Copied to
clipboard

## G = C5×C2.D8order 160 = 25·5

### Direct product of C5 and C2.D8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C5×C2.D8
 Chief series C1 — C2 — C22 — C2×C4 — C2×C20 — C5×C4⋊C4 — C5×C2.D8
 Lower central C1 — C2 — C4 — C5×C2.D8
 Upper central C1 — C2×C10 — C2×C20 — C5×C2.D8

Generators and relations for C5×C2.D8
G = < a,b,c,d | a5=b2=c8=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Smallest permutation representation of C5×C2.D8
Regular action on 160 points
Generators in S160
(1 17 13 52 63)(2 18 14 53 64)(3 19 15 54 57)(4 20 16 55 58)(5 21 9 56 59)(6 22 10 49 60)(7 23 11 50 61)(8 24 12 51 62)(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)(89 122 140 104 132)(90 123 141 97 133)(91 124 142 98 134)(92 125 143 99 135)(93 126 144 100 136)(94 127 137 101 129)(95 128 138 102 130)(96 121 139 103 131)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 79)(10 80)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 87)(18 88)(19 81)(20 82)(21 83)(22 84)(23 85)(24 86)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 49)(41 93)(42 94)(43 95)(44 96)(45 89)(46 90)(47 91)(48 92)(57 69)(58 70)(59 71)(60 72)(61 65)(62 66)(63 67)(64 68)(97 145)(98 146)(99 147)(100 148)(101 149)(102 150)(103 151)(104 152)(105 129)(106 130)(107 131)(108 132)(109 133)(110 134)(111 135)(112 136)(113 137)(114 138)(115 139)(116 140)(117 141)(118 142)(119 143)(120 144)(121 159)(122 160)(123 153)(124 154)(125 155)(126 156)(127 157)(128 158)
(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)
(1 112 27 136)(2 111 28 135)(3 110 29 134)(4 109 30 133)(5 108 31 132)(6 107 32 131)(7 106 25 130)(8 105 26 129)(9 160 79 122)(10 159 80 121)(11 158 73 128)(12 157 74 127)(13 156 75 126)(14 155 76 125)(15 154 77 124)(16 153 78 123)(17 41 87 93)(18 48 88 92)(19 47 81 91)(20 46 82 90)(21 45 83 89)(22 44 84 96)(23 43 85 95)(24 42 86 94)(33 138 50 114)(34 137 51 113)(35 144 52 120)(36 143 53 119)(37 142 54 118)(38 141 55 117)(39 140 56 116)(40 139 49 115)(57 146 69 98)(58 145 70 97)(59 152 71 104)(60 151 72 103)(61 150 65 102)(62 149 66 101)(63 148 67 100)(64 147 68 99)

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 20A ··· 20H 20I ··· 20X 40A ··· 40P order 1 2 2 2 4 4 4 4 4 4 5 5 5 5 8 8 8 8 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 4 4 4 4 1 1 1 1 2 2 2 2 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + - + + - image C1 C2 C2 C4 C5 C10 C10 C20 Q8 D4 D8 Q16 C5×Q8 C5×D4 C5×D8 C5×Q16 kernel C5×C2.D8 C5×C4⋊C4 C2×C40 C40 C2.D8 C4⋊C4 C2×C8 C8 C20 C2×C10 C10 C10 C4 C22 C2 C2 # reps 1 2 1 4 4 8 4 16 1 1 2 2 4 4 8 8

Matrix representation of C5×C2.D8 in GL4(𝔽41) generated by

 10 0 0 0 0 10 0 0 0 0 18 0 0 0 0 18
,
 40 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 32 0 0 0 40 9 0 0 0 0 12 29 0 0 12 12
,
 39 36 0 0 1 2 0 0 0 0 9 17 0 0 17 32
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,18,0,0,0,0,18],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[32,40,0,0,0,9,0,0,0,0,12,12,0,0,29,12],[39,1,0,0,36,2,0,0,0,0,9,17,0,0,17,32] >;

C5×C2.D8 in GAP, Magma, Sage, TeX

C_5\times C_2.D_8
% in TeX

G:=Group("C5xC2.D8");
// GroupNames label

G:=SmallGroup(160,57);
// by ID

G=gap.SmallGroup(160,57);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-2,240,265,607,2403,117]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^8=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

Export

׿
×
𝔽