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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

Aliases: C5×D4.C8, D4.C40, Q8.C40, C40.107D4, M5(2)⋊4C10, M4(2).3C20, (C2×C80)⋊6C2, (C2×C16)⋊2C10, C4.3(C2×C40), (C5×D4).5C8, C8.27(C5×D4), (C5×Q8).5C8, C20.66(C2×C8), C8○D4.2C10, C4○D4.1C20, (C5×M5(2))⋊12C2, C10.42(C22⋊C8), (C2×C40).442C22, (C5×M4(2)).11C4, (C2×C10).26M4(2), C20.160(C22⋊C4), C22.1(C5×M4(2)), (C5×C4○D4).9C4, (C5×C8○D4).5C2, C2.8(C5×C22⋊C8), (C2×C4).42(C2×C20), (C2×C8).96(C2×C10), C4.30(C5×C22⋊C4), (C2×C20).436(C2×C4), SmallGroup(320,155)

Series: Derived Chief Lower central Upper central

C1C4 — C5×D4.C8
C1C2C4C8C2×C8C2×C40C2×C80 — C5×D4.C8
C1C2C4 — C5×D4.C8
C1C40C2×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 >

2C2
4C2
2C4
2C22
2C10
4C10
2C2×C4
2D4
2C8
2C2×C10
2C20
2C2×C8
2C16
2M4(2)
2C16
2C5×D4
2C2×C20
2C40
2C5×M4(2)
2C80
2C80
2C2×C40

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 2A2B2C4A4B4C4D5A5B5C5D8A8B8C8D8E8F8G8H10A10B10C10D10E10F10G10H10I10J10K10L16A···16H16I16J16K16L20A···20H20I20J20K20L20M20N20O20P40A···40P40Q···40X40Y···40AF80A···80AF80AG···80AV
order1222444455558888888810101010101010101010101016···161616161620···20202020202020202040···4040···4040···4080···8080···80
size112411241111111122441111222244442···244441···1222244441···12···24···42···24···4

140 irreducible representations

dim1111111111111111222222
type+++++
imageC1C2C2C2C4C4C5C8C8C10C10C10C20C20C40C40D4M4(2)C5×D4D4.C8C5×M4(2)C5×D4.C8
kernelC5×D4.C8C2×C80C5×M5(2)C5×C8○D4C5×M4(2)C5×C4○D4D4.C8C5×D4C5×Q8C2×C16M5(2)C8○D4M4(2)C4○D4D4Q8C40C2×C10C8C5C22C1
# reps1111224444448816162288832

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

2050
0205
,
8181
222233
,
8181
230233
,
18832
21968
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

Export

Subgroup lattice of C5×D4.C8 in TeX

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