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G = D7×C31order 434 = 2·7·31

Direct product of C31 and D7

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

Aliases: D7×C31, C7⋊C62, C2173C2, SmallGroup(434,1)

Series: Derived Chief Lower central Upper central

C1C7 — D7×C31
C1C7C217 — D7×C31
C7 — D7×C31
C1C31

Generators and relations for D7×C31
 G = < a,b,c | a31=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

7C2
7C62

Smallest permutation representation of D7×C31
On 217 points
Generators in S217
(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)
(1 190 68 184 136 112 54)(2 191 69 185 137 113 55)(3 192 70 186 138 114 56)(4 193 71 156 139 115 57)(5 194 72 157 140 116 58)(6 195 73 158 141 117 59)(7 196 74 159 142 118 60)(8 197 75 160 143 119 61)(9 198 76 161 144 120 62)(10 199 77 162 145 121 32)(11 200 78 163 146 122 33)(12 201 79 164 147 123 34)(13 202 80 165 148 124 35)(14 203 81 166 149 94 36)(15 204 82 167 150 95 37)(16 205 83 168 151 96 38)(17 206 84 169 152 97 39)(18 207 85 170 153 98 40)(19 208 86 171 154 99 41)(20 209 87 172 155 100 42)(21 210 88 173 125 101 43)(22 211 89 174 126 102 44)(23 212 90 175 127 103 45)(24 213 91 176 128 104 46)(25 214 92 177 129 105 47)(26 215 93 178 130 106 48)(27 216 63 179 131 107 49)(28 217 64 180 132 108 50)(29 187 65 181 133 109 51)(30 188 66 182 134 110 52)(31 189 67 183 135 111 53)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 32)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 51)(30 52)(31 53)(63 131)(64 132)(65 133)(66 134)(67 135)(68 136)(69 137)(70 138)(71 139)(72 140)(73 141)(74 142)(75 143)(76 144)(77 145)(78 146)(79 147)(80 148)(81 149)(82 150)(83 151)(84 152)(85 153)(86 154)(87 155)(88 125)(89 126)(90 127)(91 128)(92 129)(93 130)(94 203)(95 204)(96 205)(97 206)(98 207)(99 208)(100 209)(101 210)(102 211)(103 212)(104 213)(105 214)(106 215)(107 216)(108 217)(109 187)(110 188)(111 189)(112 190)(113 191)(114 192)(115 193)(116 194)(117 195)(118 196)(119 197)(120 198)(121 199)(122 200)(123 201)(124 202)

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

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

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

155 conjugacy classes

class 1  2 7A7B7C31A···31AD62A···62AD217A···217CL
order1277731···3162···62217···217
size172221···17···72···2

155 irreducible representations

dim111122
type+++
imageC1C2C31C62D7D7×C31
kernelD7×C31C217D7C7C31C1
# reps113030390

Matrix representation of D7×C31 in GL2(𝔽1303) generated by

7840
0784
,
9361
1190824
,
8241302
112479
G:=sub<GL(2,GF(1303))| [784,0,0,784],[936,1190,1,824],[824,112,1302,479] >;

D7×C31 in GAP, Magma, Sage, TeX

D_7\times C_{31}
% in TeX

G:=Group("D7xC31");
// GroupNames label

G:=SmallGroup(434,1);
// by ID

G=gap.SmallGroup(434,1);
# by ID

G:=PCGroup([3,-2,-31,-7,3350]);
// Polycyclic

G:=Group<a,b,c|a^31=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×C31 in TeX

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