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G = D7×C30order 420 = 22·3·5·7

Direct product of C30 and D7

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

Aliases: D7×C30, C706C6, C2105C2, C422C10, C143C30, C10512C22, C73(C2×C30), C358(C2×C6), C213(C2×C10), SmallGroup(420,34)

Series: Derived Chief Lower central Upper central

C1C7 — D7×C30
C1C7C35C105D7×C15 — D7×C30
C7 — D7×C30
C1C30

Generators and relations for D7×C30
 G = < a,b,c | a30=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

7C2
7C2
7C22
7C6
7C6
7C10
7C10
7C2×C6
7C2×C10
7C30
7C30
7C2×C30

Smallest permutation representation of D7×C30
On 210 points
Generators in S210
(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)
(1 162 61 188 110 135 43)(2 163 62 189 111 136 44)(3 164 63 190 112 137 45)(4 165 64 191 113 138 46)(5 166 65 192 114 139 47)(6 167 66 193 115 140 48)(7 168 67 194 116 141 49)(8 169 68 195 117 142 50)(9 170 69 196 118 143 51)(10 171 70 197 119 144 52)(11 172 71 198 120 145 53)(12 173 72 199 91 146 54)(13 174 73 200 92 147 55)(14 175 74 201 93 148 56)(15 176 75 202 94 149 57)(16 177 76 203 95 150 58)(17 178 77 204 96 121 59)(18 179 78 205 97 122 60)(19 180 79 206 98 123 31)(20 151 80 207 99 124 32)(21 152 81 208 100 125 33)(22 153 82 209 101 126 34)(23 154 83 210 102 127 35)(24 155 84 181 103 128 36)(25 156 85 182 104 129 37)(26 157 86 183 105 130 38)(27 158 87 184 106 131 39)(28 159 88 185 107 132 40)(29 160 89 186 108 133 41)(30 161 90 187 109 134 42)
(1 58)(2 59)(3 60)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(61 95)(62 96)(63 97)(64 98)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)(71 105)(72 106)(73 107)(74 108)(75 109)(76 110)(77 111)(78 112)(79 113)(80 114)(81 115)(82 116)(83 117)(84 118)(85 119)(86 120)(87 91)(88 92)(89 93)(90 94)(121 163)(122 164)(123 165)(124 166)(125 167)(126 168)(127 169)(128 170)(129 171)(130 172)(131 173)(132 174)(133 175)(134 176)(135 177)(136 178)(137 179)(138 180)(139 151)(140 152)(141 153)(142 154)(143 155)(144 156)(145 157)(146 158)(147 159)(148 160)(149 161)(150 162)(181 196)(182 197)(183 198)(184 199)(185 200)(186 201)(187 202)(188 203)(189 204)(190 205)(191 206)(192 207)(193 208)(194 209)(195 210)

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

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

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

150 conjugacy classes

class 1 2A2B2C3A3B5A5B5C5D6A6B6C6D6E6F7A7B7C10A10B10C10D10E···10L14A14B14C15A···15H21A···21F30A···30H30I···30X35A···35L42A···42F70A···70L105A···105X210A···210X
order12223355556666667771010101010···1014141415···1521···2130···3030···3035···3542···4270···70105···105210···210
size117711111111777722211117···72221···12···21···17···72···22···22···22···22···2

150 irreducible representations

dim11111111111122222222
type+++++
imageC1C2C2C3C5C6C6C10C10C15C30C30D7D14C3×D7C5×D7C6×D7C10×D7D7×C15D7×C30
kernelD7×C30D7×C15C210C10×D7C6×D7C5×D7C70C3×D7C42D14D7C14C30C15C10C6C5C3C2C1
# reps1212442848168336126122424

Matrix representation of D7×C30 in GL3(𝔽211) generated by

21000
01370
00137
,
100
0811
02100
,
100
001
010
G:=sub<GL(3,GF(211))| [210,0,0,0,137,0,0,0,137],[1,0,0,0,81,210,0,1,0],[1,0,0,0,0,1,0,1,0] >;

D7×C30 in GAP, Magma, Sage, TeX

D_7\times C_{30}
% in TeX

G:=Group("D7xC30");
// GroupNames label

G:=SmallGroup(420,34);
// by ID

G=gap.SmallGroup(420,34);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-7,9004]);
// Polycyclic

G:=Group<a,b,c|a^30=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×C30 in TeX

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