Copied to
clipboard

G = C22.D40order 320 = 26·5

3rd non-split extension by C22 of D40 acting via D40/D20=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22.4D40, C23.39D20, C22⋊C84D5, C405C44C2, (C2×C8).4D10, C2.8(C2×D40), (C2×C10).5D8, C10.6(C2×D8), (C2×C4).34D20, (C2×C20).45D4, D205C46C2, C207D4.3C2, (C2×C40).4C22, (C22×C10).56D4, (C22×C4).86D10, C51(C22.D8), C20.283(C4○D4), (C2×C20).746C23, (C2×D20).13C22, C22.109(C2×D20), C4.107(D42D5), C2.13(C8.D10), C10.10(C8.C22), C4⋊Dic5.271C22, (C22×C20).53C22, C2.14(C22.D20), C10.18(C22.D4), (C5×C22⋊C8)⋊6C2, (C2×C4⋊Dic5)⋊6C2, (C2×C10).129(C2×D4), (C2×C4).691(C22×D5), SmallGroup(320,363)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C22.D40
C1C5C10C20C2×C20C2×D20C207D4 — C22.D40
C5C10C2×C20 — C22.D40
C1C22C22×C4C22⋊C8

Generators and relations for C22.D40
 G = < a,b,c,d | a2=b2=c40=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=bc-1 >

Subgroups: 542 in 114 conjugacy classes, 43 normal (25 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C5, C8 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, D5, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4 [×4], C2×C8 [×2], C22×C4, C22×C4, C2×D4 [×2], Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, D4⋊C4 [×2], C2.D8 [×2], C2×C4⋊C4, C4⋊D4, C40 [×2], D20 [×2], C2×Dic5 [×5], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C22.D8, C4⋊Dic5, C4⋊Dic5 [×2], C4⋊Dic5, D10⋊C4, C2×C40 [×2], C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, C405C4 [×2], D205C4 [×2], C5×C22⋊C8, C2×C4⋊Dic5, C207D4, C22.D40
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C2×D8, C8.C22, D20 [×2], C22×D5, C22.D8, D40 [×2], C2×D20, D42D5 [×2], C22.D20, C2×D40, C8.D10, C22.D40

Smallest permutation representation of C22.D40
On 160 points
Generators in S160
(1 134)(2 79)(3 136)(4 41)(5 138)(6 43)(7 140)(8 45)(9 142)(10 47)(11 144)(12 49)(13 146)(14 51)(15 148)(16 53)(17 150)(18 55)(19 152)(20 57)(21 154)(22 59)(23 156)(24 61)(25 158)(26 63)(27 160)(28 65)(29 122)(30 67)(31 124)(32 69)(33 126)(34 71)(35 128)(36 73)(37 130)(38 75)(39 132)(40 77)(42 82)(44 84)(46 86)(48 88)(50 90)(52 92)(54 94)(56 96)(58 98)(60 100)(62 102)(64 104)(66 106)(68 108)(70 110)(72 112)(74 114)(76 116)(78 118)(80 120)(81 137)(83 139)(85 141)(87 143)(89 145)(91 147)(93 149)(95 151)(97 153)(99 155)(101 157)(103 159)(105 121)(107 123)(109 125)(111 127)(113 129)(115 131)(117 133)(119 135)
(1 118)(2 119)(3 120)(4 81)(5 82)(6 83)(7 84)(8 85)(9 86)(10 87)(11 88)(12 89)(13 90)(14 91)(15 92)(16 93)(17 94)(18 95)(19 96)(20 97)(21 98)(22 99)(23 100)(24 101)(25 102)(26 103)(27 104)(28 105)(29 106)(30 107)(31 108)(32 109)(33 110)(34 111)(35 112)(36 113)(37 114)(38 115)(39 116)(40 117)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(49 145)(50 146)(51 147)(52 148)(53 149)(54 150)(55 151)(56 152)(57 153)(58 154)(59 155)(60 156)(61 157)(62 158)(63 159)(64 160)(65 121)(66 122)(67 123)(68 124)(69 125)(70 126)(71 127)(72 128)(73 129)(74 130)(75 131)(76 132)(77 133)(78 134)(79 135)(80 136)
(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)
(1 117 118 40)(2 39 119 116)(3 115 120 38)(4 37 81 114)(5 113 82 36)(6 35 83 112)(7 111 84 34)(8 33 85 110)(9 109 86 32)(10 31 87 108)(11 107 88 30)(12 29 89 106)(13 105 90 28)(14 27 91 104)(15 103 92 26)(16 25 93 102)(17 101 94 24)(18 23 95 100)(19 99 96 22)(20 21 97 98)(41 130 137 74)(42 73 138 129)(43 128 139 72)(44 71 140 127)(45 126 141 70)(46 69 142 125)(47 124 143 68)(48 67 144 123)(49 122 145 66)(50 65 146 121)(51 160 147 64)(52 63 148 159)(53 158 149 62)(54 61 150 157)(55 156 151 60)(56 59 152 155)(57 154 153 58)(75 136 131 80)(76 79 132 135)(77 134 133 78)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222224444444455888810···101010101020···202020202040···40
size1111224022420202020402244442···244442···244444···4

59 irreducible representations

dim1111112222222222444
type+++++++++++++++---
imageC1C2C2C2C2C2D4D4D5C4○D4D8D10D10D20D20D40C8.C22D42D5C8.D10
kernelC22.D40C405C4D205C4C5×C22⋊C8C2×C4⋊Dic5C207D4C2×C20C22×C10C22⋊C8C20C2×C10C2×C8C22×C4C2×C4C23C22C10C4C2
# reps12211111244424416144

Matrix representation of C22.D40 in GL4(𝔽41) generated by

133100
252800
0010
0001
,
40000
04000
0010
0001
,
32000
34900
00235
0063
,
32000
03200
00396
00202
G:=sub<GL(4,GF(41))| [13,25,0,0,31,28,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[32,34,0,0,0,9,0,0,0,0,2,6,0,0,35,3],[32,0,0,0,0,32,0,0,0,0,39,20,0,0,6,2] >;

C22.D40 in GAP, Magma, Sage, TeX

C_2^2.D_{40}
% in TeX

G:=Group("C2^2.D40");
// GroupNames label

G:=SmallGroup(320,363);
// by ID

G=gap.SmallGroup(320,363);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,219,310,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^40=1,d^2=b,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations

׿
×
𝔽