direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×D5⋊C16, D10⋊3C16, D5⋊(C2×C16), C10⋊1(C2×C16), C5⋊C16⋊4C22, C5⋊1(C22×C16), (C4×D5).7C8, (C2×C8).24F5, C8.41(C2×F5), C40.38(C2×C4), C20.23(C2×C8), (C2×C40).22C4, (C8×D5).10C4, C4.18(D5⋊C8), D10.16(C2×C8), C10.8(C22×C8), (C22×D5).7C8, C4.45(C22×F5), C20.85(C22×C4), C5⋊2C8.35C23, Dic5.16(C2×C8), (C2×Dic5).12C8, (C8×D5).61C22, C22.12(D5⋊C8), (C2×C5⋊C16)⋊9C2, C2.2(C2×D5⋊C8), (C2×C4×D5).45C4, (D5×C2×C8).29C2, (C2×C10).10(C2×C8), C5⋊2C8.51(C2×C4), (C4×D5).91(C2×C4), (C2×C4).163(C2×F5), (C2×C20).172(C2×C4), (C2×C5⋊2C8).348C22, SmallGroup(320,1051)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C2×D5⋊C16 |
Subgroups: 250 in 98 conjugacy classes, 60 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C22, C22 [×6], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, D5 [×4], C10, C10 [×2], C16 [×4], C2×C8, C2×C8 [×5], C22×C4, Dic5 [×2], C20 [×2], D10 [×6], C2×C10, C2×C16 [×6], C22×C8, C5⋊2C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C22×C16, C5⋊C16 [×4], C8×D5 [×4], C2×C5⋊2C8, C2×C40, C2×C4×D5, D5⋊C16 [×4], C2×C5⋊C16 [×2], D5×C2×C8, C2×D5⋊C16
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C16 [×4], C2×C8 [×6], C22×C4, F5, C2×C16 [×6], C22×C8, C2×F5 [×3], C22×C16, D5⋊C8 [×2], C22×F5, D5⋊C16 [×2], C2×D5⋊C8, C2×D5⋊C16
Generators and relations
G = < a,b,c,d | a2=b5=c2=d16=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b2c >
►On 160 points
Generators in S160
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 81)(8 82)(9 83)(10 84)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 109)(18 110)(19 111)(20 112)(21 97)(22 98)(23 99)(24 100)(25 101)(26 102)(27 103)(28 104)(29 105)(30 106)(31 107)(32 108)(33 147)(34 148)(35 149)(36 150)(37 151)(38 152)(39 153)(40 154)(41 155)(42 156)(43 157)(44 158)(45 159)(46 160)(47 145)(48 146)(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)(113 138)(114 139)(115 140)(116 141)(117 142)(118 143)(119 144)(120 129)(121 130)(122 131)(123 132)(124 133)(125 134)(126 135)(127 136)(128 137)
(1 69 34 121 101)(2 122 70 102 35)(3 103 123 36 71)(4 37 104 72 124)(5 73 38 125 105)(6 126 74 106 39)(7 107 127 40 75)(8 41 108 76 128)(9 77 42 113 109)(10 114 78 110 43)(11 111 115 44 79)(12 45 112 80 116)(13 65 46 117 97)(14 118 66 98 47)(15 99 119 48 67)(16 33 100 68 120)(17 83 61 156 138)(18 157 84 139 62)(19 140 158 63 85)(20 64 141 86 159)(21 87 49 160 142)(22 145 88 143 50)(23 144 146 51 89)(24 52 129 90 147)(25 91 53 148 130)(26 149 92 131 54)(27 132 150 55 93)(28 56 133 94 151)(29 95 57 152 134)(30 153 96 135 58)(31 136 154 59 81)(32 60 137 82 155)
(1 101)(2 35)(3 71)(4 124)(5 105)(6 39)(7 75)(8 128)(9 109)(10 43)(11 79)(12 116)(13 97)(14 47)(15 67)(16 120)(17 83)(18 139)(19 158)(21 87)(22 143)(23 146)(25 91)(26 131)(27 150)(29 95)(30 135)(31 154)(33 68)(36 103)(37 72)(40 107)(41 76)(44 111)(45 80)(48 99)(49 142)(51 89)(52 147)(53 130)(55 93)(56 151)(57 134)(59 81)(60 155)(61 138)(63 85)(64 159)(65 117)(69 121)(73 125)(77 113)(82 137)(84 157)(86 141)(88 145)(90 129)(92 149)(94 133)(96 153)(98 118)(102 122)(106 126)(110 114)
