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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 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×6], C5, C8 [×4], C2×C4 [×3], C2×C4 [×13], C23, D5 [×4], C10 [×3], C42, C42 [×3], C2×C8 [×2], C2×C8 [×6], C22×C4 [×3], Dic5 [×4], Dic5, C20 [×2], C20 [×2], C20, D10 [×6], C2×C10, C4⋊C8, C4⋊C8 [×3], C2×C42, C22×C8 [×2], C52C8 [×2], C40 [×2], C4×D5 [×8], C4×D5 [×2], C2×Dic5 [×3], C2×C20 [×3], C22×D5, C2×C4⋊C8, C8×D5 [×4], C2×C52C8 [×2], C4×Dic5 [×3], C4×C20, C2×C40 [×2], C2×C4×D5 [×3], C203C8, C20.8Q8 [×2], C5×C4⋊C8, D5×C42, D5×C2×C8 [×2], D5×C4⋊C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4, C2×Q8, D10 [×3], C4⋊C8 [×4], C2×C4⋊C4, C22×C8, C2×M4(2), C4×D5 [×2], C22×D5, C2×C4⋊C8, C8×D5 [×2], 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 35)(2 74 98 122 36)(3 75 99 123 37)(4 76 100 124 38)(5 77 101 125 39)(6 78 102 126 40)(7 79 103 127 33)(8 80 104 128 34)(9 152 116 81 54)(10 145 117 82 55)(11 146 118 83 56)(12 147 119 84 49)(13 148 120 85 50)(14 149 113 86 51)(15 150 114 87 52)(16 151 115 88 53)(17 72 141 93 63)(18 65 142 94 64)(19 66 143 95 57)(20 67 144 96 58)(21 68 137 89 59)(22 69 138 90 60)(23 70 139 91 61)(24 71 140 92 62)(25 47 108 135 154)(26 48 109 136 155)(27 41 110 129 156)(28 42 111 130 157)(29 43 112 131 158)(30 44 105 132 159)(31 45 106 133 160)(32 46 107 134 153)
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 85)(10 86)(11 87)(12 88)(13 81)(14 82)(15 83)(16 84)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(25 29)(26 30)(27 31)(28 32)(41 160)(42 153)(43 154)(44 155)(45 156)(46 157)(47 158)(48 159)(49 53)(50 54)(51 55)(52 56)(65 90)(66 91)(67 92)(68 93)(69 94)(70 95)(71 96)(72 89)(73 125)(74 126)(75 127)(76 128)(77 121)(78 122)(79 123)(80 124)(97 101)(98 102)(99 103)(100 104)(105 136)(106 129)(107 130)(108 131)(109 132)(110 133)(111 134)(112 135)(113 145)(114 146)(115 147)(116 148)(117 149)(118 150)(119 151)(120 152)(137 141)(138 142)(139 143)(140 144)
(1 135 119 22)(2 23 120 136)(3 129 113 24)(4 17 114 130)(5 131 115 18)(6 19 116 132)(7 133 117 20)(8 21 118 134)(9 44 126 95)(10 96 127 45)(11 46 128 89)(12 90 121 47)(13 48 122 91)(14 92 123 41)(15 42 124 93)(16 94 125 43)(25 49 138 97)(26 98 139 50)(27 51 140 99)(28 100 141 52)(29 53 142 101)(30 102 143 54)(31 55 144 103)(32 104 137 56)(33 106 145 58)(34 59 146 107)(35 108 147 60)(36 61 148 109)(37 110 149 62)(38 63 150 111)(39 112 151 64)(40 57 152 105)(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)| (1,73,97,121,35)(2,74,98,122,36)(3,75,99,123,37)(4,76,100,124,38)(5,77,101,125,39)(6,78,102,126,40)(7,79,103,127,33)(8,80,104,128,34)(9,152,116,81,54)(10,145,117,82,55)(11,146,118,83,56)(12,147,119,84,49)(13,148,120,85,50)(14,149,113,86,51)(15,150,114,87,52)(16,151,115,88,53)(17,72,141,93,63)(18,65,142,94,64)(19,66,143,95,57)(20,67,144,96,58)(21,68,137,89,59)(22,69,138,90,60)(23,70,139,91,61)(24,71,140,92,62)(25,47,108,135,154)(26,48,109,136,155)(27,41,110,129,156)(28,42,111,130,157)(29,43,112,131,158)(30,44,105,132,159)(31,45,106,133,160)(32,46,107,134,153), (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,85)(10,86)(11,87)(12,88)(13,81)(14,82)(15,83)(16,84)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,57)(24,58)(25,29)(26,30)(27,31)(28,32)(41,160)(42,153)(43,154)(44,155)(45,156)(46,157)(47,158)(48,159)(49,53)(50,54)(51,55)(52,56)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,89)(73,125)(74,126)(75,127)(76,128)(77,121)(78,122)(79,123)(80,124)(97,101)(98,102)(99,103)(100,104)(105,136)(106,129)(107,130)(108,131)(109,132)(110,133)(111,134)(112,135)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152)(137,141)(138,142)(139,143)(140,144), (1,135,119,22)(2,23,120,136)(3,129,113,24)(4,17,114,130)(5,131,115,18)(6,19,116,132)(7,133,117,20)(8,21,118,134)(9,44,126,95)(10,96,127,45)(11,46,128,89)(12,90,121,47)(13,48,122,91)(14,92,123,41)