direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×D4○C16, D4.2C40, Q8.2C40, M5(2)⋊7C10, C80.29C22, C40.83C23, M4(2).4C20, (C2×C16)⋊9C10, (C2×C80)⋊19C2, C4.5(C2×C40), (C5×D4).6C8, (C5×Q8).6C8, C8.12(C2×C20), C16.8(C2×C10), C20.68(C2×C8), C4○D4.3C20, C8○D4.3C10, C40.109(C2×C4), C22.1(C2×C40), C2.7(C22×C40), (C5×M5(2))⋊15C2, C10.60(C22×C8), C4.36(C22×C20), C8.16(C22×C10), C20.253(C22×C4), (C2×C40).449C22, (C5×M4(2)).12C4, (C5×C8○D4).6C2, (C2×C10).36(C2×C8), (C2×C4).51(C2×C20), (C5×C4○D4).11C4, (C2×C20).445(C2×C4), (C2×C8).103(C2×C10), SmallGroup(320,1005)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 90 in 84 conjugacy classes, 78 normal (18 characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×3], C5, C8, C8 [×3], C2×C4 [×3], D4 [×3], Q8, C10, C10 [×3], C16, C16 [×3], C2×C8 [×3], M4(2) [×3], C4○D4, C20, C20 [×3], C2×C10 [×3], C2×C16 [×3], M5(2) [×3], C8○D4, C40, C40 [×3], C2×C20 [×3], C5×D4 [×3], C5×Q8, D4○C16, C80, C80 [×3], C2×C40 [×3], C5×M4(2) [×3], C5×C4○D4, C2×C80 [×3], C5×M5(2) [×3], C5×C8○D4, C5×D4○C16
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C5, C8 [×4], C2×C4 [×6], C23, C10 [×7], C2×C8 [×6], C22×C4, C20 [×4], C2×C10 [×7], C22×C8, C40 [×4], C2×C20 [×6], C22×C10, D4○C16, C2×C40 [×6], C22×C20, C22×C40, C5×D4○C16
Generators and relations
G = < a,b,c,d | a5=b4=c2=1, d8=b2, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, cd=dc >
►On 160 points
Generators in S160
(1 89 117 73 27)(2 90 118 74 28)(3 91 119 75 29)(4 92 120 76 30)(5 93 121 77 31)(6 94 122 78 32)(7 95 123 79 17)(8 96 124 80 18)(9 81 125 65 19)(10 82 126 66 20)(11 83 127 67 21)(12 84 128 68 22)(13 85 113 69 23)(14 86 114 70 24)(15 87 115 71 25)(16 88 116 72 26)(33 98 60 140 158)(34 99 61 141 159)(35 100 62 142 160)(36 101 63 143 145)(37 102 64 144 146)(38 103 49 129 147)(39 104 50 130 148)(40 105 51 131 149)(41 106 52 132 150)(42 107 53 133 151)(43 108 54 134 152)(44 109 55 135 153)(45 110 56 136 154)(46 111 57 137 155)(47 112 58 138 156)(48 97 59 139 157)
(1 147 9 155)(2 148 10 156)(3 149 11 157)(4 150 12 158)(5 151 13 159)(6 152 14 160)(7 153 15 145)(8 154 16 146)(17 135 25 143)(18 136 26 144)(19 137 27 129)(20 138 28 130)(21 139 29 131)(22 140 30 132)(23 141 31 133)(24 142 32 134)(33 92 41 84)(34 93 42 85)(35 94 43 86)(36 95 44 87)(37 96 45 88)(38 81 46 89)(39 82 47 90)(40 83 48 91)(49 65 57 73)(50 66 58 74)(51 67 59 75)(52 68 60 76)(53 69 61 77)(54 70 62 78)(55 71 63 79)(56 72 64 80)(97 119 105 127)(98 120 106 128)(99 121 107 113)(100 122 108 114)(101 123 109 115)(102 124 110 116)(103 125 111 117)(104 126 112 118)
(1 155)(2 156)(3 157)(4 158)(5 159)(6 160)(7 145)(8 146)(9 147)(10 148)(11 149)(12 150)(13 151)(14 152)(15 153)(16 154)(17 143)(18 144)(19 129)(20 130)(21 131)(22 132)(23 133)(24 134)(25 135)(26 136)(27 137)(28 138)(29 139)(30 140)(31 141)(32 142)(33 92)(34 93)(35 94)(36 95)(37 96)(38 81)(39 82)(40 83)(41 84)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)(55 71)(56 72)(57 73)(58 74)(59 75)(60 76)(61 77)(62 78)(63 79)(64 80)(97 119)(98 120)(99 121)(100 122)(101 123)(102 124)(103 125)(104 126)(105 127)(106 128)(107 113)(108 114)(109 115)(110 116)(111 117)(112 118)
(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)
