Copied to
clipboard

## G = D5×C2×C16order 320 = 26·5

### Direct product of C2×C16 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C2×C16
 Chief series C1 — C5 — C10 — C20 — C40 — C8×D5 — D5×C2×C8 — D5×C2×C16
 Lower central C5 — D5×C2×C16
 Upper central C1 — C2×C16

Generators and relations for D5×C2×C16
G = < a,b,c,d | a2=b16=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 238 in 98 conjugacy classes, 63 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, C16, C16, C2×C8, C2×C8, C22×C4, Dic5, C20, D10, C2×C10, C2×C16, C2×C16, C22×C8, C52C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×C16, C52C16, C80, C8×D5, C2×C52C8, C2×C40, C2×C4×D5, D5×C16, C2×C52C16, C2×C80, D5×C2×C8, D5×C2×C16
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D5, C16, C2×C8, C22×C4, D10, C2×C16, C22×C8, C4×D5, C22×D5, C22×C16, C8×D5, C2×C4×D5, D5×C16, D5×C2×C8, D5×C2×C16

Smallest permutation representation of D5×C2×C16
On 160 points
Generators in S160
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 81)(14 82)(15 83)(16 84)(17 113)(18 114)(19 115)(20 116)(21 117)(22 118)(23 119)(24 120)(25 121)(26 122)(27 123)(28 124)(29 125)(30 126)(31 127)(32 128)(33 148)(34 149)(35 150)(36 151)(37 152)(38 153)(39 154)(40 155)(41 156)(42 157)(43 158)(44 159)(45 160)(46 145)(47 146)(48 147)(49 139)(50 140)(51 141)(52 142)(53 143)(54 144)(55 129)(56 130)(57 131)(58 132)(59 133)(60 134)(61 135)(62 136)(63 137)(64 138)(65 98)(66 99)(67 100)(68 101)(69 102)(70 103)(71 104)(72 105)(73 106)(74 107)(75 108)(76 109)(77 110)(78 111)(79 112)(80 97)
(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 54 26 152 75)(2 55 27 153 76)(3 56 28 154 77)(4 57 29 155 78)(5 58 30 156 79)(6 59 31 157 80)(7 60 32 158 65)(8 61 17 159 66)(9 62 18 160 67)(10 63 19 145 68)(11 64 20 146 69)(12 49 21 147 70)(13 50 22 148 71)(14 51 23 149 72)(15 52 24 150 73)(16 53 25 151 74)(33 104 81 140 118)(34 105 82 141 119)(35 106 83 142 120)(36 107 84 143 121)(37 108 85 144 122)(38 109 86 129 123)(39 110 87 130 124)(40 111 88 131 125)(41 112 89 132 126)(42 97 90 133 127)(43 98 91 134 128)(44 99 92 135 113)(45 100 93 136 114)(46 101 94 137 115)(47 102 95 138 116)(48 103 96 139 117)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 65)(16 66)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 132)(34 133)(35 134)(36 135)(37 136)(38 137)(39 138)(40 139)(41 140)(42 141)(43 142)(44 143)(45 144)(46 129)(47 130)(48 131)(49 155)(50 156)(51 157)(52 158)(53 159)(54 160)(55 145)(56 146)(57 147)(58 148)(59 149)(60 150)(61 151)(62 152)(63 153)(64 154)(81 112)(82 97)(83 98)(84 99)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 127)(120 128)

