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G = Dic61order 244 = 22·61

Dicyclic group

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

Aliases: Dic61, C612C4, C2.D61, C122.C2, SmallGroup(244,1)

Series: Derived Chief Lower central Upper central

C1C61 — Dic61
C1C61C122 — Dic61
C61 — Dic61
C1C2

Generators and relations for Dic61
 G = < a,b | a122=1, b2=a61, bab-1=a-1 >

61C4

Smallest permutation representation of Dic61
Regular action on 244 points
Generators in S244
(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)(123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244)
(1 162 62 223)(2 161 63 222)(3 160 64 221)(4 159 65 220)(5 158 66 219)(6 157 67 218)(7 156 68 217)(8 155 69 216)(9 154 70 215)(10 153 71 214)(11 152 72 213)(12 151 73 212)(13 150 74 211)(14 149 75 210)(15 148 76 209)(16 147 77 208)(17 146 78 207)(18 145 79 206)(19 144 80 205)(20 143 81 204)(21 142 82 203)(22 141 83 202)(23 140 84 201)(24 139 85 200)(25 138 86 199)(26 137 87 198)(27 136 88 197)(28 135 89 196)(29 134 90 195)(30 133 91 194)(31 132 92 193)(32 131 93 192)(33 130 94 191)(34 129 95 190)(35 128 96 189)(36 127 97 188)(37 126 98 187)(38 125 99 186)(39 124 100 185)(40 123 101 184)(41 244 102 183)(42 243 103 182)(43 242 104 181)(44 241 105 180)(45 240 106 179)(46 239 107 178)(47 238 108 177)(48 237 109 176)(49 236 110 175)(50 235 111 174)(51 234 112 173)(52 233 113 172)(53 232 114 171)(54 231 115 170)(55 230 116 169)(56 229 117 168)(57 228 118 167)(58 227 119 166)(59 226 120 165)(60 225 121 164)(61 224 122 163)

G:=sub<Sym(244)| (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)(123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244), (1,162,62,223)(2,161,63,222)(3,160,64,221)(4,159,65,220)(5,158,66,219)(6,157,67,218)(7,156,68,217)(8,155,69,216)(9,154,70,215)(10,153,71,214)(11,152,72,213)(12,151,73,212)(13,150,74,211)(14,149,75,210)(15,148,76,209)(16,147,77,208)(17,146,78,207)(18,145,79,206)(19,144,80,205)(20,143,81,204)(21,142,82,203)(22,141,83,202)(23,140,84,201)(24,139,85,200)(25,138,86,199)(26,137,87,198)(27,136,88,197)(28,135,89,196)(29,134,90,195)(30,133,91,194)(31,132,92,193)(32,131,93,192)(33,130,94,191)(34,129,95,190)(35,128,96,189)(36,127,97,188)(37,126,98,187)(38,125,99,186)(39,124,100,185)(40,123,101,184)(41,244,102,183)(42,243,103,182)(43,242,104,181)(44,241,105,180)(45,240,106,179)(46,239,107,178)(47,238,108,177)(48,237,109,176)(49,236,110,175)(50,235,111,174)(51,234,112,173)(52,233,113,172)(53,232,114,171)(54,231,115,170)(55,230,116,169)(56,229,117,168)(57,228,118,167)(58,227,119,166)(59,226,120,165)(60,225,121,164)(61,224,122,163)>;

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)(123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244), (1,162,62,223)(2,161,63,222)(3,160,64,221)(4,159,65,220)(5,158,66,219)(6,157,67,218)(7,156,68,217)(8,155,69,216)(9,154,70,215)(10,153,71,214)(11,152,72,213)(12,151,73,212)(13,150,74,211)(14,149,75,210)(15,148,76,209)(16,147,77,208)(17,146,78,207)(18,145,79,206)(19,144,80,205)(20,143,81,204)(21,142,82,203)(22,141,83,202)(23,140,84,201)(24,139,85,200)(25,138,86,199)(26,137,87,198)(27,136,88,197)(28,135,89,196)(29,134,90,195)(30,133,91,194)(31,132,92,193)(32,131,93,192)(33,130,94,191)(34,129,95,190)(35,128,96,189)(36,127,97,188)(37,126,98,187)(38,125,99,186)(39,124,100,185)(40,123,101,184)(41,244,102,183)(42,243,103,182)(43,242,104,181)(44,241,105,180)(45,240,106,179)(46,239,107,178)(47,238,108,177)(48,237,109,176)(49,236,110,175)(50,235,111,174)(51,234,112,173)(52,233,113,172)(53,232,114,171)(54,231,115,170)(55,230,116,169)(56,229,117,168)(57,228,118,167)(58,227,119,166)(59,226,120,165)(60,225,121,164)(61,224,122,163) );

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),(123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244)], [(1,162,62,223),(2,161,63,222),(3,160,64,221),(4,159,65,220),(5,158,66,219),(6,157,67,218),(7,156,68,217),(8,155,69,216),(9,154,70,215),(10,153,71,214),(11,152,72,213),(12,151,73,212),(13,150,74,211),(14,149,75,210),(15,148,76,209),(16,147,77,208),(17,146,78,207),(18,145,79,206),(19,144,80,205),(20,143,81,204),(21,142,82,203),(22,141,83,202),(23,140,84,201),(24,139,85,200),(25,138,86,199),(26,137,87,198),(27,136,88,197),(28,135,89,196),(29,134,90,195),(30,133,91,194),(31,132,92,193),(32,131,93,192),(33,130,94,191),(34,129,95,190),(35,128,96,189),(36,127,97,188),(37,126,98,187),(38,125,99,186),(39,124,100,185),(40,123,101,184),(41,244,102,183),(42,243,103,182),(43,242,104,181),(44,241,105,180),(45,240,106,179),(46,239,107,178),(47,238,108,177),(48,237,109,176),(49,236,110,175),(50,235,111,174),(51,234,112,173),(52,233,113,172),(53,232,114,171),(54,231,115,170),(55,230,116,169),(56,229,117,168),(57,228,118,167),(58,227,119,166),(59,226,120,165),(60,225,121,164),(61,224,122,163)])

Dic61 is a maximal subgroup of   C61⋊C8  Dic122  C4×D61  C61⋊D4
Dic61 is a maximal quotient of   C612C8

64 conjugacy classes

class 1  2 4A4B61A···61AD122A···122AD
order124461···61122···122
size1161612···22···2

64 irreducible representations

dim11122
type+++-
imageC1C2C4D61Dic61
kernelDic61C122C61C2C1
# reps1123030

Matrix representation of Dic61 in GL3(𝔽733) generated by

73200
0695732
010
,
35300
090162
0407643
G:=sub<GL(3,GF(733))| [732,0,0,0,695,1,0,732,0],[353,0,0,0,90,407,0,162,643] >;

Dic61 in GAP, Magma, Sage, TeX

{\rm Dic}_{61}
% in TeX

G:=Group("Dic61");
// GroupNames label

G:=SmallGroup(244,1);
// by ID

G=gap.SmallGroup(244,1);
# by ID

G:=PCGroup([3,-2,-2,-61,6,2162]);
// Polycyclic

G:=Group<a,b|a^122=1,b^2=a^61,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic61 in TeX

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