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## G = C6×C42order 252 = 22·32·7

### Abelian group of type [6,42]

Aliases: C6×C42, SmallGroup(252,46)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6×C42
 Chief series C1 — C7 — C21 — C3×C21 — C3×C42 — C6×C42
 Lower central C1 — C6×C42
 Upper central C1 — C6×C42

Generators and relations for C6×C42
G = < a,b | a6=b42=1, ab=ba >

Subgroups: 60, all normal (8 characteristic)
C1, C2 [×3], C3 [×4], C22, C6 [×12], C7, C32, C2×C6 [×4], C14 [×3], C3×C6 [×3], C21 [×4], C2×C14, C62, C42 [×12], C3×C21, C2×C42 [×4], C3×C42 [×3], C6×C42
Quotients: C1, C2 [×3], C3 [×4], C22, C6 [×12], C7, C32, C2×C6 [×4], C14 [×3], C3×C6 [×3], C21 [×4], C2×C14, C62, C42 [×12], C3×C21, C2×C42 [×4], C3×C42 [×3], C6×C42

Smallest permutation representation of C6×C42
Regular action on 252 points
Generators in S252
(1 115 215 75 138 197)(2 116 216 76 139 198)(3 117 217 77 140 199)(4 118 218 78 141 200)(5 119 219 79 142 201)(6 120 220 80 143 202)(7 121 221 81 144 203)(8 122 222 82 145 204)(9 123 223 83 146 205)(10 124 224 84 147 206)(11 125 225 43 148 207)(12 126 226 44 149 208)(13 85 227 45 150 209)(14 86 228 46 151 210)(15 87 229 47 152 169)(16 88 230 48 153 170)(17 89 231 49 154 171)(18 90 232 50 155 172)(19 91 233 51 156 173)(20 92 234 52 157 174)(21 93 235 53 158 175)(22 94 236 54 159 176)(23 95 237 55 160 177)(24 96 238 56 161 178)(25 97 239 57 162 179)(26 98 240 58 163 180)(27 99 241 59 164 181)(28 100 242 60 165 182)(29 101 243 61 166 183)(30 102 244 62 167 184)(31 103 245 63 168 185)(32 104 246 64 127 186)(33 105 247 65 128 187)(34 106 248 66 129 188)(35 107 249 67 130 189)(36 108 250 68 131 190)(37 109 251 69 132 191)(38 110 252 70 133 192)(39 111 211 71 134 193)(40 112 212 72 135 194)(41 113 213 73 136 195)(42 114 214 74 137 196)
(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)

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

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

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

252 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 6A ··· 6X 7A ··· 7F 14A ··· 14R 21A ··· 21AV 42A ··· 42EN order 1 2 2 2 3 ··· 3 6 ··· 6 7 ··· 7 14 ··· 14 21 ··· 21 42 ··· 42 size 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

252 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C6 C7 C14 C21 C42 kernel C6×C42 C3×C42 C2×C42 C42 C62 C3×C6 C2×C6 C6 # reps 1 3 8 24 6 18 48 144

Matrix representation of C6×C42 in GL2(𝔽43) generated by

 7 0 0 42
,
 30 0 0 23
G:=sub<GL(2,GF(43))| [7,0,0,42],[30,0,0,23] >;

C6×C42 in GAP, Magma, Sage, TeX

C_6\times C_{42}
% in TeX

G:=Group("C6xC42");
// GroupNames label

G:=SmallGroup(252,46);
// by ID

G=gap.SmallGroup(252,46);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-7]);
// Polycyclic

G:=Group<a,b|a^6=b^42=1,a*b=b*a>;
// generators/relations

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