direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C8⋊9D4, C40⋊35D4, C8⋊9(C5×D4), C4⋊C8⋊15C10, C4⋊C4.7C20, C8⋊C4⋊9C10, (C2×D4).8C20, (C4×D4).2C10, C2.10(D4×C20), C4.80(D4×C10), C22⋊C8⋊13C10, (C22×C40)⋊25C2, (C22×C8)⋊11C10, (D4×C20).17C2, (D4×C10).33C4, C10.142(C4×D4), C20.485(C2×D4), C22⋊C4.4C20, C42.7(C2×C10), C10.73(C8○D4), (C2×C10)⋊10M4(2), C23.19(C2×C20), C2.9(C10×M4(2)), C22⋊1(C5×M4(2)), C20.354(C4○D4), (C10×M4(2))⋊32C2, (C2×M4(2))⋊14C10, (C4×C20).248C22, (C2×C20).991C23, (C2×C40).446C22, C10.87(C2×M4(2)), C22.47(C22×C20), (C22×C20).417C22, (C5×C4⋊C8)⋊34C2, C2.7(C5×C8○D4), (C5×C4⋊C4).32C4, (C5×C8⋊C4)⋊23C2, C4.52(C5×C4○D4), (C5×C22⋊C8)⋊30C2, (C2×C4).28(C2×C20), (C2×C8).52(C2×C10), (C2×C20).374(C2×C4), (C5×C22⋊C4).19C4, (C22×C4).96(C2×C10), (C22×C10).91(C2×C4), (C2×C4).159(C22×C10), (C2×C10).342(C22×C4), SmallGroup(320,936)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8⋊9D4
G = < a,b,c,d | a5=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b5, dcd=c-1 >
Subgroups: 178 in 124 conjugacy classes, 74 normal (66 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C20, C20, C2×C10, C2×C10, C2×C10, C8⋊C4, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C40, C40, C2×C20, C2×C20, C5×D4, C22×C10, C8⋊9D4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C2×C40, C5×M4(2), C22×C20, D4×C10, C5×C8⋊C4, C5×C22⋊C8, C5×C4⋊C8, D4×C20, C22×C40, C10×M4(2), C5×C8⋊9D4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, M4(2), C22×C4, C2×D4, C4○D4, C20, C2×C10, C4×D4, C2×M4(2), C8○D4, C2×C20, C5×D4, C22×C10, C8⋊9D4, C5×M4(2), C22×C20, D4×C10, C5×C4○D4, D4×C20, C10×M4(2), C5×C8○D4, C5×C8⋊9D4
►On 160 points
Generators in S160
(1 51 14 57 17)(2 52 15 58 18)(3 53 16 59 19)(4 54 9 60 20)(5 55 10 61 21)(6 56 11 62 22)(7 49 12 63 23)(8 50 13 64 24)(25 81 73 33 65)(26 82 74 34 66)(27 83 75 35 67)(28 84 76 36 68)(29 85 77 37 69)(30 86 78 38 70)(31 87 79 39 71)(32 88 80 40 72)(41 157 117 149 109)(42 158 118 150 110)(43 159 119 151 111)(44 160 120 152 112)(45 153 113 145 105)(46 154 114 146 106)(47 155 115 147 107)(48 156 116 148 108)(89 126 138 98 130)(90 127 139 99 131)(91 128 140 100 132)(92 121 141 101 133)(93 122 142 102 134)(94 123 143 103 135)(95 124 144 104 136)(96 125 137 97 129)
(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)
(1 129 31 111)(2 134 32 108)(3 131 25 105)(4 136 26 110)(5 133 27 107)(6 130 28 112)(7 135 29 109)(8 132 30 106)(9 124 74 158)(10 121 75 155)(11 126 76 160)(12 123 77 157)(13 128 78 154)(14 125 79 159)(15 122 80 156)(16 127 73 153)(17 97 71 151)(18 102 72 148)(19 99 65 145)(20 104 66 150)(21 101 67 147)(22 98 68 152)(23 103 69 149)(24 100 70 146)(33 113 59 139)(34 118 60 144)(35 115 61 141)(36 120 62 138)(37 117 63 143)(38 114 64 140)(39 119 57 137)(40 116 58 142)(41 49 94 85)(42 54 95 82)(43 51 96 87)(44 56 89 84)(45 53 90 81)(46 50 91 86)(47 55 92 83)(48 52 93 88)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)(34 38)(36 40)(41 94)(42 91)(43 96)(44 93)(45 90)(46 95)(47 92)(48 89)(50 54)(52 56)(58 62)(60 64)(66 70)(68 72)(74 78)(76 80)(82 86)(84 88)(97 151)(98 148)(99 145)(100 150)(101 147)(102 152)(103 149)(104 146)(105 131)(106 136)(107 133)(108 130)(109 135)(110 132)(111 129)(112 134)(113 139)(114 144)(115 141)(116 138)(117 143)(118 140)(119 137)(120 142)(121 155)(122 160)(123 157)(124 154)(125 159)(126 156)(127 153)(128 158)
