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## G = C5×D4.C8order 320 = 26·5

### Direct product of C5 and D4.C8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C5×D4.C8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×C40 — C2×C80 — C5×D4.C8
 Lower central C1 — C2 — C4 — C5×D4.C8
 Upper central C1 — C40 — C2×C40 — C5×D4.C8

Generators and relations for C5×D4.C8
G = < a,b,c,d | a5=b4=c2=1, d8=b2, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=bc >

Smallest permutation representation of C5×D4.C8
On 160 points
Generators in S160
(1 146 86 113 41)(2 147 87 114 42)(3 148 88 115 43)(4 149 89 116 44)(5 150 90 117 45)(6 151 91 118 46)(7 152 92 119 47)(8 153 93 120 48)(9 154 94 121 33)(10 155 95 122 34)(11 156 96 123 35)(12 157 81 124 36)(13 158 82 125 37)(14 159 83 126 38)(15 160 84 127 39)(16 145 85 128 40)(17 68 106 129 62)(18 69 107 130 63)(19 70 108 131 64)(20 71 109 132 49)(21 72 110 133 50)(22 73 111 134 51)(23 74 112 135 52)(24 75 97 136 53)(25 76 98 137 54)(26 77 99 138 55)(27 78 100 139 56)(28 79 101 140 57)(29 80 102 141 58)(30 65 103 142 59)(31 66 104 143 60)(32 67 105 144 61)
(1 104 9 112)(2 105 10 97)(3 106 11 98)(4 107 12 99)(5 108 13 100)(6 109 14 101)(7 110 15 102)(8 111 16 103)(17 123 25 115)(18 124 26 116)(19 125 27 117)(20 126 28 118)(21 127 29 119)(22 128 30 120)(23 113 31 121)(24 114 32 122)(33 74 41 66)(34 75 42 67)(35 76 43 68)(36 77 44 69)(37 78 45 70)(38 79 46 71)(39 80 47 72)(40 65 48 73)(49 83 57 91)(50 84 58 92)(51 85 59 93)(52 86 60 94)(53 87 61 95)(54 88 62 96)(55 89 63 81)(56 90 64 82)(129 156 137 148)(130 157 138 149)(131 158 139 150)(132 159 140 151)(133 160 141 152)(134 145 142 153)(135 146 143 154)(136 147 144 155)
(1 112)(2 10)(3 106)(5 100)(6 14)(7 110)(9 104)(11 98)(13 108)(15 102)(17 115)(18 26)(19 125)(21 119)(22 30)(23 113)(25 123)(27 117)(29 127)(31 121)(33 66)(34 42)(35 76)(37 70)(38 46)(39 80)(41 74)(43 68)(45 78)(47 72)(50 92)(51 59)(52 86)(54 96)(55 63)(56 90)(58 84)(60 94)(62 88)(64 82)(65 73)(69 77)(83 91)(87 95)(99 107)(103 111)(114 122)(118 126)(129 148)(130 138)(131 158)(133 152)(134 142)(135 146)(137 156)(139 150)(141 160)(143 154)(147 155)(151 159)
(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)| (1,146,86,113,41)(2,147,87,114,42)(3,148,88,115,43)(4,149,89,116,44)(5,150,90,117,45)(6,151,91,118,46)(7,152,92,119,47)(8,153,93,120,48)(9,154,94,121,33)(10,155,95,122,34)(11,156,96,123,35)(12,157,81,124,36)(13,158,82,125,37)(14,159,83,126,38)(15,160,84,127,39)(16,145,85,128,40)(17,68,106,129,62)(18,69,107,130,63)(19,70,108,131,64)(20,71,109,132,49)(21,72,110,133,50)(22,73,111,134,51)(23,74,112,135,52)(24,75,97,136,53)(25,76,98,137,54)(26,77,99,138,55)(27,78,100,139,56)(28,79,101,140,57)(29,80,102,141,58)(30,65,103,142,59)(31,66,104,143,60)(32,67,105,144,61), (1,104,9,112)(2,105,10,97)(3,106,11,98)(4,107,12,99)(5,108,13,100)(6,109,14,101)(7,110,15,102)(8,111,16,103)(17,123,25,115)(18,124,26,116)(19,125,27,117)(20,126,28,118)(21,127,29,119)(22,128,30,120)(23,113,31,121)(24,114,32,122)(33,74,41,66)(34,75,42,67)(35,76,43,68)(36,77,44,69)(37,78,45,70)(38,79,46,71)(39,80,47,72)(40,65,48,73)(49,83,57,91)(50,84,58,92)(51,85,59,93)(52,86,60,94)(53,87,61,95)(54,88,62,96)(55,89,63,81)(56,90,64,82)(129,156,137,148)(130,157,138,149)(131,158,139,150)(132,159,140,151)(133,160,141,152)(134,145,142,153)(135,146,143,154)(136,147,144,155), (1,112)(2,10)(3,106)(5,100)(6,14)(7,110)(9,104)(11,98)(13,108)(15,102)(17,115)(18,26)(19,125)(21,119)(22,30)(23,113)(25,123)(27,117)(29,127)(31,121)(33,66)(34,42)(35,76)(37,70)(38,46)(39,80)(41,74)(43,68)(45,78)(47,72)(50,92)(51,59)(52,86)(54,96)(55,63)(56,90)(58,84)(60,94)(62,88)(64,82)(65,73)(69,77)(83,91)