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G = Dic63order 252 = 22·32·7

Dicyclic group

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

Aliases: Dic63, C9⋊Dic7, C7⋊Dic9, C631C4, C18.D7, C14.D9, C2.D63, C42.1S3, C6.1D21, C3.Dic21, C126.1C2, C21.1Dic3, SmallGroup(252,5)

Series: Derived Chief Lower central Upper central

C1C63 — Dic63
C1C3C21C63C126 — Dic63
C63 — Dic63
C1C2

Generators and relations for Dic63
 G = < a,b | a126=1, b2=a63, bab-1=a-1 >

63C4
21Dic3
9Dic7
7Dic9
3Dic21

Smallest permutation representation of Dic63
Regular action on 252 points
Generators in S252
(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)(127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252)
(1 137 64 200)(2 136 65 199)(3 135 66 198)(4 134 67 197)(5 133 68 196)(6 132 69 195)(7 131 70 194)(8 130 71 193)(9 129 72 192)(10 128 73 191)(11 127 74 190)(12 252 75 189)(13 251 76 188)(14 250 77 187)(15 249 78 186)(16 248 79 185)(17 247 80 184)(18 246 81 183)(19 245 82 182)(20 244 83 181)(21 243 84 180)(22 242 85 179)(23 241 86 178)(24 240 87 177)(25 239 88 176)(26 238 89 175)(27 237 90 174)(28 236 91 173)(29 235 92 172)(30 234 93 171)(31 233 94 170)(32 232 95 169)(33 231 96 168)(34 230 97 167)(35 229 98 166)(36 228 99 165)(37 227 100 164)(38 226 101 163)(39 225 102 162)(40 224 103 161)(41 223 104 160)(42 222 105 159)(43 221 106 158)(44 220 107 157)(45 219 108 156)(46 218 109 155)(47 217 110 154)(48 216 111 153)(49 215 112 152)(50 214 113 151)(51 213 114 150)(52 212 115 149)(53 211 116 148)(54 210 117 147)(55 209 118 146)(56 208 119 145)(57 207 120 144)(58 206 121 143)(59 205 122 142)(60 204 123 141)(61 203 124 140)(62 202 125 139)(63 201 126 138)

G:=sub<Sym(252)| (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)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252), (1,137,64,200)(2,136,65,199)(3,135,66,198)(4,134,67,197)(5,133,68,196)(6,132,69,195)(7,131,70,194)(8,130,71,193)(9,129,72,192)(10,128,73,191)(11,127,74,190)(12,252,75,189)(13,251,76,188)(14,250,77,187)(15,249,78,186)(16,248,79,185)(17,247,80,184)(18,246,81,183)(19,245,82,182)(20,244,83,181)(21,243,84,180)(22,242,85,179)(23,241,86,178)(24,240,87,177)(25,239,88,176)(26,238,89,175)(27,237,90,174)(28,236,91,173)(29,235,92,172)(30,234,93,171)(31,233,94,170)(32,232,95,169)(33,231,96,168)(34,230,97,167)(35,229,98,166)(36,228,99,165)(37,227,100,164)(38,226,101,163)(39,225,102,162)(40,224,103,161)(41,223,104,160)(42,222,105,159)(43,221,106,158)(44,220,107,157)(45,219,108,156)(46,218,109,155)(47,217,110,154)(48,216,111,153)(49,215,112,152)(50,214,113,151)(51,213,114,150)(52,212,115,149)(53,211,116,148)(54,210,117,147)(55,209,118,146)(56,208,119,145)(57,207,120,144)(58,206,121,143)(59,205,122,142)(60,204,123,141)(61,203,124,140)(62,202,125,139)(63,201,126,138)>;