G:=sub<Sym(160)| (1,51,14,57,17)(2,52,15,58,18)(3,53,16,59,19)(4,54,9,60,20)(5,55,10,61,21)(6,56,11,62,22)(7,49,12,63,23)(8,50,13,64,24)(25,81,73,33,65)(26,82,74,34,66)(27,83,75,35,67)(28,84,76,36,68)(29,85,77,37,69)(30,86,78,38,70)(31,87,79,39,71)(32,88,80,40,72)(41,157,117,149,109)(42,158,118,150,110)(43,159,119,151,111)(44,160,120,152,112)(45,153,113,145,105)(46,154,114,146,106)(47,155,115,147,107)(48,156,116,148,108)(89,126,138,98,130)(90,127,139,99,131)(91,128,140,100,132)(92,121,141,101,133)(93,122,142,102,134)(94,123,143,103,135)(95,124,144,104,136)(96,125,137,97,129), (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), (1,129,31,111)(2,134,32,108)(3,131,25,105)(4,136,26,110)(5,133,27,107)(6,130,28,112)(7,135,29,109)(8,132,30,106)(9,124,74,158)(10,121,75,155)(11,126,76,160)(12,123,77,157)(13,128,78,154)(14,125,79,159)(15,122,80,156)(16,127,73,153)(17,97,71,151)(18,102,72,148)(19,99,65,145)(20,104,66,150)(21,101,67,147)(22,98,68,152)(23,103,69,149)(24,100,70,146)(33,113,59,139)(34,118,60,144)(35,115,61,141)(36,120,62,138)(37,117,63,143)(38,114,64,140)(39,119,57,137)(40,116,58,142)(41,49,94,85)(42,54,95,82)(43,51,96,87)(44,56,89,84)(45,53,90,81)(46,50,91,86)(47,55,92,83)(48,52,93,88), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(26,30)(28,32)(34,38)(36,40)(41,94)(42,91)(43,96)(44,93)(45,90)(46,95)(47,92)(48,89)(50,54)(52,56)(58,62)(60,64)(66,70)(68,72)(74,78)(76,80)(82,86)(84,88)(97,151)(98,148)(99,145)(100,150)(101,147)(102,152)(103,149)(104,146)(105,131)(106,136)(107,133)(108,130)(109,135)(110,132)(111,129)(112,134)(113,139)(114,144)(115,141)(116,138)(117,143)(118,140)(119,137)(120,142)(121,155)(122,160)(123,157)(124,154)(125,159)(126,156)(127,153)(128,158)>;
G:=Group( (1,51,14,57,17)(2,52,15,58,18)(3,53,16,59,19)(4,54,9,60,20)(5,55,10,61,21)(6,56,11,62,22)(7,49,12,63,23)(8,50,13,64,24)(25,81,73,33,65)(26,82,74,34,66)(27,83,75,35,67)(28,84,76,36,68)(29,85,77,37,69)(30,86,78,38,70)(31,87,79,39,71)(32,88,80,40,72)(41,157,117,149,109)(42,158,118,150,110)(43,159,119,151,111)(44,160,120,152,112)(45,153,113,145,105)(46,154,114,146,106)(47,155,115,147,107)(48,156,116,148,108)(89,126,138,98,130)(90,127,139,99,131)(91,128,140,100,132)(92,121,141,101,133)(93,122,142,102,134)(94,123,143,103,135)(95,124,144,104,136)(96,125,137,97,129), (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), (1,129,31,111)(2,134,32,108)(3,131,25,105)(4,136,26,110)(5,133,27,107)(6,130,28,112)(7,135,29,109)(8,132,30,106)(9,124,74,158)(10,121,75,155)(11,126,76,160)(12,123,77,157)(13,128,78,154)(14,125,79,159)(15,122,80,156)(16,127,73,153)(17,97,71,151)(18,102,72,148)(19,99,65,145)(20,104,66,150)(21,101,67,147)(22,98,68,152)(23,103,69,149)(24,100,70,146)(33,113,59,139)(34,118,60,144)(35,115,61,141)(36,120,62,138)(37,117,63,143)(38,114,64,140)(39,119,57,137)(40,116,58,142)(41,49,94,85)(42,54,95,82)(43,51,96,87)(44,56,89,84)(45,53,90,81)(46,50,91,86)(47,55,92,83)(48,52,93,88), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(26,30)(28,32)(34,38)(36,40)(41,94)(42,91)(43,96)(44,93)(45,90)(46,95)(47,92)(48,89)(50,54)(52,56)(58,62)(60,64)(66,70)(68,72)(74,78)(76,80)(82,86)(84,88)(97,151)(98,148)(99,145)(100,150)(101,147)(102,152)(103,149)(104,146)(105,131)(106,136)(107,133)(108,130)(109,135)(110,132)(111,129)(112,134)(113,139)(114,144)(115,141)(116,138)(117,143)(118,140)(119,137)(120,142)(121,155)(122,160)(123,157)(124,154)(125,159)(126,156)(127,153)(128,158) );
