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G = C4×D41order 328 = 23·41

Direct product of C4 and D41

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×D41, C1642C2, C2.1D82, D82.2C2, Dic412C2, C82.2C22, C412(C2×C4), SmallGroup(328,5)

Series: Derived Chief Lower central Upper central

C1C41 — C4×D41
C1C41C82D82 — C4×D41
C41 — C4×D41
C1C4

Generators and relations for C4×D41
 G = < a,b,c | a4=b41=c2=1, ab=ba, ac=ca, cbc=b-1 >

41C2
41C2
41C22
41C4
41C2×C4

Smallest permutation representation of C4×D41
On 164 points
Generators in S164
(1 132 61 122)(2 133 62 123)(3 134 63 83)(4 135 64 84)(5 136 65 85)(6 137 66 86)(7 138 67 87)(8 139 68 88)(9 140 69 89)(10 141 70 90)(11 142 71 91)(12 143 72 92)(13 144 73 93)(14 145 74 94)(15 146 75 95)(16 147 76 96)(17 148 77 97)(18 149 78 98)(19 150 79 99)(20 151 80 100)(21 152 81 101)(22 153 82 102)(23 154 42 103)(24 155 43 104)(25 156 44 105)(26 157 45 106)(27 158 46 107)(28 159 47 108)(29 160 48 109)(30 161 49 110)(31 162 50 111)(32 163 51 112)(33 164 52 113)(34 124 53 114)(35 125 54 115)(36 126 55 116)(37 127 56 117)(38 128 57 118)(39 129 58 119)(40 130 59 120)(41 131 60 121)
(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)(124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164)
(1 41)(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)(42 79)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(49 72)(50 71)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(80 82)(83 119)(84 118)(85 117)(86 116)(87 115)(88 114)(89 113)(90 112)(91 111)(92 110)(93 109)(94 108)(95 107)(96 106)(97 105)(98 104)(99 103)(100 102)(120 123)(121 122)(124 139)(125 138)(126 137)(127 136)(128 135)(129 134)(130 133)(131 132)(140 164)(141 163)(142 162)(143 161)(144 160)(145 159)(146 158)(147 157)(148 156)(149 155)(150 154)(151 153)

G:=sub<Sym(164)| (1,132,61,122)(2,133,62,123)(3,134,63,83)(4,135,64,84)(5,136,65,85)(6,137,66,86)(7,138,67,87)(8,139,68,88)(9,140,69,89)(10,141,70,90)(11,142,71,91)(12,143,72,92)(13,144,73,93)(14,145,74,94)(15,146,75,95)(16,147,76,96)(17,148,77,97)(18,149,78,98)(19,150,79,99)(20,151,80,100)(21,152,81,101)(22,153,82,102)(23,154,42,103)(24,155,43,104)(25,156,44,105)(26,157,45,106)(27,158,46,107)(28,159,47,108)(29,160,48,109)(30,161,49,110)(31,162,50,111)(32,163,51,112)(33,164,52,113)(34,124,53,114)(35,125,54,115)(36,126,55,116)(37,127,56,117)(38,128,57,118)(39,129,58,119)(40,130,59,120)(41,131,60,121), (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)(124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164), (1,41)(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(80,82)(83,119)(84,118)(85,117)(86,116)(87,115)(88,114)(89,113)(90,112)(91,111)(92,110)(93,109)(94,108)(95,107)(96,106)(97,105)(98,104)(99,103)(100,102)(120,123)(121,122)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(131,132)(140,164)(141,163)(142,162)(143,161)(144,160)(145,159)(146,158)(147,157)(148,156)(149,155)(150,154)(151,153)>;