G:=sub<Sym(160)| (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,81)(14,82)(15,83)(16,84)(17,113)(18,114)(19,115)(20,116)(21,117)(22,118)(23,119)(24,120)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,127)(32,128)(33,148)(34,149)(35,150)(36,151)(37,152)(38,153)(39,154)(40,155)(41,156)(42,157)(43,158)(44,159)(45,160)(46,145)(47,146)(48,147)(49,139)(50,140)(51,141)(52,142)(53,143)(54,144)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,135)(62,136)(63,137)(64,138)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,111)(79,112)(80,97), (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), (1,54,26,152,75)(2,55,27,153,76)(3,56,28,154,77)(4,57,29,155,78)(5,58,30,156,79)(6,59,31,157,80)(7,60,32,158,65)(8,61,17,159,66)(9,62,18,160,67)(10,63,19,145,68)(11,64,20,146,69)(12,49,21,147,70)(13,50,22,148,71)(14,51,23,149,72)(15,52,24,150,73)(16,53,25,151,74)(33,104,81,140,118)(34,105,82,141,119)(35,106,83,142,120)(36,107,84,143,121)(37,108,85,144,122)(38,109,86,129,123)(39,110,87,130,124)(40,111,88,131,125)(41,112,89,132,126)(42,97,90,133,127)(43,98,91,134,128)(44,99,92,135,113)(45,100,93,136,114)(46,101,94,137,115)(47,102,95,138,116)(48,103,96,139,117), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,65)(16,66)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,132)(34,133)(35,134)(36,135)(37,136)(38,137)(39,138)(40,139)(41,140)(42,141)(43,142)(44,143)(45,144)(46,129)(47,130)(48,131)(49,155)(50,156)(51,157)(52,158)(53,159)(54,160)(55,145)(56,146)(57,147)(58,148)(59,149)(60,150)(61,151)(62,152)(63,153)(64,154)(81,112)(82,97)(83,98)(84,99)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)>;

G:=Group( (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,81)(14,82)(15,83)(16,84)(17,113)(18,114)(19,115)(20,116)(21,117)(22,118)(23,119)(24,120)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,127)(32,128)(33,148)(34,149)(35,150)(36,151)(37,152)(38,153)(39,154)(40,155)(41,156)(42,157)(43,158)(44,159)(45,160)(46,145)(47,146)(48,147)(49,139)(50,140)(51,141)(52,142)(53,143)(54,144)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,135)(62,136)(63,137)(64,138)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,111)(79,112)(80,97), (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), (1,54,26,152,75)(2,55,27,153,76)(3,56,28,154,77)(4,57,29,155,78)(5,58,30,156,79)(6,59,31,157,80)(7,60,32,158,65)(8,61,17,159,66)(9,62,18,160,67)(10,63,19,145,68)(11,64,20,146,69)(12,49,21,147,70)(13,50,22,148,71)(14,51,23,149,72)(15,52,24,150,73)(16,53,25,151,74)(33,104,81,140,118)(34,105,82,141,119)(35,106,83,142,120)(36,107,84,143,121)(37,108,85,144,122)(38,109,86,129,123)(39,110,87,130,124)(40,111,88,131,125)(41,112,89,132,126)(42,97,90,133,127)(43,98,91,134,128)(44,99,92,135,113)(45,100,93,136,114)(46,101,94,137,115)(47,102,95,138,116)(48,103,96,139,117), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,65)(16,66)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,132)(34,133)(35,134)(36,135)(37,136)(38,137)(39,138)(40,139)(41,140)(42,141)(43,142)(44,143)(45,144)(46,129)(47,130)(48,131)(49,155)(50,156)(51,157)(52,158)(53,159)(54,160)(55,145)(56,146)(57,147)(58,148)(59,149)(60,150)(61,151)(62,152)(63,153)(64,154)(81,112)(82,97)(83,98)(84,99)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128) );

