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G = Dic53order 212 = 22·53

Dicyclic group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: Dic53, C532C4, C2.D53, C106.C2, SmallGroup(212,1)

Series: Derived Chief Lower central Upper central

C1C53 — Dic53
C1C53C106 — Dic53
C53 — Dic53
C1C2

Generators and relations for Dic53
 G = < a,b | a106=1, b2=a53, bab-1=a-1 >

53C4

Smallest permutation representation of Dic53
Regular action on 212 points
Generators in S212
(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)(107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212)
(1 153 54 206)(2 152 55 205)(3 151 56 204)(4 150 57 203)(5 149 58 202)(6 148 59 201)(7 147 60 200)(8 146 61 199)(9 145 62 198)(10 144 63 197)(11 143 64 196)(12 142 65 195)(13 141 66 194)(14 140 67 193)(15 139 68 192)(16 138 69 191)(17 137 70 190)(18 136 71 189)(19 135 72 188)(20 134 73 187)(21 133 74 186)(22 132 75 185)(23 131 76 184)(24 130 77 183)(25 129 78 182)(26 128 79 181)(27 127 80 180)(28 126 81 179)(29 125 82 178)(30 124 83 177)(31 123 84 176)(32 122 85 175)(33 121 86 174)(34 120 87 173)(35 119 88 172)(36 118 89 171)(37 117 90 170)(38 116 91 169)(39 115 92 168)(40 114 93 167)(41 113 94 166)(42 112 95 165)(43 111 96 164)(44 110 97 163)(45 109 98 162)(46 108 99 161)(47 107 100 160)(48 212 101 159)(49 211 102 158)(50 210 103 157)(51 209 104 156)(52 208 105 155)(53 207 106 154)

G:=sub<Sym(212)| (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)(107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212), (1,153,54,206)(2,152,55,205)(3,151,56,204)(4,150,57,203)(5,149,58,202)(6,148,59,201)(7,147,60,200)(8,146,61,199)(9,145,62,198)(10,144,63,197)(11,143,64,196)(12,142,65,195)(13,141,66,194)(14,140,67,193)(15,139,68,192)(16,138,69,191)(17,137,70,190)(18,136,71,189)(19,135,72,188)(20,134,73,187)(21,133,74,186)(22,132,75,185)(23,131,76,184)(24,130,77,183)(25,129,78,182)(26,128,79,181)(27,127,80,180)(28,126,81,179)(29,125,82,178)(30,124,83,177)(31,123,84,176)(32,122,85,175)(33,121,86,174)(34,120,87,173)(35,119,88,172)(36,118,89,171)(37,117,90,170)(38,116,91,169)(39,115,92,168)(40,114,93,167)(41,113,94,166)(42,112,95,165)(43,111,96,164)(44,110,97,163)(45,109,98,162)(46,108,99,161)(47,107,100,160)(48,212,101,159)(49,211,102,158)(50,210,103,157)(51,209,104,156)(52,208,105,155)(53,207,106,154)>;

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)(107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212), (1,153,54,206)(2,152,55,205)(3,151,56,204)(4,150,57,203)(5,149,58,202)(6,148,59,201)(7,147,60,200)(8,146,61,199)(9,145,62,198)(10,144,63,197)(11,143,64,196)(12,142,65,195)(13,141,66,194)(14,140,67,193)(15,139,68,192)(16,138,69,191)(17,137,70,190)(18,136,71,189)(19,135,72,188)(20,134,73,187)(21,133,74,186)(22,132,75,185)(23,131,76,184)(24,130,77,183)(25,129,78,182)(26,128,79,181)(27,127,80,180)(28,126,81,179)(29,125,82,178)(30,124,83,177)(31,123,84,176)(32,122,85,175)(33,121,86,174)(34,120,87,173)(35,119,88,172)(36,118,89,171)(37,117,90,170)(38,116,91,169)(39,115,92,168)(40,114,93,167)(41,113,94,166)(42,112,95,165)(43,111,96,164)(44,110,97,163)(45,109,98,162)(46,108,99,161)(47,107,100,160)(48,212,101,159)(49,211,102,158)(50,210,103,157)(51,209,104,156)(52,208,105,155)(53,207,106,154) );

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),(107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212)], [(1,153,54,206),(2,152,55,205),(3,151,56,204),(4,150,57,203),(5,149,58,202),(6,148,59,201),(7,147,60,200),(8,146,61,199),(9,145,62,198),(10,144,63,197),(11,143,64,196),(12,142,65,195),(13,141,66,194),(14,140,67,193),(15,139,68,192),(16,138,69,191),(17,137,70,190),(18,136,71,189),(19,135,72,188),(20,134,73,187),(21,133,74,186),(22,132,75,185),(23,131,76,184),(24,130,77,183),(25,129,78,182),(26,128,79,181),(27,127,80,180),(28,126,81,179),(29,125,82,178),(30,124,83,177),(31,123,84,176),(32,122,85,175),(33,121,86,174),(34,120,87,173),(35,119,88,172),(36,118,89,171),(37,117,90,170),(38,116,91,169),(39,115,92,168),(40,114,93,167),(41,113,94,166),(42,112,95,165),(43,111,96,164),(44,110,97,163),(45,109,98,162),(46,108,99,161),(47,107,100,160),(48,212,101,159),(49,211,102,158),(50,210,103,157),(51,209,104,156),(52,208,105,155),(53,207,106,154)])

Dic53 is a maximal subgroup of   C53⋊C8  Dic106  C4×D53  C53⋊D4
Dic53 is a maximal quotient of   C532C8

56 conjugacy classes

class 1  2 4A4B53A···53Z106A···106Z
order124453···53106···106
size1153532···22···2

56 irreducible representations

dim11122
type+++-
imageC1C2C4D53Dic53
kernelDic53C106C53C2C1
# reps1122626

Matrix representation of Dic53 in GL2(𝔽1061) generated by

3691
10600
,
1023224
45338
G:=sub<GL(2,GF(1061))| [369,1060,1,0],[1023,453,224,38] >;

Dic53 in GAP, Magma, Sage, TeX

{\rm Dic}_{53}
% in TeX

G:=Group("Dic53");
// GroupNames label

G:=SmallGroup(212,1);
// by ID

G=gap.SmallGroup(212,1);
# by ID

G:=PCGroup([3,-2,-2,-53,6,1874]);
// Polycyclic

G:=Group<a,b|a^106=1,b^2=a^53,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic53 in TeX

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