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G = C2×C20.53D4order 320 = 26·5

Direct product of C2 and C20.53D4

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

Aliases: C2×C20.53D4, M4(2).29D10, C23.12Dic10, C20.80(C4⋊C4), (C2×C20).27Q8, (C2×C20).168D4, C20.440(C2×D4), C104(C8.C4), (C2×C4).36Dic10, (C22×C10).15Q8, C20.126(C22×C4), (C2×C20).414C23, (C22×C4).348D10, (C2×M4(2)).15D5, C22.3(C2×Dic10), C4.20(C10.D4), (C10×M4(2)).26C2, C4.Dic5.40C22, (C22×C20).182C22, (C5×M4(2)).32C22, C22.26(C10.D4), C4.89(C2×C4×D5), C56(C2×C8.C4), C10.74(C2×C4⋊C4), (C2×C52C8).12C4, C52C8.42(C2×C4), (C2×C4).158(C4×D5), C4.130(C2×C5⋊D4), (C2×C10).10(C2×Q8), (C2×C10).81(C4⋊C4), (C2×C20).275(C2×C4), (C2×C4).277(C5⋊D4), (C22×C52C8).12C2, C2.18(C2×C10.D4), (C2×C4).510(C22×D5), (C2×C4.Dic5).23C2, (C2×C52C8).274C22, SmallGroup(320,750)

Series: Derived Chief Lower central Upper central

C1C20 — C2×C20.53D4
C1C5C10C20C2×C20C2×C52C8C22×C52C8 — C2×C20.53D4
C5C10C20 — C2×C20.53D4
C1C2×C4C22×C4C2×M4(2)

Generators and relations for C2×C20.53D4
 G = < a,b,c,d | a2=b20=1, c4=b10, d2=b5, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b9, dcd-1=b10c3 >

Subgroups: 238 in 106 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C22 [×3], C22 [×2], C5, C8 [×8], C2×C4 [×6], C23, C10, C10 [×2], C10 [×2], C2×C8 [×8], M4(2) [×2], M4(2) [×4], C22×C4, C20 [×4], C2×C10 [×3], C2×C10 [×2], C8.C4 [×4], C22×C8, C2×M4(2), C2×M4(2), C52C8 [×4], C52C8 [×2], C40 [×2], C2×C20 [×6], C22×C10, C2×C8.C4, C2×C52C8 [×2], C2×C52C8 [×4], C2×C52C8, C4.Dic5 [×2], C4.Dic5, C2×C40, C5×M4(2) [×2], C5×M4(2), C22×C20, C20.53D4 [×4], C22×C52C8, C2×C4.Dic5, C10×M4(2), C2×C20.53D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C8.C4 [×2], C2×C4⋊C4, Dic10 [×2], C4×D5 [×2], C5⋊D4 [×2], C22×D5, C2×C8.C4, C10.D4 [×4], C2×Dic10, C2×C4×D5, C2×C5⋊D4, C20.53D4 [×2], C2×C10.D4, C2×C20.53D4

Smallest permutation representation of C2×C20.53D4
On 160 points
Generators in S160
(1 135)(2 136)(3 137)(4 138)(5 139)(6 140)(7 121)(8 122)(9 123)(10 124)(11 125)(12 126)(13 127)(14 128)(15 129)(16 130)(17 131)(18 132)(19 133)(20 134)(21 68)(22 69)(23 70)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(33 80)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(57 97)(58 98)(59 99)(60 100)(101 151)(102 152)(103 153)(104 154)(105 155)(106 156)(107 157)(108 158)(109 159)(110 160)(111 141)(112 142)(113 143)(114 144)(115 145)(116 146)(117 147)(118 148)(119 149)(120 150)
(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 79 130 27 11 69 140 37)(2 68 131 36 12 78 121 26)(3 77 132 25 13 67 122 35)(4 66 133 34 14 76 123 24)(5 75 134 23 15 65 124 33)(6 64 135 32 16 74 125 22)(7 73 136 21 17 63 126 31)(8 62 137 30 18 72 127 40)(9 71 138 39 19 61 128 29)(10 80 139 28 20 70 129 38)(41 143 86 118 51 153 96 108)(42 152 87 107 52 142 97 117)(43 141 88 116 53 151 98 106)(44 150 89 105 54 160 99 115)(45 159 90 114 55 149 100 104)(46 148 91 103 56 158 81 113)(47 157 92 112 57 147 82 102)(48 146 93 101 58 156 83 111)(49 155 94 110 59 145 84 120)(50 144 95 119 60 154 85 109)
(1 104 6 109 11 114 16 119)(2 113 7 118 12 103 17 108)(3 102 8 107 13 112 18 117)(4 111 9 116 14 101 19 106)(5 120 10 105 15 110 20 115)(21 41 26 46 31 51 36 56)(22 50 27 55 32 60 37 45)(23 59 28 44 33 49 38 54)(24 48 29 53 34 58 39 43)(25 57 30 42 35 47 40 52)(61 98 66 83 71 88 76 93)(62 87 67 92 72 97 77 82)(63 96 68 81 73 86 78 91)(64 85 69 90 74 95 79 100)(65 94 70 99 75 84 80 89)(121 148 126 153 131 158 136 143)(122 157 127 142 132 147 137 152)(123 146 128 151 133 156 138 141)(124 155 129 160 134 145 139 150)(125 144 130 149 135 154 140 159)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F5A5B8A8B8C8D8E···8L8M8N8O8P10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order1222224444445588888···8888810···101010101020···202020202040···40
size11112211112222444410···10202020202···244442···244444···4

68 irreducible representations

dim111111222222222224
type++++++--+++--
imageC1C2C2C2C2C4D4Q8Q8D5D10D10C8.C4Dic10C4×D5C5⋊D4Dic10C20.53D4
kernelC2×C20.53D4C20.53D4C22×C52C8C2×C4.Dic5C10×M4(2)C2×C52C8C2×C20C2×C20C22×C10C2×M4(2)M4(2)C22×C4C10C2×C4C2×C4C2×C4C23C2
# reps141118211242848848

Matrix representation of C2×C20.53D4 in GL4(𝔽41) generated by

40000
04000
00400
00040
,
353600
404000
0090
0009
,
271700
271400
0030
00014
,
181000
212300
00027
00140
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[35,40,0,0,36,40,0,0,0,0,9,0,0,0,0,9],[27,27,0,0,17,14,0,0,0,0,3,0,0,0,0,14],[18,21,0,0,10,23,0,0,0,0,0,14,0,0,27,0] >;

C2×C20.53D4 in GAP, Magma, Sage, TeX

C_2\times C_{20}._{53}D_4
% in TeX

G:=Group("C2xC20.53D4");
// GroupNames label

G:=SmallGroup(320,750);
// by ID

G=gap.SmallGroup(320,750);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,422,58,136,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=1,c^4=b^10,d^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^9,d*c*d^-1=b^10*c^3>;
// generators/relations

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