G=PermutationGroup([[(1,85),(2,86),(3,87),(4,88),(5,89),(6,90),(7,91),(8,92),(9,93),(10,94),(11,95),(12,96),(13,81),(14,82),(15,83),(16,84),(17,113),(18,114),(19,115),(20,116),(21,117),(22,118),(23,119),(24,120),(25,121),(26,122),(27,123),(28,124),(29,125),(30,126),(31,127),(32,128),(33,148),(34,149),(35,150),(36,151),(37,152),(38,153),(39,154),(40,155),(41,156),(42,157),(43,158),(44,159),(45,160),(46,145),(47,146),(48,147),(49,139),(50,140),(51,141),(52,142),(53,143),(54,144),(55,129),(56,130),(57,131),(58,132),(59,133),(60,134),(61,135),(62,136),(63,137),(64,138),(65,98),(66,99),(67,100),(68,101),(69,102),(70,103),(71,104),(72,105),(73,106),(74,107),(75,108),(76,109),(77,110),(78,111),(79,112),(80,97)], [(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)], [(1,54,26,152,75),(2,55,27,153,76),(3,56,28,154,77),(4,57,29,155,78),(5,58,30,156,79),(6,59,31,157,80),(7,60,32,158,65),(8,61,17,159,66),(9,62,18,160,67),(10,63,19,145,68),(11,64,20,146,69),(12,49,21,147,70),(13,50,22,148,71),(14,51,23,149,72),(15,52,24,150,73),(16,53,25,151,74),(33,104,81,140,118),(34,105,82,141,119),(35,106,83,142,120),(36,107,84,143,121),(37,108,85,144,122),(38,109,86,129,123),(39,110,87,130,124),(40,111,88,131,125),(41,112,89,132,126),(42,97,90,133,127),(43,98,91,134,128),(44,99,92,135,113),(45,100,93,136,114),(46,101,94,137,115),(47,102,95,138,116),(48,103,96,139,117)], [(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,73),(8,74),(9,75),(10,76),(11,77),(12,78),(13,79),(14,80),(15,65),(16,66),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,132),(34,133),(35,134),(36,135),(37,136),(38,137),(39,138),(40,139),(41,140),(42,141),(43,142),(44,143),(45,144),(46,129),(47,130),(48,131),(49,155),(50,156),(51,157),(52,158),(53,159),(54,160),(55,145),(56,146),(57,147),(58,148),(59,149),(60,150),(61,151),(62,152),(63,153),(64,154),(81,112),(82,97),(83,98),(84,99),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105),(91,106),(92,107),(93,108),(94,109),(95,110),(96,111),(113,121),(114,122),(115,123),(116,124),(117,125),(118,126),(119,127),(120,128)]])

128 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A ··· 8H 8I ··· 8P 10A ··· 10F 16A ··· 16P 16Q ··· 16AF 20A ··· 20H 40A ··· 40P 80A ··· 80AF order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 ··· 8 8 ··· 8 10 ··· 10 16 ··· 16 16 ··· 16 20 ··· 20 40 ··· 40 80 ··· 80 size 1 1 1 1 5 5 5 5 1 1 1 1 5 5 5 5 2 2 1 ··· 1 5 ··· 5 2 ··· 2 1 ··· 1 5 ··· 5 2 ··· 2 2 ··· 2 2 ··· 2

128 irreducible representations

 dim 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 C4 C4 C4 C8 C8 C8 C16 D5 D10 D10 C4×D5 C4×D5 C8×D5 C8×D5 D5×C16 kernel D5×C2×C16 D5×C16 C2×C5⋊2C16 C2×C80 D5×C2×C8 C8×D5 C2×C5⋊2C8 C2×C4×D5 C4×D5 C2×Dic5 C22×D5 D10 C2×C16 C16 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 8 4 4 32 2 4 2 4 4 8 8 32

Matrix representation of D5×C2×C16 in GL3(𝔽241) generated by

 240 0 0 0 240 0 0 0 240
,
 130 0 0 0 8 0 0 0 8
,
 1 0 0 0 189 1 0 240 0
,
 1 0 0 0 1 189 0 0 240
G:=sub<GL(3,GF(241))| [240,0,0,0,240,0,0,0,240],[130,0,0,0,8,0,0,0,8],[1,0,0,0,189,240,0,1,0],[1,0,0,0,1,0,0,189,240] >;

D5×C2×C16 in GAP, Magma, Sage, TeX

D_5\times C_2\times C_{16}
% in TeX

G:=Group("D5xC2xC16");
// GroupNames label

G:=SmallGroup(320,526);
// by ID

G=gap.SmallGroup(320,526);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,58,80,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^16=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