(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)| (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,109)(18,110)(19,111)(20,112)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,147)(34,148)(35,149)(36,150)(37,151)(38,152)(39,153)(40,154)(41,155)(42,156)(43,157)(44,158)(45,159)(46,160)(47,145)(48,146)(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)(113,138)(114,139)(115,140)(116,141)(117,142)(118,143)(119,144)(120,129)(121,130)(122,131)(123,132)(124,133)(125,134)(126,135)(127,136)(128,137), (1,69,34,121,101)(2,122,70,102,35)(3,103,123,36,71)(4,37,104,72,124)(5,73,38,125,105)(6,126,74,106,39)(7,107,127,40,75)(8,41,108,76,128)(9,77,42,113,109)(10,114,78,110,43)(11,111,115,44,79)(12,45,112,80,116)(13,65,46,117,97)(14,118,66,98,47)(15,99,119,48,67)(16,33,100,68,120)(17,83,61,156,138)(18,157,84,139,62)(19,140,158,63,85)(20,64,141,86,159)(21,87,49,160,142)(22,145,88,143,50)(23,144,146,51,89)(24,52,129,90,147)(25,91,53,148,130)(26,149,92,131,54)(27,132,150,55,93)(28,56,133,94,151)(29,95,57,152,134)(30,153,96,135,58)(31,136,154,59,81)(32,60,137,82,155), (1,101)(2,35)(3,71)(4,124)(5,105)(6,39)(7,75)(8,128)(9,109)(10,43)(11,79)(12,116)(13,97)(14,47)(15,67)(16,120)(17,83)(18,139)(19,158)(21,87)(22,143)(23,146)(25,91)(26,131)(27,150)(29,95)(30,135)(31,154)(33,68)(36,103)(37,72)(40,107)(41,76)(44,111)(45,80)(48,99)(49,142)(51,89)(52,147)(53,130)(55,93)(56,151)(57,134)(59,81)(60,155)(61,138)(63,85)(64,159)(65,117)(69,121)(73,125)(77,113)(82,137)(84,157)(86,141)(88,145)(90,129)(92,149)(94,133)(96,153)(98,118)(102,122)(106,126)(110,114), (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( (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,109)(18,110)(19,111)(20,112)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,147)(34,148)(35,149)(36,150)(37,151)(38,152)(39,153)(40,154)(41,155)(42,156)(43,157)(44,158)(45,159)(46,160)(47,145)(48,146)(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)(113,138)(114,139)(115,140)(116,141)(117,142)(118,143)(119,144)(120,129)(121,130)(122,131)(123,132)(124,133)(125,134)(126,135)(127,136)(128,137), (1,69,34,121,101)(2,122,70,102,35)(3,103,123,36,71)(4,37,104,72,124)(5,73,38,125,105)(6,126,74,106,39)(7,107,127,40,75)(8,41,108,76,128)(9,77,42,113,109)(10,114,78,110,43)(11,111,115,44,79)(12,45,112,80,116)(13,65,46,117,97)(14,118,66,98,47)(15,99,119,48,67)(16,33,100,68,120)(17,83,61,156,138)(18,157,84,139,62)(19,140,158,63,85)(20,64,141,86,159)(21,87,49,160,142)(22,145,88,143,50)(23,144,146,51,89)(24,52,129,90,147)(25,91,53,148,130)(26,149,92,131,54)(27,132,150,55,93)(28,56,133,94,151)(29,95,57,152,134)(30,153,96,135,58)(31,136,154,59,81)(32,60,137,82,155), (1,101)(2,35)(3,71)(4,124)(5,105)(6,39)(7,75)(8,128)(9,109)(10,43)(11,79)(12,116)(13,97)(14,47)(15,67)(16,120)(17,83)(18,139)(19,158)(21,87)(22,143)(23,146)(25,91)(26,131)(27,150)(29,95)(30,135)(31,154)(33,68)(36,103)(37,72)(40,107)(41,76)(44,111)(45,80)(48,99)(49,142)(51,89)(52,147)(53,130)(55,93)(56,151)(57,134)(59,81)(60,155)(61,138)(63,85)(64,159)(65,117)(69,121)(73,125)(77,113)(82,137)(84,157)(86,141)(88,145)(90,129)(92,149)(94,133)(96,153)(98,118)(102,122)(106,126)(110,114), (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([(1,91),(2,92),(3,93),(4,94),(5,95),(6,96),(7,81),(8,82),(9,83),(10,84),(11,85),(12,86),(13,87),(14,88),(15,89),(16,90),(17,109),(18,110),(19,111),(20,112),(21,97),(22,98),(23,99),(24