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G = C11×C22order 242 = 2·112

Abelian group of type [11,22]

direct product, abelian, monomial, 11-elementary

Aliases: C11×C22, SmallGroup(242,5)

Series: Derived Chief Lower central Upper central

C1 — C11×C22
C1C11C112 — C11×C22
C1 — C11×C22
C1 — C11×C22

Generators and relations for C11×C22
 G = < a,b | a11=b22=1, ab=ba >


Smallest permutation representation of C11×C22
Regular action on 242 points
Generators in S242
(1 221 90 29 122 74 152 194 202 160 59)(2 222 91 30 123 75 153 195 203 161 60)(3 223 92 31 124 76 154 196 204 162 61)(4 224 93 32 125 77 133 197 205 163 62)(5 225 94 33 126 78 134 198 206 164 63)(6 226 95 34 127 79 135 177 207 165 64)(7 227 96 35 128 80 136 178 208 166 65)(8 228 97 36 129 81 137 179 209 167 66)(9 229 98 37 130 82 138 180 210 168 45)(10 230 99 38 131 83 139 181 211 169 46)(11 231 100 39 132 84 140 182 212 170 47)(12 232 101 40 111 85 141 183 213 171 48)(13 233 102 41 112 86 142 184 214 172 49)(14 234 103 42 113 87 143 185 215 173 50)(15 235 104 43 114 88 144 186 216 174 51)(16 236 105 44 115 67 145 187 217 175 52)(17 237 106 23 116 68 146 188 218 176 53)(18 238 107 24 117 69 147 189 219 155 54)(19 239 108 25 118 70 148 190 220 156 55)(20 240 109 26 119 71 149 191 199 157 56)(21 241 110 27 120 72 150 192 200 158 57)(22 242 89 28 121 73 151 193 201 159 58)
(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)

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

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

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

C11×C22 is a maximal subgroup of   C11⋊Dic11

242 conjugacy classes

class 1  2 11A···11DP22A···22DP
order1211···1122···22
size111···11···1

242 irreducible representations

dim1111
type++
imageC1C2C11C22
kernelC11×C22C112C22C11
# reps11120120

Matrix representation of C11×C22 in GL2(𝔽23) generated by

180
02
,
60
011
G:=sub<GL(2,GF(23))| [18,0,0,2],[6,0,0,11] >;

C11×C22 in GAP, Magma, Sage, TeX

C_{11}\times C_{22}
% in TeX

G:=Group("C11xC22");
// GroupNames label

G:=SmallGroup(242,5);
// by ID

G=gap.SmallGroup(242,5);
# by ID

G:=PCGroup([3,-2,-11,-11]);
// Polycyclic

G:=Group<a,b|a^11=b^22=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C11×C22 in TeX

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