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G = D5×C8⋊C4order 320 = 26·5

Direct product of D5 and C8⋊C4

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

Aliases: D5×C8⋊C4, D10.17C42, C42.181D10, D10.12M4(2), Dic5.17C42, (C8×D5)⋊9C4, C815(C4×D5), C4026(C2×C4), C408C424C2, C2.9(D5×C42), (C2×C8).269D10, C2.1(D5×M4(2)), C10.28(C2×C42), (C4×Dic5).17C4, (D5×C42).13C2, (C2×C40).225C22, (C4×C20).226C22, (C2×C20).810C23, C20.184(C22×C4), C42.D517C2, C10.49(C2×M4(2)), (C4×Dic5).297C22, C54(C2×C8⋊C4), C4.99(C2×C4×D5), (D5×C2×C8).26C2, (C2×C4×D5).18C4, (C5×C8⋊C4)⋊6C2, C52C831(C2×C4), C22.39(C2×C4×D5), (C4×D5).99(C2×C4), (C2×C4).126(C4×D5), (C2×C20).318(C2×C4), (C2×C4×D5).418C22, (C2×C4).752(C22×D5), (C2×C10).166(C22×C4), (C2×C52C8).302C22, (C2×Dic5).203(C2×C4), (C22×D5).138(C2×C4), SmallGroup(320,331)

Series: Derived Chief Lower central Upper central

C1C10 — D5×C8⋊C4
C1C5C10C20C2×C20C2×C4×D5D5×C42 — D5×C8⋊C4
C5C10 — D5×C8⋊C4
C1C2×C4C8⋊C4

Generators and relations for D5×C8⋊C4
 G = < a,b,c,d | a5=b2=c8=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 398 in 146 conjugacy classes, 83 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×6], C22, C22 [×6], C5, C8 [×4], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×11], C23, D5 [×4], C10, C10 [×2], C42, C42 [×3], C2×C8 [×2], C2×C8 [×10], C22×C4 [×3], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C8⋊C4, C8⋊C4 [×3], C2×C42, C22×C8 [×2], C52C8 [×4], C40 [×4], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C2×C8⋊C4, C8×D5 [×8], C2×C52C8 [×2], C4×Dic5, C4×Dic5 [×2], C4×C20, C2×C40 [×2], C2×C4×D5, C2×C4×D5 [×2], C42.D5, C408C4 [×2], C5×C8⋊C4, D5×C42, D5×C2×C8 [×2], D5×C8⋊C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, D5, C42 [×4], M4(2) [×4], C22×C4 [×3], D10 [×3], C8⋊C4 [×4], C2×C42, C2×M4(2) [×2], C4×D5 [×6], C22×D5, C2×C8⋊C4, C2×C4×D5 [×3], D5×C42, D5×M4(2) [×2], D5×C8⋊C4

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

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P5A5B8A···8H8I···8P10A···10F20A···20H20I···20P40A···40P
order122222224444444444444444558···88···810···1020···2020···2040···40
size1111555511112222555510101010222···210···102···22···24···44···4

80 irreducible representations

dim1111111112222224
type+++++++++
imageC1C2C2C2C2C2C4C4C4D5M4(2)D10D10C4×D5C4×D5D5×M4(2)
kernelD5×C8⋊C4C42.D5C408C4C5×C8⋊C4D5×C42D5×C2×C8C8×D5C4×Dic5C2×C4×D5C8⋊C4D10C42C2×C8C8C2×C4C2
# reps112112164428241688

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

40100
53500
0010
0001
,
1000
364000
0010
0001
,
1000
0100
0097
001932
,
9000
0900
003440
0097
G:=sub<GL(4,GF(41))| [40,5,0,0,1,35,0,0,0,0,1,0,0,0,0,1],[1,36,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,9,19,0,0,7,32],[9,0,0,0,0,9,0,0,0,0,34,9,0,0,40,7] >;

D5×C8⋊C4 in GAP, Magma, Sage, TeX

D_5\times C_8\rtimes C_4
% in TeX

G:=Group("D5xC8:C4");
// GroupNames label

G:=SmallGroup(320,331);
// by ID

G=gap.SmallGroup(320,331);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,387,58,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^8=d^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations

׿
×
𝔽