,100),(25,101),(26,102),(27,103),(28,104),(29,105),(30,106),(31,107),(32,108),(33,147),(34,148),(35,149),(36,150),(37,151),(38,152),(39,153),(40,154),(41,155),(42,156),(43,157),(44,158),(45,159),(46,160),(47,145),(48,146),(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),(113,138),(114,139),(115,140),(116,141),(117,142),(118,143),(119,144),(120,129),(121,130),(122,131),(123,132),(124,133),(125,134),(126,135),(127,136),(128,137)], [(1,69,34,121,101),(2,122,70,102,35),(3,103,123,36,71),(4,37,104,72,124),(5,73,38,125,105),(6,126,74,106,39),(7,107,127,40,75),(8,41,108,76,128),(9,77,42,113,109),(10,114,78,110,43),(11,111,115,44,79),(12,45,112,80,116),(13,65,46,117,97),(14,118,66,98,47),(15,99,119,48,67),(16,33,100,68,120),(17,83,61,156,138),(18,157,84,139,62),(19,140,158,63,85),(20,64,141,86,159),(21,87,49,160,142),(22,145,88,143,50),(23,144,146,51,89),(24,52,129,90,147),(25,91,53,148,130),(26,149,92,131,54),(27,132,150,55,93),(28,56,133,94,151),(29,95,57,152,134),(30,153,96,135,58),(31,136,154,59,81),(32,60,137,82,155)], [(1,101),(2,35),(3,71),(4,124),(5,105),(6,39),(7,75),(8,128),(9,109),(10,43),(11,79),(12,116),(13,97),(14,47),(15,67),(16,120),(17,83),(18,139),(19,158),(21,87),(22,143),(23,146),(25,91),(26,131),(27,150),(29,95),(30,135),(31,154),(33,68),(36,103),(37,72),(40,107),(41,76),(44,111),(45,80),(48,99),(49,142),(51,89),(52,147),(53,130),(55,93),(56,151),(57,134),(59,81),(60,155),(61,138),(63,85),(64,159),(65,117),(69,121),(73,125),(77,113),(82,137),(84,157),(86,141),(88,145),(90,129),(92,149),(94,133),(96,153),(98,118),(102,122),(106,126),(110,114)], [(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 ⊆ GL5(𝔽241)
| 240 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 240 | 240 | 240 | 240 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 240 | 0 | 0 | 0 | 0 |
| 0 | 240 | 240 | 240 | 240 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 |
| 0 | 154 | 0 | 76 | 76 |
| 0 | 76 | 76 | 0 | 154 |
| 0 | 165 | 78 | 165 | 0 |
| 0 | 87 | 163 | 163 | 87 |
G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,240],[1,0,0,0,0,0,240,1,0,0,0,240,0,1,0,0,240,0,0,1,0,240,0,0,0],[240,0,0,0,0,0,240,0,0,0,0,240,0,0,1,0,240,0,1,0,0,240,1,0,0],[240,0,0,0,0,0,154,76,165,87,0,0,76,78,163,0,76,0,165,163,0,76,154,0,87] >;
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 8A | ··· | 8H | 8I | ··· | 8P | 10A | 10B | 10C | 16A | ··· | 16AF | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 4 | 1 | ··· | 1 | 5 | ··· | 5 | 4 | 4 | 4 | 5 | ··· | 5 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | C16 | F5 | C2×F5 | C2×F5 | D5⋊C8 | D5⋊C8 | D5⋊C16 |
| kernel | C2×D5⋊C16 | D5⋊C16 | C2×C5⋊C16 | D5×C2×C8 | C8×D5 | C2×C40 | C2×C4×D5 | C4×D5 | C2×Dic5 | C22×D5 | D10 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 8 | 4 | 4 | 32 | 1 | 2 | 1 | 2 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times D_5\rtimes C_{16} % in TeX
G:=Group("C2xD5:C16"); // GroupNames label
G:=SmallGroup(320,1051);
// by ID
G=gap.SmallGroup(320,1051);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,100,80,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^5=c^2=d^16=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^2*c>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