G:=sub<Sym(160)| (1,89,117,73,27)(2,90,118,74,28)(3,91,119,75,29)(4,92,120,76,30)(5,93,121,77,31)(6,94,122,78,32)(7,95,123,79,17)(8,96,124,80,18)(9,81,125,65,19)(10,82,126,66,20)(11,83,127,67,21)(12,84,128,68,22)(13,85,113,69,23)(14,86,114,70,24)(15,87,115,71,25)(16,88,116,72,26)(33,98,60,140,158)(34,99,61,141,159)(35,100,62,142,160)(36,101,63,143,145)(37,102,64,144,146)(38,103,49,129,147)(39,104,50,130,148)(40,105,51,131,149)(41,106,52,132,150)(42,107,53,133,151)(43,108,54,134,152)(44,109,55,135,153)(45,110,56,136,154)(46,111,57,137,155)(47,112,58,138,156)(48,97,59,139,157), (1,147,9,155)(2,148,10,156)(3,149,11,157)(4,150,12,158)(5,151,13,159)(6,152,14,160)(7,153,15,145)(8,154,16,146)(17,135,25,143)(18,136,26,144)(19,137,27,129)(20,138,28,130)(21,139,29,131)(22,140,30,132)(23,141,31,133)(24,142,32,134)(33,92,41,84)(34,93,42,85)(35,94,43,86)(36,95,44,87)(37,96,45,88)(38,81,46,89)(39,82,47,90)(40,83,48,91)(49,65,57,73)(50,66,58,74)(51,67,59,75)(52,68,60,76)(53,69,61,77)(54,70,62,78)(55,71,63,79)(56,72,64,80)(97,119,105,127)(98,120,106,128)(99,121,107,113)(100,122,108,114)(101,123,109,115)(102,124,110,116)(103,125,111,117)(104,126,112,118), (1,155)(2,156)(3,157)(4,158)(5,159)(6,160)(7,145)(8,146)(9,147)(10,148)(11,149)(12,150)(13,151)(14,152)(15,153)(16,154)(17,143)(18,144)(19,129)(20,130)(21,131)(22,132)(23,133)(24,134)(25,135)(26,136)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,92)(34,93)(35,94)(36,95)(37,96)(38,81)(39,82)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76)(61,77)(62,78)(63,79)(64,80)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(105,127)(106,128)(107,113)(108,114)(109,115)(110,116)(111,117)(112,118), (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)>;
G:=Group( (1,89,117,73,27)(2,90,118,74,28)(3,91,119,75,29)(4,92,120,76,30)(5,93,121,77,31)(6,94,122,78,32)(7,95,123,79,17)(8,96,124,80,18)(9,81,125,65,19)(10,82,126,66,20)(11,83,127,67,21)(12,84,128,68,22)(13,85,113,69,23)(14,86,114,70,24)(15,87,115,71,25)(16,88,116,72,26)(33,98,60,140,158)(34,99,61,141,159)(35,100,62,142,160)(36,101,63,143,145)(37,102,64,144,146)(38,103,49,129,147)(39,104,50,130,148)(40,105,51,131,149)(41,106,52,132,150)(42,107,53,133,151)(43,108,54,134,152)(44,109,55,135,153)(45,110,56,136,154)(46,111,57,137,155)(47,112,58,138,156)(48,97,59,139,157), (1,147,9,155)(2,148,10,156)(3,149,11,157)(4,150,12,158)(5,151,13,159)(6,152,14,160)(7,153,15,145)(8,154,16,146)(17,135,25,143)(18,136,26,144)(19,137,27,129)(20,138,28,130)(21,139,29,131)(22,140,30,132)(23,141,31,133)(24,142,32,134)(33,92,41,84)(34,93,42,85)(35,94,43,86)(36,95,44,87)(37,96,45,88)(38,81,46,89)(39,82,47,90)(40,83,48,91)(49,65,57,73)(50,66,58,74)(51,67,59,75)(52,68,60,76)(53,69,61,77)(54,70,62,78)(55,71,63,79)(56,72,64,80)(97,119,105,127)(98,120,106,128)(99,121,107,113)(100,122,108,114)(101,123,109,115)(102,124,110,116)(103,125,111,117)(104,126,112,118), (1,155)(2,156)(3,157)(4,158)(5,159)(6,160)(7,145)(8,146)(9,147)(10,148)(11,149)(12,150)(13,151)(14,152)(15,153)(16,154)(17,143)(18,144)(19,129)(20,130)(21,131)(22,132)(23,133)(24,134)(25,135)(26,136)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,92)(34,93)(35,94)(36,95)(37,96)(38,81)(39,82)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76)(61,77)(62,78)(63,79)(64,80)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(105,127)(106,128)(107,113)(108,114)(109,115)(110,116)(111,117)(112,118), (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) );
