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G = Dic42order 168 = 23·3·7

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic42, C4.D21, C212Q8, C72Dic6, C84.1C2, C28.1S3, C12.1D7, C6.8D14, C2.3D42, C14.8D6, C32Dic14, C42.8C22, Dic21.1C2, SmallGroup(168,34)

Series: Derived Chief Lower central Upper central

C1C42 — Dic42
C1C7C21C42Dic21 — Dic42
C21C42 — Dic42
C1C2C4

Generators and relations for Dic42
 G = < a,b | a84=1, b2=a42, bab-1=a-1 >

21C4
21C4
21Q8
7Dic3
7Dic3
3Dic7
3Dic7
7Dic6
3Dic14

Smallest permutation representation of Dic42
Regular action on 168 points
Generators in S168
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)
(1 132 43 90)(2 131 44 89)(3 130 45 88)(4 129 46 87)(5 128 47 86)(6 127 48 85)(7 126 49 168)(8 125 50 167)(9 124 51 166)(10 123 52 165)(11 122 53 164)(12 121 54 163)(13 120 55 162)(14 119 56 161)(15 118 57 160)(16 117 58 159)(17 116 59 158)(18 115 60 157)(19 114 61 156)(20 113 62 155)(21 112 63 154)(22 111 64 153)(23 110 65 152)(24 109 66 151)(25 108 67 150)(26 107 68 149)(27 106 69 148)(28 105 70 147)(29 104 71 146)(30 103 72 145)(31 102 73 144)(32 101 74 143)(33 100 75 142)(34 99 76 141)(35 98 77 140)(36 97 78 139)(37 96 79 138)(38 95 80 137)(39 94 81 136)(40 93 82 135)(41 92 83 134)(42 91 84 133)

G:=sub<Sym(168)| (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,132,43,90)(2,131,44,89)(3,130,45,88)(4,129,46,87)(5,128,47,86)(6,127,48,85)(7,126,49,168)(8,125,50,167)(9,124,51,166)(10,123,52,165)(11,122,53,164)(12,121,54,163)(13,120,55,162)(14,119,56,161)(15,118,57,160)(16,117,58,159)(17,116,59,158)(18,115,60,157)(19,114,61,156)(20,113,62,155)(21,112,63,154)(22,111,64,153)(23,110,65,152)(24,109,66,151)(25,108,67,150)(26,107,68,149)(27,106,69,148)(28,105,70,147)(29,104,71,146)(30,103,72,145)(31,102,73,144)(32,101,74,143)(33,100,75,142)(34,99,76,141)(35,98,77,140)(36,97,78,139)(37,96,79,138)(38,95,80,137)(39,94,81,136)(40,93,82,135)(41,92,83,134)(42,91,84,133)>;

G:=Group( (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,132,43,90)(2,131,44,89)(3,130,45,88)(4,129,46,87)(5,128,47,86)(6,127,48,85)(7,126,49,168)(8,125,50,167)(9,124,51,166)(10,123,52,165)(11,122,53,164)(12,121,54,163)(13,120,55,162)(14,119,56,161)(15,118,57,160)(16,117,58,159)(17,116,59,158)(18,115,60,157)(19,114,61,156)(20,113,62,155)(21,112,63,154)(22,111,64,153)(23,110,65,152)(24,109,66,151)(25,108,67,150)(26,107,68,149)(27,106,69,148)(28,105,70,147)(29,104,71,146)(30,103,72,145)(31,102,73,144)(32,101,74,143)(33,100,75,142)(34,99,76,141)(35,98,77,140)(36,97,78,139)(37,96,79,138)(38,95,80,137)(39,94,81,136)(40,93,82,135)(41,92,83,134)(42,91,84,133) );

G=PermutationGroup([(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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)], [(1,132,43,90),(2,131,44,89),(3,130,45,88),(4,129,46,87),(5,128,47,86),(6,127,48,85),(7,126,49,168),(8,125,50,167),(9,124,51,166),(10,123,52,165),(11,122,53,164),(12,121,54,163),(13,120,55,162),(14,119,56,161),(15,118,57,160),(16,117,58,159),(17,116,59,158),(18,115,60,157),(19,114,61,156),(20,113,62,155),(21,112,63,154),(22,111,64,153),(23,110,65,152),(24,109,66,151),(25,108,67,150),(26,107,68,149),(27,106,69,148),(28,105,70,147),(29,104,71,146),(30,103,72,145),(31,102,73,144),(32,101,74,143),(33,100,75,142),(34,99,76,141),(35,98,77,140),(36,97,78,139),(37,96,79,138),(38,95,80,137),(39,94,81,136),(40,93,82,135),(41,92,83,134),(42,91,84,133)])

Dic42 is a maximal subgroup of
C6.D28  D12.D7  C3⋊Dic28  C7⋊Dic12  C8⋊D21  Dic84  D4.D21  C217Q16  D7×Dic6  D285S3  S3×Dic14  D125D7  D8411C2  D42D21  Q8×D21
Dic42 is a maximal quotient of
C42.4Q8  C84⋊C4

45 conjugacy classes

class 1  2  3 4A4B4C 6 7A7B7C12A12B14A14B14C21A···21F28A···28F42A···42F84A···84L
order1234446777121214141421···2128···2842···4284···84
size112242422222222222···22···22···22···2

45 irreducible representations

dim1112222222222
type++++-++-++-+-
imageC1C2C2S3Q8D6D7Dic6D14D21Dic14D42Dic42
kernelDic42Dic21C84C28C21C14C12C7C6C4C3C2C1
# reps12111132366612

Matrix representation of Dic42 in GL2(𝔽337) generated by

144120
133277
,
126323
99211
G:=sub<GL(2,GF(337))| [144,133,120,277],[126,99,323,211] >;

Dic42 in GAP, Magma, Sage, TeX

{\rm Dic}_{42}
% in TeX

G:=Group("Dic42");
// GroupNames label

G:=SmallGroup(168,34);
// by ID

G=gap.SmallGroup(168,34);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-7,20,61,26,323,3604]);
// Polycyclic

G:=Group<a,b|a^84=1,b^2=a^42,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic42 in TeX

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