(87,95)(99,107)(103,111)(114,122)(118,126)(129,148)(130,138)(131,158)(133,152)(134,142)(135,146)(137,156)(139,150)(141,160)(143,154)(147,155)(151,159), (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( (1,146,86,113,41)(2,147,87,114,42)(3,148,88,115,43)(4,149,89,116,44)(5,150,90,117,45)(6,151,91,118,46)(7,152,92,119,47)(8,153,93,120,48)(9,154,94,121,33)(10,155,95,122,34)(11,156,96,123,35)(12,157,81,124,36)(13,158,82,125,37)(14,159,83,126,38)(15,160,84,127,39)(16,145,85,128,40)(17,68,106,129,62)(18,69,107,130,63)(19,70,108,131,64)(20,71,109,132,49)(21,72,110,133,50)(22,73,111,134,51)(23,74,112,135,52)(24,75,97,136,53)(25,76,98,137,54)(26,77,99,138,55)(27,78,100,139,56)(28,79,101,140,57)(29,80,102,141,58)(30,65,103,142,59)(31,66,104,143,60)(32,67,105,144,61), (1,104,9,112)(2,105,10,97)(3,106,11,98)(4,107,12,99)(5,108,13,100)(6,109,14,101)(7,110,15,102)(8,111,16,103)(17,123,25,115)(18,124,26,116)(19,125,27,117)(20,126,28,118)(21,127,29,119)(22,128,30,120)(23,113,31,121)(24,114,32,122)(33,74,41,66)(34,75,42,67)(35,76,43,68)(36,77,44,69)(37,78,45,70)(38,79,46,71)(39,80,47,72)(40,65,48,73)(49,83,57,91)(50,84,58,92)(51,85,59,93)(52,86,60,94)(53,87,61,95)(54,88,62,96)(55,89,63,81)(56,90,64,82)(129,156,137,148)(130,157,138,149)(131,158,139,150)(132,159,140,151)(133,160,141,152)(134,145,142,153)(135,146,143,154)(136,147,144,155), (1,112)(2,10)(3,106)(5,100)(6,14)(7,110)(9,104)(11,98)(13,108)(15,102)(17,115)(18,26)(19,125)(21,119)(22,30)(23,113)(25,123)(27,117)(29,127)(31,121)(33,66)(34,42)(35,76)(37,70)(38,46)(39,80)(41,74)(43,68)(45,78)(47,72)(50,92)(51,59)(52,86)(54,96)(55,63)(56,90)(58,84)(60,94)(62,88)(64,82)(65,73)(69,77)(83,91)(87,95)(99,107)(103,111)(114,122)(118,126)(129,148)(130,138)(131,158)(133,152)(134,142)(135,146)(137,156)(139,150)(141,160)(143,154)(147,155)(151,159), (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([(1,146,86,113,41),(2,147,87,114,42),(3,148,88,115,43),(4,149,89,116,44),(5,150,90,117,45),(6,151,91,118,46),(7,152,92,119,47),(8,153,93,120,48),(9,154,94,121,33),(10,155,95,122,34),(11,156,96,123,35),(12,157,81,124,36),(13,158,82,125,37),(14,159,83,126,38),(15,160,84,127,39),(16,145,85,128,40),(17,68,106,129,62),(18,69,107,130,63),(19,70,108,131,64),(20,71,109,132,49),(21,72,110,133,50),(22,73,111,134,51),(23,74,112,135,52),(24,75,97,136,53),(25,76,98,137,54),(26,77,99,138,55),(27,78,100,139,56),(28,79,101,140,57),(29,80,102,141,58),(30,65,103,142,59),(31,66,104,143,60),(32,67,105,144,61)], [(1,104,9,112),(2,105,10,97),(3,106,11,98),(4,107,12,99),(5,108,13,100),(6,109,14,101),(7,110,15,102),(8,111,16,103),(17,123,25,115),(18,124,26,116),(19,125,27,117),(20,126,28,118),(21,127,29,119),(22,128,30,120),(23,113,31,121),(24,114,32,122),(33,74,41,66),(34,75,42,67),(35,76,43,68),(36,77,44,69),(37,78,45,70),(38,79,46,71),(39,80,47,72),(40,65,48,73),(49,83,57,91),(50,84,58,92),(51,85,59,93),(52,86,60,94),(53,87,61,95),(54,88,62,96),(55,89,63,81),(56,90,64,82),(129,156,137,148),(130,157,138,149),(131,158,139,150),(132,159,140,151),(133,160,141,152),(134,145,142,153),(135,146,143,154),(136,147,144,155)], [(1,112),(2,10),(3,106),(5,100),(6,14),(7,110),(9,104),(11,98),(13,108),(15,102),(17,115),(18,26),(19,125),(21,119),(22,30),(23,113),(25,123),(27,117),(29,127),(31,121),(33,66),(34,42),(35,76),(37,70),(38,46),(39,80),(41,74),(43,68),(45,78),(47,72),(50,92),(51,59),(52,86),(54,96),(55,63),(56,90),(58,84),(60,94),(62,88),(64,82),(65,73),(69,77),(83,91),(87,95),(99,107),(103,111),(114,122),(118,126),(129,148),(130,138),(131,158),(133,152),(134,142),(135,146),(137,156),(139,150),(141,160),(143,154),(147,155),(151,159)], [(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)])

