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G = D5×C4×C8order 320 = 26·5

Direct product of C4×C8 and D5

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D5×C4×C8, D10.16C42, C42.281D10, Dic5.16C42, (C4×C40)⋊18C2, C4034(C2×C4), C2011(C2×C8), C2.1(D5×C42), (C8×Dic5)⋊33C2, Dic510(C2×C8), D10.19(C2×C8), (C2×C8).338D10, C10.26(C22×C8), C10.25(C2×C42), (C4×Dic5).50C4, (D5×C42).33C2, (C2×C40).404C22, C20.181(C22×C4), (C4×C20).337C22, (C2×C20).802C23, (C4×Dic5).356C22, C54(C2×C4×C8), C2.1(D5×C2×C8), C4.96(C2×C4×D5), (D5×C2×C8).35C2, (C2×C4×D5).48C4, (C4×C52C8)⋊27C2, C52C838(C2×C4), C22.34(C2×C4×D5), (C4×D5).98(C2×C4), (C2×C4).172(C4×D5), (C2×C20).418(C2×C4), (C2×C4×D5).417C22, (C2×C4).744(C22×D5), (C2×C10).158(C22×C4), (C2×C52C8).351C22, (C2×Dic5).202(C2×C4), (C22×D5).137(C2×C4), SmallGroup(320,311)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C4×C8
C1C5C10C20C2×C20C2×C4×D5D5×C42 — D5×C4×C8
C5 — D5×C4×C8
C1C4×C8

Generators and relations for D5×C4×C8
 G = < a,b,c,d | a4=b8=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 398 in 162 conjugacy classes, 103 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×6], C4 [×6], C22, C22 [×6], C5, C8 [×4], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×15], C23, D5 [×4], C10, C10 [×2], C42, C42 [×3], C2×C8 [×2], C2×C8 [×10], C22×C4 [×3], Dic5 [×6], C20 [×6], D10 [×6], C2×C10, C4×C8, C4×C8 [×3], C2×C42, C22×C8 [×2], C52C8 [×4], C40 [×4], C4×D5 [×12], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C2×C4×C8, C8×D5 [×8], C2×C52C8 [×2], C4×Dic5, C4×Dic5 [×2], C4×C20, C2×C40 [×2], C2×C4×D5, C2×C4×D5 [×2], C4×C52C8, C8×Dic5 [×2], C4×C40, D5×C42, D5×C2×C8 [×2], D5×C4×C8
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], C23, D5, C42 [×4], C2×C8 [×12], C22×C4 [×3], D10 [×3], C4×C8 [×4], C2×C42, C22×C8 [×2], C4×D5 [×6], C22×D5, C2×C4×C8, C8×D5 [×4], C2×C4×D5 [×3], D5×C42, D5×C2×C8 [×2], D5×C4×C8

Smallest permutation representation of D5×C4×C8
On 160 points
Generators in S160
(1 94 48 53)(2 95 41 54)(3 96 42 55)(4 89 43 56)(5 90 44 49)(6 91 45 50)(7 92 46 51)(8 93 47 52)(9 26 152 109)(10 27 145 110)(11 28 146 111)(12 29 147 112)(13 30 148 105)(14 31 149 106)(15 32 150 107)(16 25 151 108)(17 128 130 34)(18 121 131 35)(19 122 132 36)(20 123 133 37)(21 124 134 38)(22 125 135 39)(23 126 136 40)(24 127 129 33)(57 74 83 157)(58 75 84 158)(59 76 85 159)(60 77 86 160)(61 78 87 153)(62 79 88 154)(63 80 81 155)(64 73 82 156)(65 104 141 116)(66 97 142 117)(67 98 143 118)(68 99 144 119)(69 100 137 120)(70 101 138 113)(71 102 139 114)(72 103 140 115)
(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 84 117 35 147)(2 85 118 36 148)(3 86 119 37 149)(4 87 120 38 150)(5 88 113 39 151)(6 81 114 40 152)(7 82 115 33 145)(8 83 116 34 146)(9 45 63 102 126)(10 46 64 103 127)(11 47 57 104 128)(12 48 58 97 121)(13 41 59 98 122)(14 42 60 99 123)(15 43 61 100 124)(16 44 62 101 125)(17 111 93 157 65)(18 112 94 158 66)(19 105 95 159 67)(20 106 96 160 68)(21 107 89 153 69)(22 108 90 154 70)(23 109 91 155 71)(24 110 92 156 72)(25 49 79 138 135)(26 50 80 139 136)(27 51 73 140 129)(28 52 74 141 130)(29 53 75 142 131)(30 54 76 143 132)(31 55 77 144 133)(32 56 78 137 134)
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 78)(18 79)(19 80)(20 73)(21 74)(22 75)(23 76)(24 77)(25 94)(26 95)(27 96)(28 89)(29 90)(30 91)(31 92)(32 93)(33 60)(34 61)(35 62)(36 63)(37 64)(38 57)(39 58)(40 59)(41 152)(42 145)(43 146)(44 147)(45 148)(46 149)(47 150)(48 151)(49 112)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)(81 122)(82 123)(83 124)(84 125)(85 126)(86 127)(87 128)(88 121)(97 113)(98 114)(99 115)(100 116)(101 117)(102 118)(103 119)(104 120)(129 160)(130 153)(131 154)(132 155)(133 156)(134 157)(135 158)(136 159)

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

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

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

128 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4L4M···4X5A5B8A···8P8Q···8AF10A···10F20A···20X40A···40AF
order122222224···44···4558···88···810···1020···2040···40
size111155551···15···5221···15···52···22···22···2

128 irreducible representations

dim1111111111222222
type+++++++++
imageC1C2C2C2C2C2C4C4C4C8D5D10D10C4×D5C4×D5C8×D5
kernelD5×C4×C8C4×C52C8C8×Dic5C4×C40D5×C42D5×C2×C8C8×D5C4×Dic5C2×C4×D5C4×D5C4×C8C42C2×C8C8C2×C4C4
# reps11211216443222416832

Matrix representation of D5×C4×C8 in GL4(𝔽41) generated by

40000
0900
00400
00040
,
14000
01400
0090
0009
,
1000
0100
0001
004034
,
1000
04000
0001
0010
G:=sub<GL(4,GF(41))| [40,0,0,0,0,9,0,0,0,0,40,0,0,0,0,40],[14,0,0,0,0,14,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,0,40,0,0,1,34],[1,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0] >;

D5×C4×C8 in GAP, Magma, Sage, TeX

D_5\times C_4\times C_8
% in TeX

G:=Group("D5xC4xC8");
// GroupNames label

G:=SmallGroup(320,311);
// by ID

G=gap.SmallGroup(320,311);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,58,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^8=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