Copied to
clipboard

G = C7×D31order 434 = 2·7·31

Direct product of C7 and D31

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

Aliases: C7×D31, C31⋊C14, C2172C2, SmallGroup(434,2)

Series: Derived Chief Lower central Upper central

C1C31 — C7×D31
C1C31C217 — C7×D31
C31 — C7×D31
C1C7

Generators and relations for C7×D31
 G = < a,b,c | a7=b31=c2=1, ab=ba, ac=ca, cbc=b-1 >

31C2
31C14

Smallest permutation representation of C7×D31
On 217 points
Generators in S217
(1 209 158 139 98 78 60)(2 210 159 140 99 79 61)(3 211 160 141 100 80 62)(4 212 161 142 101 81 32)(5 213 162 143 102 82 33)(6 214 163 144 103 83 34)(7 215 164 145 104 84 35)(8 216 165 146 105 85 36)(9 217 166 147 106 86 37)(10 187 167 148 107 87 38)(11 188 168 149 108 88 39)(12 189 169 150 109 89 40)(13 190 170 151 110 90 41)(14 191 171 152 111 91 42)(15 192 172 153 112 92 43)(16 193 173 154 113 93 44)(17 194 174 155 114 63 45)(18 195 175 125 115 64 46)(19 196 176 126 116 65 47)(20 197 177 127 117 66 48)(21 198 178 128 118 67 49)(22 199 179 129 119 68 50)(23 200 180 130 120 69 51)(24 201 181 131 121 70 52)(25 202 182 132 122 71 53)(26 203 183 133 123 72 54)(27 204 184 134 124 73 55)(28 205 185 135 94 74 56)(29 206 186 136 95 75 57)(30 207 156 137 96 76 58)(31 208 157 138 97 77 59)
(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 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)(41 47)(42 46)(43 45)(57 62)(58 61)(59 60)(63 92)(64 91)(65 90)(66 89)(67 88)(68 87)(69 86)(70 85)(71 84)(72 83)(73 82)(74 81)(75 80)(76 79)(77 78)(94 101)(95 100)(96 99)(97 98)(102 124)(103 123)(104 122)(105 121)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)(125 152)(126 151)(127 150)(128 149)(129 148)(130 147)(131 146)(132 145)(133 144)(134 143)(135 142)(136 141)(137 140)(138 139)(153 155)(156 159)(157 158)(160 186)(161 185)(162 184)(163 183)(164 182)(165 181)(166 180)(167 179)(168 178)(169 177)(170 176)(171 175)(172 174)(187 199)(188 198)(189 197)(190 196)(191 195)(192 194)(200 217)(201 216)(202 215)(203 214)(204 213)(205 212)(206 211)(207 210)(208 209)

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

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

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

119 conjugacy classes

class 1  2 7A···7F14A···14F31A···31O217A···217CL
order127···714···1431···31217···217
size1311···131···312···22···2

119 irreducible representations

dim111122
type+++
imageC1C2C7C14D31C7×D31
kernelC7×D31C217D31C31C7C1
# reps11661590

Matrix representation of C7×D31 in GL2(𝔽1303) generated by

520
052
,
01
1302534
,
01
10
G:=sub<GL(2,GF(1303))| [52,0,0,52],[0,1302,1,534],[0,1,1,0] >;

C7×D31 in GAP, Magma, Sage, TeX

C_7\times D_{31}
% in TeX

G:=Group("C7xD31");
// GroupNames label

G:=SmallGroup(434,2);
// by ID

G=gap.SmallGroup(434,2);
# by ID

G:=PCGroup([3,-2,-7,-31,3782]);
// Polycyclic

G:=Group<a,b,c|a^7=b^31=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C7×D31 in TeX

׿
×
𝔽