140 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 16A ··· 16H 16I 16J 16K 16L 20A ··· 20H 20I 20J 20K 20L 20M 20N 20O 20P 40A ··· 40P 40Q ··· 40X 40Y ··· 40AF 80A ··· 80AF 80AG ··· 80AV order 1 2 2 2 4 4 4 4 5 5 5 5 8 8 8 8 8 8 8 8 10 10 10 10 10 10 10 10 10 10 10 10 16 ··· 16 16 16 16 16 20 ··· 20 20 20 20 20 20 20 20 20 40 ··· 40 40 ··· 40 40 ··· 40 80 ··· 80 80 ··· 80 size 1 1 2 4 1 1 2 4 1 1 1 1 1 1 1 1 2 2 4 4 1 1 1 1 2 2 2 2 4 4 4 4 2 ··· 2 4 4 4 4 1 ··· 1 2 2 2 2 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C2 C2 C4 C4 C5 C8 C8 C10 C10 C10 C20 C20 C40 C40 D4 M4(2) C5×D4 D4.C8 C5×M4(2) C5×D4.C8 kernel C5×D4.C8 C2×C80 C5×M5(2) C5×C8○D4 C5×M4(2) C5×C4○D4 D4.C8 C5×D4 C5×Q8 C2×C16 M5(2) C8○D4 M4(2) C4○D4 D4 Q8 C40 C2×C10 C8 C5 C22 C1 # reps 1 1 1 1 2 2 4 4 4 4 4 4 8 8 16 16 2 2 8 8 8 32

Matrix representation of C5×D4.C8 in GL2(𝔽241) generated by

 205 0 0 205
,
 8 181 222 233
,
 8 181 230 233
,
 188 32 219 68
G:=sub<GL(2,GF(241))| [205,0,0,205],[8,222,181,233],[8,230,181,233],[188,219,32,68] >;

C5×D4.C8 in GAP, Magma, Sage, TeX

C_5\times D_4.C_8
% in TeX

G:=Group("C5xD4.C8");
// GroupNames label

G:=SmallGroup(320,155);
// by ID

G=gap.SmallGroup(320,155);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2530,248,3511,102,124]);
// 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,d*c*d^-1=b*c>;
// generators/relations

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