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G = D5×C4⋊C8order 320 = 26·5

Direct product of D5 and C4⋊C8

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

Aliases: D5×C4⋊C8, C42.199D10, D10.14M4(2), C43(C8×D5), C205(C2×C8), (C4×D5)⋊1C8, C4.54(Q8×D5), Dic54(C2×C8), C203C811C2, C4.204(D4×D5), (C4×D5).34Q8, (C4×D5).123D4, D10.20(C2×C8), C20.363(C2×D4), (C2×C8).214D10, C20.112(C2×Q8), (C4×Dic5).8C4, (D5×C42).1C2, C2.5(D5×M4(2)), D10.35(C4⋊C4), (C4×C20).58C22, C10.33(C22×C8), C20.8Q822C2, Dic5.37(C4⋊C4), (C2×C40).208C22, (C2×C20).829C23, C10.59(C2×M4(2)), (C4×Dic5).306C22, C54(C2×C4⋊C8), (C5×C4⋊C8)⋊13C2, C2.3(D5×C4⋊C4), (C2×C4×D5).9C4, C2.11(D5×C2×C8), (D5×C2×C8).17C2, C10.30(C2×C4⋊C4), C22.46(C2×C4×D5), (C2×C4).144(C4×D5), (C2×C20).241(C2×C4), (C2×C4×D5).420C22, (C2×C4).771(C22×D5), (C2×C10).185(C22×C4), (C2×C52C8).312C22, (C2×Dic5).142(C2×C4), (C22×D5).139(C2×C4), SmallGroup(320,459)

Series: Derived Chief Lower central Upper central

C1C10 — D5×C4⋊C8
C1C5C10C20C2×C20C2×C4×D5D5×C42 — D5×C4⋊C8
C5C10 — D5×C4⋊C8
C1C2×C4C4⋊C8

Generators and relations for D5×C4⋊C8
 G = < a,b,c,d | a5=b2=c4=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 398 in 138 conjugacy classes, 73 normal (37 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C42, C42, C2×C8, C2×C8, C22×C4, Dic5, Dic5, C20, C20, C20, D10, C2×C10, C4⋊C8, C4⋊C8, C2×C42, C22×C8, C52C8, C40, C4×D5, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C4⋊C8, C8×D5, C2×C52C8, C4×Dic5, C4×C20, C2×C40, C2×C4×D5, C203C8, C20.8Q8, C5×C4⋊C8, D5×C42, D5×C2×C8, D5×C4⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, D5, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, D10, C4⋊C8, C2×C4⋊C4, C22×C8, C2×M4(2), C4×D5, C22×D5, C2×C4⋊C8, C8×D5, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, D5×C2×C8, D5×M4(2), D5×C4⋊C8

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

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

dim11111111122222222444
type+++++++-++++-
imageC1C2C2C2C2C2C4C4C8D4Q8D5M4(2)D10D10C4×D5C8×D5D4×D5Q8×D5D5×M4(2)
kernelD5×C4⋊C8C203C8C20.8Q8C5×C4⋊C8D5×C42D5×C2×C8C4×Dic5C2×C4×D5C4×D5C4×D5C4×D5C4⋊C8D10C42C2×C8C2×C4C4C4C4C2
# reps1121124416222424816224

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

6100
40000
0010
0001
,
403500
0100
0010
0001
,
1000
0100
003128
003310
,
27000
02700
00939
00032
G:=sub<GL(4,GF(41))| [6,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,35,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,31,33,0,0,28,10],[27,0,0,0,0,27,0,0,0,0,9,0,0,0,39,32] >;

D5×C4⋊C8 in GAP, Magma, Sage, TeX

D_5\times C_4\rtimes C_8
% in TeX

G:=Group("D5xC4:C8");
// GroupNames label

G:=SmallGroup(320,459);
// by ID

G=gap.SmallGroup(320,459);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,58,136,12550]);
// Polycyclic

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

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×
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