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G = C7×D33order 462 = 2·3·7·11

Direct product of C7 and D33

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

Aliases: C7×D33, C772S3, C331C14, C2313C2, C213D11, C11⋊(S3×C7), C3⋊(C7×D11), SmallGroup(462,9)

Series: Derived Chief Lower central Upper central

C1C33 — C7×D33
C1C11C33C231 — C7×D33
C33 — C7×D33
C1C7

Generators and relations for C7×D33
 G = < a,b,c | a7=b33=c2=1, ab=ba, ac=ca, cbc=b-1 >

33C2
11S3
33C14
3D11
11S3×C7
3C7×D11

Smallest permutation representation of C7×D33
On 231 points
Generators in S231
(1 227 182 161 100 75 38)(2 228 183 162 101 76 39)(3 229 184 163 102 77 40)(4 230 185 164 103 78 41)(5 231 186 165 104 79 42)(6 199 187 133 105 80 43)(7 200 188 134 106 81 44)(8 201 189 135 107 82 45)(9 202 190 136 108 83 46)(10 203 191 137 109 84 47)(11 204 192 138 110 85 48)(12 205 193 139 111 86 49)(13 206 194 140 112 87 50)(14 207 195 141 113 88 51)(15 208 196 142 114 89 52)(16 209 197 143 115 90 53)(17 210 198 144 116 91 54)(18 211 166 145 117 92 55)(19 212 167 146 118 93 56)(20 213 168 147 119 94 57)(21 214 169 148 120 95 58)(22 215 170 149 121 96 59)(23 216 171 150 122 97 60)(24 217 172 151 123 98 61)(25 218 173 152 124 99 62)(26 219 174 153 125 67 63)(27 220 175 154 126 68 64)(28 221 176 155 127 69 65)(29 222 177 156 128 70 66)(30 223 178 157 129 71 34)(31 224 179 158 130 72 35)(32 225 180 159 131 73 36)(33 226 181 160 132 74 37)
(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)
(1 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(34 41)(35 40)(36 39)(37 38)(42 66)(43 65)(44 64)(45 63)(46 62)(47 61)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(67 82)(68 81)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)(83 99)(84 98)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)(100 132)(101 131)(102 130)(103 129)(104 128)(105 127)(106 126)(107 125)(108 124)(109 123)(110 122)(111 121)(112 120)(113 119)(114 118)(115 117)(133 155)(134 154)(135 153)(136 152)(137 151)(138 150)(139 149)(140 148)(141 147)(142 146)(143 145)(156 165)(157 164)(158 163)(159 162)(160 161)(166 197)(167 196)(168 195)(169 194)(170 193)(171 192)(172 191)(173 190)(174 189)(175 188)(176 187)(177 186)(178 185)(179 184)(180 183)(181 182)(199 221)(200 220)(201 219)(202 218)(203 217)(204 216)(205 215)(206 214)(207 213)(208 212)(209 211)(222 231)(223 230)(224 229)(225 228)(226 227)

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

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

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

126 conjugacy classes

class 1  2  3 7A···7F11A···11E14A···14F21A···21F33A···33J77A···77AD231A···231BH
order1237···711···1114···1421···2133···3377···77231···231
size13321···12···233···332···22···22···22···2

126 irreducible representations

dim1111222222
type+++++
imageC1C2C7C14S3D11S3×C7D33C7×D11C7×D33
kernelC7×D33C231D33C33C77C21C11C7C3C1
# reps1166156103060

Matrix representation of C7×D33 in GL2(𝔽463) generated by

3080
0308
,
44559
448152
,
174252
330289
G:=sub<GL(2,GF(463))| [308,0,0,308],[445,448,59,152],[174,330,252,289] >;

C7×D33 in GAP, Magma, Sage, TeX

C_7\times D_{33}
% in TeX

G:=Group("C7xD33");
// GroupNames label

G:=SmallGroup(462,9);
// by ID

G=gap.SmallGroup(462,9);
# by ID

G:=PCGroup([4,-2,-7,-3,-11,338,6723]);
// Polycyclic

G:=Group<a,b,c|a^7=b^33=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C7×D33 in TeX

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