(15,42,124,93)(16,94,125,43)(25,49,138,97)(26,98,139,50)(27,51,140,99)(28,100,141,52)(29,53,142,101)(30,102,143,54)(31,55,144,103)(32,104,137,56)(33,106,145,58)(34,59,146,107)(35,108,147,60)(36,61,148,109)(37,110,149,62)(38,63,150,111)(39,112,151,64)(40,57,152,105)(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( (1,73,97,121,35)(2,74,98,122,36)(3,75,99,123,37)(4,76,100,124,38)(5,77,101,125,39)(6,78,102,126,40)(7,79,103,127,33)(8,80,104,128,34)(9,152,116,81,54)(10,145,117,82,55)(11,146,118,83,56)(12,147,119,84,49)(13,148,120,85,50)(14,149,113,86,51)(15,150,114,87,52)(16,151,115,88,53)(17,72,141,93,63)(18,65,142,94,64)(19,66,143,95,57)(20,67,144,96,58)(21,68,137,89,59)(22,69,138,90,60)(23,70,139,91,61)(24,71,140,92,62)(25,47,108,135,154)(26,48,109,136,155)(27,41,110,129,156)(28,42,111,130,157)(29,43,112,131,158)(30,44,105,132,159)(31,45,106,133,160)(32,46,107,134,153), (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,85)(10,86)(11,87)(12,88)(13,81)(14,82)(15,83)(16,84)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,57)(24,58)(25,29)(26,30)(27,31)(28,32)(41,160)(42,153)(43,154)(44,155)(45,156)(46,157)(47,158)(48,159)(49,53)(50,54)(51,55)(52,56)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,89)(73,125)(74,126)(75,127)(76,128)(77,121)(78,122)(79,123)(80,124)(97,101)(98,102)(99,103)(100,104)(105,136)(106,129)(107,130)(108,131)(109,132)(110,133)(111,134)(112,135)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152)(137,141)(138,142)(139,143)(140,144), (1,135,119,22)(2,23,120,136)(3,129,113,24)(4,17,114,130)(5,131,115,18)(6,19,116,132)(7,133,117,20)(8,21,118,134)(9,44,126,95)(10,96,127,45)(11,46,128,89)(12,90,121,47)(13,48,122,91)(14,92,123,41)(15,42,124,93)(16,94,125,43)(25,49,138,97)(26,98,139,50)(27,51,140,99)(28,100,141,52)(29,53,142,101)(30,102,143,54)(31,55,144,103)(32,104,137,56)(33,106,145,58)(34,59,146,107)(35,108,147,60)(36,61,148,109)(37,110,149,62)(38,63,150,111)(39,112,151,64)(40,57,152,105)(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([(1,73,97,121,35),(2,74,98,122,36),(3,75,99,123,37),(4,76,100,124,38),(5,77,101,125,39),(6,78,102,126,40),(7,79,103,127,33),(8,80,104,128,34),(9,152,116,81,54),(10,145,117,82,55),(11,146,118,83,56),(12,147,119,84,49),(13,148,120,85,50),(14,149,113,86,51),(15,150,114,87,52),(16,151,115,88,53),(17,72,141,93,63),(18,65,142,94,64),(19,66,143,95,57),(20,67,144,96,58),(21,68,137,89,59),(22,69,138,90,60),(23,70,139,91,61),(24,71,140,92,62),(25,47,108,135,154),(26,48,109,136,155),(27,41,110,129,156),(28,42,111,130,157),(29,43,112,131,158),(30,44,105,132,159),(31,45,106,133,160),(32,46,107,134,153)], [(1,39),(2,40),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,85),(10,86),(11,87),(12,88),(13,81),(14,82),(15,83),(16,84),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,57),(24,58),(25,29),(26,30),(27,31),(28,32),(41,160),(42,153),(43,154),(44,155),(45,156),(46,157),(47,158),(48,159),(49,53),(50,54),(51,55),(52,56),(65,90),(66,91),(67,92),(68,93),(69,94),(70,95),(71,96),(72,89),(73,125),(74,126),(75,127),(76,128),(77,121),(78,122),(79,123),(80,124),(97,101),(98,102),(99,103),(100,104),(105,136),(106,129),(107,130),(108,131),(109,132),(110,133),(111,134),(112,135),(113,145),(114,146),(115,147),(116,148),(117,149),(118,150),(119,151),(120,152),(137,141),(138,142),(139,143),(140,144)], [(1,135,119,22),(2,23,120,136),(3,129,113,24),(4,17,114,130),(5,131,115,18),(6,19,116,132),(7,133,117,20),(8,21,118,134),(9,44,126,95),(10,96,127,45),(11,46,128,89),(12,90,121,47),(13,48,122,91),(14,92,123,41),(15,42,124,93),(16,94,125,43),(25,49,138,97),(26,98,139,50),(27,51,140,99),(28,100,141,52),(29,53,142,101),(30,102,143,54),(31,55,144,103),(32,104,137,56),(33,106,145,58),(34,59,146,107),(35,108,147,60),(36,61,148,109),(37,110,149,62),(38,63,150,111),(39,112,151,64),(40,57,152,105),(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

׿
×
𝔽