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)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252), (1,137,64,200)(2,136,65,199)(3,135,66,198)(4,134,67,197)(5,133,68,196)(6,132,69,195)(7,131,70,194)(8,130,71,193)(9,129,72,192)(10,128,73,191)(11,127,74,190)(12,252,75,189)(13,251,76,188)(14,250,77,187)(15,249,78,186)(16,248,79,185)(17,247,80,184)(18,246,81,183)(19,245,82,182)(20,244,83,181)(21,243,84,180)(22,242,85,179)(23,241,86,178)(24,240,87,177)(25,239,88,176)(26,238,89,175)(27,237,90,174)(28,236,91,173)(29,235,92,172)(30,234,93,171)(31,233,94,170)(32,232,95,169)(33,231,96,168)(34,230,97,167)(35,229,98,166)(36,228,99,165)(37,227,100,164)(38,226,101,163)(39,225,102,162)(40,224,103,161)(41,223,104,160)(42,222,105,159)(43,221,106,158)(44,220,107,157)(45,219,108,156)(46,218,109,155)(47,217,110,154)(48,216,111,153)(49,215,112,152)(50,214,113,151)(51,213,114,150)(52,212,115,149)(53,211,116,148)(54,210,117,147)(55,209,118,146)(56,208,119,145)(57,207,120,144)(58,206,121,143)(59,205,122,142)(60,204,123,141)(61,203,124,140)(62,202,125,139)(63,201,126,138) );

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),(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252)], [(1,137,64,200),(2,136,65,199),(3,135,66,198),(4,134,67,197),(5,133,68,196),(6,132,69,195),(7,131,70,194),(8,130,71,193),(9,129,72,192),(10,128,73,191),(11,127,74,190),(12,252,75,189),(13,251,76,188),(14,250,77,187),(15,249,78,186),(16,248,79,185),(17,247,80,184),(18,246,81,183),(19,245,82,182),(20,244,83,181),(21,243,84,180),(22,242,85,179),(23,241,86,178),(24,240,87,177),(25,239,88,176),(26,238,89,175),(27,237,90,174),(28,236,91,173),(29,235,92,172),(30,234,93,171),(31,233,94,170),(32,232,95,169),(33,231,96,168),(34,230,97,167),(35,229,98,166),(36,228,99,165),(37,227,100,164),(38,226,101,163),(39,225,102,162),(40,224,103,161),(41,223,104,160),(42,222,105,159),(43,221,106,158),(44,220,107,157),(45,219,108,156),(46,218,109,155),(47,217,110,154),(48,216,111,153),(49,215,112,152),(50,214,113,151),(51,213,114,150),(52,212,115,149),(53,211,116,148),(54,210,117,147),(55,209,118,146),(56,208,119,145),(57,207,120,144),(58,206,121,143),(59,205,122,142),(60,204,123,141),(61,203,124,140),(62,202,125,139),(63,201,126,138)])

66 conjugacy classes

class 1  2  3 4A4B 6 7A7B7C9A9B9C14A14B14C18A18B18C21A···21F42A···42F63A···63R126A···126R
order12344677799914141418181821···2142···4263···63126···126
size112636322222222222222···22···22···22···2

66 irreducible representations

dim1112222222222
type+++-++--+-+-
imageC1C2C4S3Dic3D7D9Dic7Dic9D21Dic21D63Dic63
kernelDic63C126C63C42C21C18C14C9C7C6C3C2C1
# reps112113333661818

Matrix representation of Dic63 in GL3(𝔽757) generated by

75600
0167311
0446613
,
8700
0342600
0258415
G:=sub<GL(3,GF(757))| [756,0,0,0,167,446,0,311,613],[87,0,0,0,342,258,0,600,415] >;

Dic63 in GAP, Magma, Sage, TeX

{\rm Dic}_{63}
% in TeX

G:=Group("Dic63");
// GroupNames label

G:=SmallGroup(252,5);
// by ID

G=gap.SmallGroup(252,5);
# by ID

G:=PCGroup([5,-2,-2,-3,-7,-3,10,1382,642,1443,4204]);
// Polycyclic

G:=Group<a,b|a^126=1,b^2=a^63,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic63 in TeX

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