G=PermutationGroup([[(1,51,14,57,17),(2,52,15,58,18),(3,53,16,59,19),(4,54,9,60,20),(5,55,10,61,21),(6,56,11,62,22),(7,49,12,63,23),(8,50,13,64,24),(25,81,73,33,65),(26,82,74,34,66),(27,83,75,35,67),(28,84,76,36,68),(29,85,77,37,69),(30,86,78,38,70),(31,87,79,39,71),(32,88,80,40,72),(41,157,117,149,109),(42,158,118,150,110),(43,159,119,151,111),(44,160,120,152,112),(45,153,113,145,105),(46,154,114,146,106),(47,155,115,147,107),(48,156,116,148,108),(89,126,138,98,130),(90,127,139,99,131),(91,128,140,100,132),(92,121,141,101,133),(93,122,142,102,134),(94,123,143,103,135),(95,124,144,104,136),(96,125,137,97,129)], [(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)], [(1,129,31,111),(2,134,32,108),(3,131,25,105),(4,136,26,110),(5,133,27,107),(6,130,28,112),(7,135,29,109),(8,132,30,106),(9,124,74,158),(10,121,75,155),(11,126,76,160),(12,123,77,157),(13,128,78,154),(14,125,79,159),(15,122,80,156),(16,127,73,153),(17,97,71,151),(18,102,72,148),(19,99,65,145),(20,104,66,150),(21,101,67,147),(22,98,68,152),(23,103,69,149),(24,100,70,146),(33,113,59,139),(34,118,60,144),(35,115,61,141),(36,120,62,138),(37,117,63,143),(38,114,64,140),(39,119,57,137),(40,116,58,142),(41,49,94,85),(42,54,95,82),(43,51,96,87),(44,56,89,84),(45,53,90,81),(46,50,91,86),(47,55,92,83),(48,52,93,88)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(26,30),(28,32),(34,38),(36,40),(41,94),(42,91),(43,96),(44,93),(45,90),(46,95),(47,92),(48,89),(50,54),(52,56),(58,62),(60,64),(66,70),(68,72),(74,78),(76,80),(82,86),(84,88),(97,151),(98,148),(99,145),(100,150),(101,147),(102,152),(103,149),(104,146),(105,131),(106,136),(107,133),(108,130),(109,135),(110,132),(111,129),(112,134),(113,139),(114,144),(115,141),(116,138),(117,143),(118,140),(119,137),(120,142),(121,155),(122,160),(123,157),(124,154),(125,159),(126,156),(127,153),(128,158)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10L | 10M | ··· | 10T | 10U | 10V | 10W | 10X | 20A | ··· | 20P | 20Q | ··· | 20X | 20Y | ··· | 20AJ | 40A | ··· | 40AF | 40AG | ··· | 40AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C20 | C20 | C20 | D4 | C4○D4 | M4(2) | C8○D4 | C5×D4 | C5×C4○D4 | C5×M4(2) | C5×C8○D4 |
| kernel | C5×C8⋊9D4 | C5×C8⋊C4 | C5×C22⋊C8 | C5×C4⋊C8 | D4×C20 | C22×C40 | C10×M4(2) | C5×C22⋊C4 | C5×C4⋊C4 | D4×C10 | C8⋊9D4 | C8⋊C4 | C22⋊C8 | C4⋊C8 | C4×D4 | C22×C8 | C2×M4(2) | C22⋊C4 | C4⋊C4 | C2×D4 | C40 | C20 | C2×C10 | C10 | C8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 4 | 8 | 4 | 4 | 4 | 4 | 16 | 8 | 8 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 16 |
Matrix representation of C5×C8⋊9D4 ►in GL4(𝔽41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 8 | 39 | 0 | 0 |
| 16 | 33 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 32 | 0 |
| 40 | 0 | 0 | 0 |
| 33 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 27 | 0 |
| 1 | 0 | 0 | 0 |
| 8 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,10,0,0,0,0,10],[8,16,0,0,39,33,0,0,0,0,0,32,0,0,1,0],[40,33,0,0,0,1,0,0,0,0,0,27,0,0,3,0],[1,8,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;
C5×C8⋊9D4 in GAP, Magma, Sage, TeX
C_5\times C_8\rtimes_9D_4
% in TeX
G:=Group("C5xC8:9D4"); // GroupNames label
G:=SmallGroup(320,936);
// by ID
G=gap.SmallGroup(320,936);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,3446,436,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^5,d*c*d=c^-1>;
// generators/relations