G:=Group( (1,132,61,122)(2,133,62,123)(3,134,63,83)(4,135,64,84)(5,136,65,85)(6,137,66,86)(7,138,67,87)(8,139,68,88)(9,140,69,89)(10,141,70,90)(11,142,71,91)(12,143,72,92)(13,144,73,93)(14,145,74,94)(15,146,75,95)(16,147,76,96)(17,148,77,97)(18,149,78,98)(19,150,79,99)(20,151,80,100)(21,152,81,101)(22,153,82,102)(23,154,42,103)(24,155,43,104)(25,156,44,105)(26,157,45,106)(27,158,46,107)(28,159,47,108)(29,160,48,109)(30,161,49,110)(31,162,50,111)(32,163,51,112)(33,164,52,113)(34,124,53,114)(35,125,54,115)(36,126,55,116)(37,127,56,117)(38,128,57,118)(39,129,58,119)(40,130,59,120)(41,131,60,121), (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)(124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164), (1,41)(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(80,82)(83,119)(84,118)(85,117)(86,116)(87,115)(88,114)(89,113)(90,112)(91,111)(92,110)(93,109)(94,108)(95,107)(96,106)(97,105)(98,104)(99,103)(100,102)(120,123)(121,122)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(131,132)(140,164)(141,163)(142,162)(143,161)(144,160)(145,159)(146,158)(147,157)(148,156)(149,155)(150,154)(151,153) );

G=PermutationGroup([(1,132,61,122),(2,133,62,123),(3,134,63,83),(4,135,64,84),(5,136,65,85),(6,137,66,86),(7,138,67,87),(8,139,68,88),(9,140,69,89),(10,141,70,90),(11,142,71,91),(12,143,72,92),(13,144,73,93),(14,145,74,94),(15,146,75,95),(16,147,76,96),(17,148,77,97),(18,149,78,98),(19,150,79,99),(20,151,80,100),(21,152,81,101),(22,153,82,102),(23,154,42,103),(24,155,43,104),(25,156,44,105),(26,157,45,106),(27,158,46,107),(28,159,47,108),(29,160,48,109),(30,161,49,110),(31,162,50,111),(32,163,51,112),(33,164,52,113),(34,124,53,114),(35,125,54,115),(36,126,55,116),(37,127,56,117),(38,128,57,118),(39,129,58,119),(40,130,59,120),(41,131,60,121)], [(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),(124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164)], [(1,41),(2,40),(3,39),(4,38),(5,37),(6,36),(7,35),(8,34),(9,33),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26),(17,25),(18,24),(19,23),(20,22),(42,79),(43,78),(44,77),(45,76),(46,75),(47,74),(48,73),(49,72),(50,71),(51,70),(52,69),(53,68),(54,67),(55,66),(56,65),(57,64),(58,63),(59,62),(60,61),(80,82),(83,119),(84,118),(85,117),(86,116),(87,115),(88,114),(89,113),(90,112),(91,111),(92,110),(93,109),(94,108),(95,107),(96,106),(97,105),(98,104),(99,103),(100,102),(120,123),(121,122),(124,139),(125,138),(126,137),(127,136),(128,135),(129,134),(130,133),(131,132),(140,164),(141,163),(142,162),(143,161),(144,160),(145,159),(146,158),(147,157),(148,156),(149,155),(150,154),(151,153)])

88 conjugacy classes

class 1 2A2B2C4A4B4C4D41A···41T82A···82T164A···164AN
order1222444441···4182···82164···164
size1141411141412···22···22···2

88 irreducible representations

dim11111222
type++++++
imageC1C2C2C2C4D41D82C4×D41
kernelC4×D41Dic41C164D82D41C4C2C1
# reps11114202040

Matrix representation of C4×D41 in GL3(𝔽821) generated by

29500
010
001
,
100
001
0820532
,
82000
001
010
G:=sub<GL(3,GF(821))| [295,0,0,0,1,0,0,0,1],[1,0,0,0,0,820,0,1,532],[820,0,0,0,0,1,0,1,0] >;

C4×D41 in GAP, Magma, Sage, TeX

C_4\times D_{41}
% in TeX

G:=Group("C4xD41");
// GroupNames label

G:=SmallGroup(328,5);
// by ID

G=gap.SmallGroup(328,5);
# by ID

G:=PCGroup([4,-2,-2,-2,-41,21,5123]);
// Polycyclic

G:=Group<a,b,c|a^4=b^41=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C4×D41 in TeX

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