Copied to
clipboard

## G = C5×D4.5D4order 320 = 26·5

### Direct product of C5 and D4.5D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×D4.5D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — Q8×C10 — C10×Q16 — C5×D4.5D4
 Lower central C1 — C2 — C2×C4 — C5×D4.5D4
 Upper central C1 — C10 — C2×C20 — C5×D4.5D4

Generators and relations for C5×D4.5D4
G = < a,b,c,d,e | a5=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=b2d3 >

Subgroups: 162 in 100 conjugacy classes, 50 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C10, C10, C2×C8, C2×C8, M4(2), M4(2), M4(2), SD16, Q16, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, C40, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, D4.5D4, C2×C40, C2×C40, C5×M4(2), C5×M4(2), C5×M4(2), C5×SD16, C5×Q16, Q8×C10, C5×C4○D4, C5×C4.10D4, C5×C8.C4, C5×C8○D4, C10×Q16, C5×C8.C22, C5×D4.5D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C2×C10, C4⋊D4, C5×D4, C22×C10, D4.5D4, D4×C10, C5×C4○D4, C5×C4⋊D4, C5×D4.5D4

Smallest permutation representation of C5×D4.5D4
On 160 points
Generators in S160
(1 17 52 13 61)(2 18 53 14 62)(3 19 54 15 63)(4 20 55 16 64)(5 21 56 9 57)(6 22 49 10 58)(7 23 50 11 59)(8 24 51 12 60)(25 85 73 33 65)(26 86 74 34 66)(27 87 75 35 67)(28 88 76 36 68)(29 81 77 37 69)(30 82 78 38 70)(31 83 79 39 71)(32 84 80 40 72)(41 156 120 148 112)(42 157 113 149 105)(43 158 114 150 106)(44 159 115 151 107)(45 160 116 152 108)(46 153 117 145 109)(47 154 118 146 110)(48 155 119 147 111)(89 126 141 97 133)(90 127 142 98 134)(91 128 143 99 135)(92 121 144 100 136)(93 122 137 101 129)(94 123 138 102 130)(95 124 139 103 131)(96 125 140 104 132)
(1 31 5 27)(2 32 6 28)(3 25 7 29)(4 26 8 30)(9 35 13 39)(10 36 14 40)(11 37 15 33)(12 38 16 34)(17 83 21 87)(18 84 22 88)(19 85 23 81)(20 86 24 82)(41 123 45 127)(42 124 46 128)(43 125 47 121)(44 126 48 122)(49 76 53 80)(50 77 54 73)(51 78 55 74)(52 79 56 75)(57 67 61 71)(58 68 62 72)(59 69 63 65)(60 70 64 66)(89 111 93 107)(90 112 94 108)(91 105 95 109)(92 106 96 110)(97 119 101 115)(98 120 102 116)(99 113 103 117)(100 114 104 118)(129 151 133 147)(130 152 134 148)(131 145 135 149)(132 146 136 150)(137 159 141 155)(138 160 142 156)(139 153 143 157)(140 154 144 158)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 39)(10 40)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 87)(18 88)(19 81)(20 82)(21 83)(22 84)(23 85)(24 86)(49 80)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)(57 71)(58 72)(59 65)(60 66)(61 67)(62 68)(63 69)(64 70)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)
(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 94 5 90)(2 93 6 89)(3 92 7 96)(4 91 8 95)(9 98 13 102)(10 97 14 101)(11 104 15 100)(12 103 16 99)(17 123 21 127)(18 122 22 126)(19 121 23 125)(20 128 24 124)(25 110 29 106)(26 109 30 105)(27 108 31 112)(28 107 32 111)(33 118 37 114)(34 117 38 113)(35 116 39 120)(36 115 40 119)(41 87 45 83)(42 86 46 82)(43 85 47 81)(44 84 48 88)(49 141 53 137)(50 140 54 144)(51 139 55 143)(52 138 56 142)(57 134 61 130)(58 133 62 129)(59 132 63 136)(60 131 64 135)(65 146 69 150)(66 145 70 149)(67 152 71 148)(68 151 72 147)(73 154 77 158)(74 153 78 157)(75 160 79 156)(76 159 80 155)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 5A 5B 5C 5D 8A 8B 8C 8D 8E 8F 8G 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 20A ··· 20H 20I 20J 20K 20L 20M ··· 20T 40A ··· 40H 40I ··· 40T 40U ··· 40AB order 1 2 2 2 4 4 4 4 4 5 5 5 5 8 8 8 8 8 8 8 10 10 10 10 10 10 10 10 10 10 10 10 20 ··· 20 20 20 20 20 20 ··· 20 40 ··· 40 40 ··· 40 40 ··· 40 size 1 1 2 4 2 2 4 8 8 1 1 1 1 2 2 4 4 4 8 8 1 1 1 1 2 2 2 2 4 4 4 4 2 ··· 2 4 4 4 4 8 ··· 8 2 ··· 2 4 ··· 4 8 ··· 8

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D4 D4 D4 C4○D4 C5×D4 C5×D4 C5×D4 C5×C4○D4 D4.5D4 C5×D4.5D4 kernel C5×D4.5D4 C5×C4.10D4 C5×C8.C4 C5×C8○D4 C10×Q16 C5×C8.C22 D4.5D4 C4.10D4 C8.C4 C8○D4 C2×Q16 C8.C22 C40 C5×D4 C5×Q8 C2×C10 C8 D4 Q8 C22 C5 C1 # reps 1 2 1 1 1 2 4 8 4 4 4 8 2 1 1 2 8 4 4 8 2 8

Matrix representation of C5×D4.5D4 in GL6(𝔽41)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 22 2 0 0 0 0 24 19 0 0 0 0 0 0 29 0 12 0 0 0 0 29 0 12 0 0 29 0 29 0 0 0 0 29 0 29
,
 36 13 0 0 0 0 36 5 0 0 0 0 0 0 4 37 2 39 0 0 37 37 39 39 0 0 2 39 37 4 0 0 39 39 4 4

G:=sub<GL(6,GF(41))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[22,24,0,0,0,0,2,19,0,0,0,0,0,0,29,0,29,0,0,0,0,29,0,29,0,0,12,0,29,0,0,0,0,12,0,29],[36,36,0,0,0,0,13,5,0,0,0,0,0,0,4,37,2,39,0,0,37,37,39,39,0,0,2,39,37,4,0,0,39,39,4,4] >;

C5×D4.5D4 in GAP, Magma, Sage, TeX

C_5\times D_4._5D_4
% in TeX

G:=Group("C5xD4.5D4");
// GroupNames label

G:=SmallGroup(320,974);
// by ID

G=gap.SmallGroup(320,974);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1408,1766,7004,172,10085,2539,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=c^2=1,d^4=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=b^2*d^3>;
// generators/relations

׿
×
𝔽