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G = D7×C33order 462 = 2·3·7·11

Direct product of C33 and D7

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

Aliases: D7×C33, C73C66, C777C6, C2316C2, C212C22, SmallGroup(462,6)

Series: Derived Chief Lower central Upper central

C1C7 — D7×C33
C1C7C77C231 — D7×C33
C7 — D7×C33
C1C33

Generators and relations for D7×C33
 G = < a,b,c | a33=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

7C2
7C6
7C22
7C66

Smallest permutation representation of D7×C33
On 231 points
Generators in S231
(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 94 203 171 150 60 112)(2 95 204 172 151 61 113)(3 96 205 173 152 62 114)(4 97 206 174 153 63 115)(5 98 207 175 154 64 116)(6 99 208 176 155 65 117)(7 67 209 177 156 66 118)(8 68 210 178 157 34 119)(9 69 211 179 158 35 120)(10 70 212 180 159 36 121)(11 71 213 181 160 37 122)(12 72 214 182 161 38 123)(13 73 215 183 162 39 124)(14 74 216 184 163 40 125)(15 75 217 185 164 41 126)(16 76 218 186 165 42 127)(17 77 219 187 133 43 128)(18 78 220 188 134 44 129)(19 79 221 189 135 45 130)(20 80 222 190 136 46 131)(21 81 223 191 137 47 132)(22 82 224 192 138 48 100)(23 83 225 193 139 49 101)(24 84 226 194 140 50 102)(25 85 227 195 141 51 103)(26 86 228 196 142 52 104)(27 87 229 197 143 53 105)(28 88 230 198 144 54 106)(29 89 231 166 145 55 107)(30 90 199 167 146 56 108)(31 91 200 168 147 57 109)(32 92 201 169 148 58 110)(33 93 202 170 149 59 111)
(1 112)(2 113)(3 114)(4 115)(5 116)(6 117)(7 118)(8 119)(9 120)(10 121)(11 122)(12 123)(13 124)(14 125)(15 126)(16 127)(17 128)(18 129)(19 130)(20 131)(21 132)(22 100)(23 101)(24 102)(25 103)(26 104)(27 105)(28 106)(29 107)(30 108)(31 109)(32 110)(33 111)(34 68)(35 69)(36 70)(37 71)(38 72)(39 73)(40 74)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 81)(48 82)(49 83)(50 84)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)(57 91)(58 92)(59 93)(60 94)(61 95)(62 96)(63 97)(64 98)(65 99)(66 67)(133 219)(134 220)(135 221)(136 222)(137 223)(138 224)(139 225)(140 226)(141 227)(142 228)(143 229)(144 230)(145 231)(146 199)(147 200)(148 201)(149 202)(150 203)(151 204)(152 205)(153 206)(154 207)(155 208)(156 209)(157 210)(158 211)(159 212)(160 213)(161 214)(162 215)(163 216)(164 217)(165 218)

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

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

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

165 conjugacy classes

class 1  2 3A3B6A6B7A7B7C11A···11J21A···21F22A···22J33A···33T66A···66T77A···77AD231A···231BH
order12336677711···1121···2122···2233···3366···6677···77231···231
size1711772221···12···27···71···17···72···22···2

165 irreducible representations

dim111111112222
type+++
imageC1C2C3C6C11C22C33C66D7C3×D7C11×D7D7×C33
kernelD7×C33C231C11×D7C77C3×D7C21D7C7C33C11C3C1
# reps112210102020363060

Matrix representation of D7×C33 in GL3(𝔽463) generated by

9400
01580
00158
,
100
03201
04620
,
46200
001
010
G:=sub<GL(3,GF(463))| [94,0,0,0,158,0,0,0,158],[1,0,0,0,320,462,0,1,0],[462,0,0,0,0,1,0,1,0] >;

D7×C33 in GAP, Magma, Sage, TeX

D_7\times C_{33}
% in TeX

G:=Group("D7xC33");
// GroupNames label

G:=SmallGroup(462,6);
// by ID

G=gap.SmallGroup(462,6);
# by ID

G:=PCGroup([4,-2,-3,-11,-7,6339]);
// Polycyclic

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

Export

Subgroup lattice of D7×C33 in TeX

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