G=PermutationGroup([(1,89,117,73,27),(2,90,118,74,28),(3,91,119,75,29),(4,92,120,76,30),(5,93,121,77,31),(6,94,122,78,32),(7,95,123,79,17),(8,96,124,80,18),(9,81,125,65,19),(10,82,126,66,20),(11,83,127,67,21),(12,84,128,68,22),(13,85,113,69,23),(14,86,114,70,24),(15,87,115,71,25),(16,88,116,72,26),(33,98,60,140,158),(34,99,61,141,159),(35,100,62,142,160),(36,101,63,143,145),(37,102,64,144,146),(38,103,49,129,147),(39,104,50,130,148),(40,105,51,131,149),(41,106,52,132,150),(42,107,53,133,151),(43,108,54,134,152),(44,109,55,135,153),(45,110,56,136,154),(46,111,57,137,155),(47,112,58,138,156),(48,97,59,139,157)], [(1,147,9,155),(2,148,10,156),(3,149,11,157),(4,150,12,158),(5,151,13,159),(6,152,14,160),(7,153,15,145),(8,154,16,146),(17,135,25,143),(18,136,26,144),(19,137,27,129),(20,138,28,130),(21,139,29,131),(22,140,30,132),(23,141,31,133),(24,142,32,134),(33,92,41,84),(34,93,42,85),(35,94,43,86),(36,95,44,87),(37,96,45,88),(38,81,46,89),(39,82,47,90),(40,83,48,91),(49,65,57,73),(50,66,58,74),(51,67,59,75),(52,68,60,76),(53,69,61,77),(54,70,62,78),(55,71,63,79),(56,72,64,80),(97,119,105,127),(98,120,106,128),(99,121,107,113),(100,122,108,114),(101,123,109,115),(102,124,110,116),(103,125,111,117),(104,126,112,118)], [(1,155),(2,156),(3,157),(4,158),(5,159),(6,160),(7,145),(8,146),(9,147),(10,148),(11,149),(12,150),(13,151),(14,152),(15,153),(16,154),(17,143),(18,144),(19,129),(20,130),(21,131),(22,132),(23,133),(24,134),(25,135),(26,136),(27,137),(28,138),(29,139),(30,140),(31,141),(32,142),(33,92),(34,93),(35,94),(36,95),(37,96),(38,81),(39,82),(40,83),(41,84),(42,85),(43,86),(44,87),(45,88),(46,89),(47,90),(48,91),(49,65),(50,66),(51,67),(52,68),(53,69),(54,70),(55,71),(56,72),(57,73),(58,74),(59,75),(60,76),(61,77),(62,78),(63,79),(64,80),(97,119),(98,120),(99,121),(100,122),(101,123),(102,124),(103,125),(104,126),(105,127),(106,128),(107,113),(108,114),(109,115),(110,116),(111,117),(112,118)], [(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)])
Matrix representation ►G ⊆ GL3(𝔽241) generated by
| 91 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 240 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 240 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 240 | 0 | 0 |
| 0 | 111 | 0 |
| 0 | 0 | 111 |
G:=sub<GL(3,GF(241))| [91,0,0,0,1,0,0,0,1],[240,0,0,0,0,240,0,1,0],[1,0,0,0,0,1,0,1,0],[240,0,0,0,111,0,0,0,111] >;
200 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 10A | 10B | 10C | 10D | 10E | ··· | 10P | 16A | ··· | 16H | 16I | ··· | 16T | 20A | ··· | 20H | 20I | ··· | 20T | 40A | ··· | 40P | 40Q | ··· | 40AN | 80A | ··· | 80AF | 80AG | ··· | 80CB |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 16 | ··· | 16 | 16 | ··· | 16 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 | 80 | ··· | 80 | 80 | ··· | 80 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 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 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C8 | C8 | C10 | C10 | C10 | C20 | C20 | C40 | C40 | D4○C16 | C5×D4○C16 |
| kernel | C5×D4○C16 | C2×C80 | C5×M5(2) | C5×C8○D4 | C5×M4(2) | C5×C4○D4 | D4○C16 | C5×D4 | C5×Q8 | C2×C16 | M5(2) | C8○D4 | M4(2) | C4○D4 | D4 | Q8 | C5 | C1 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 4 | 12 | 4 | 12 | 12 | 4 | 24 | 8 | 48 | 16 | 8 | 32 |
In GAP, Magma, Sage, TeX
C_5\times D_4\circ C_{16} % in TeX
G:=Group("C5xD4oC16"); // GroupNames label
G:=SmallGroup(320,1005);
// by ID
G=gap.SmallGroup(320,1005);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,1731,102,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^2=1,d^8=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,c*d=d*c>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