320,1516);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,436]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b*c^2,c*d=d*c>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