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G = C8⋊D31order 496 = 24·31

3rd semidirect product of C8 and D31 acting via D31/C31=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C83D31, D62.C4, C2484C2, C4.13D62, Dic31.C4, C311M4(2), C124.13C22, C31⋊C84C2, C62.2(C2×C4), C2.3(C4×D31), (C4×D31).2C2, SmallGroup(496,4)

Series: Derived Chief Lower central Upper central

C1C62 — C8⋊D31
C1C31C62C124C4×D31 — C8⋊D31
C31C62 — C8⋊D31
C1C4C8

Generators and relations for C8⋊D31
 G = < a,b,c | a8=b31=c2=1, ab=ba, cac=a5, cbc=b-1 >

62C2
31C22
31C4
2D31
31C2×C4
31C8
31M4(2)

Smallest permutation representation of C8⋊D31
On 248 points
Generators in S248
(1 236 108 176 35 187 72 154)(2 237 109 177 36 188 73 155)(3 238 110 178 37 189 74 125)(4 239 111 179 38 190 75 126)(5 240 112 180 39 191 76 127)(6 241 113 181 40 192 77 128)(7 242 114 182 41 193 78 129)(8 243 115 183 42 194 79 130)(9 244 116 184 43 195 80 131)(10 245 117 185 44 196 81 132)(11 246 118 186 45 197 82 133)(12 247 119 156 46 198 83 134)(13 248 120 157 47 199 84 135)(14 218 121 158 48 200 85 136)(15 219 122 159 49 201 86 137)(16 220 123 160 50 202 87 138)(17 221 124 161 51 203 88 139)(18 222 94 162 52 204 89 140)(19 223 95 163 53 205 90 141)(20 224 96 164 54 206 91 142)(21 225 97 165 55 207 92 143)(22 226 98 166 56 208 93 144)(23 227 99 167 57 209 63 145)(24 228 100 168 58 210 64 146)(25 229 101 169 59 211 65 147)(26 230 102 170 60 212 66 148)(27 231 103 171 61 213 67 149)(28 232 104 172 62 214 68 150)(29 233 105 173 32 215 69 151)(30 234 106 174 33 216 70 152)(31 235 107 175 34 217 71 153)
(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)(218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248)
(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 37)(33 36)(34 35)(38 62)(39 61)(40 60)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)(94 121)(95 120)(96 119)(97 118)(98 117)(99 116)(100 115)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)(122 124)(125 173)(126 172)(127 171)(128 170)(129 169)(130 168)(131 167)(132 166)(133 165)(134 164)(135 163)(136 162)(137 161)(138 160)(139 159)(140 158)(141 157)(142 156)(143 186)(144 185)(145 184)(146 183)(147 182)(148 181)(149 180)(150 179)(151 178)(152 177)(153 176)(154 175)(155 174)(187 235)(188 234)(189 233)(190 232)(191 231)(192 230)(193 229)(194 228)(195 227)(196 226)(197 225)(198 224)(199 223)(200 222)(201 221)(202 220)(203 219)(204 218)(205 248)(206 247)(207 246)(208 245)(209 244)(210 243)(211 242)(212 241)(213 240)(214 239)(215 238)(216 237)(217 236)

G:=sub<Sym(248)| (1,236,108,176,35,187,72,154)(2,237,109,177,36,188,73,155)(3,238,110,178,37,189,74,125)(4,239,111,179,38,190,75,126)(5,240,112,180,39,191,76,127)(6,241,113,181,40,192,77,128)(7,242,114,182,41,193,78,129)(8,243,115,183,42,194,79,130)(9,244,116,184,43,195,80,131)(10,245,117,185,44,196,81,132)(11,246,118,186,45,197,82,133)(12,247,119,156,46,198,83,134)(13,248,120,157,47,199,84,135)(14,218,121,158,48,200,85,136)(15,219,122,159,49,201,86,137)(16,220,123,160,50,202,87,138)(17,221,124,161,51,203,88,139)(18,222,94,162,52,204,89,140)(19,223,95,163,53,205,90,141)(20,224,96,164,54,206,91,142)(21,225,97,165,55,207,92,143)(22,226,98,166,56,208,93,144)(23,227,99,167,57,209,63,145)(24,228,100,168,58,210,64,146)(25,229,101,169,59,211,65,147)(26,230,102,170,60,212,66,148)(27,231,103,171,61,213,67,149)(28,232,104,172,62,214,68,150)(29,233,105,173,32,215,69,151)(30,234,106,174,33,216,70,152)(31,235,107,175,34,217,71,153), (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)(218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248), (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)(32,37)(33,36)(34,35)(38,62)(39,61)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(94,121)(95,120)(96,119)(97,118)(98,117)(99,116)(100,115)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)(122,124)(125,173)(126,172)(127,171)(128,170)(129,169)(130,168)(131,167)(132,166)(133,165)(134,164)(135,163)(136,162)(137,161)(138,160)(139,159)(140,158)(141,157)(142,156)(143,186)(144,185)(145,184)(146,183)(147,182)(148,181)(149,180)(150,179)(151,178)(152,177)(153,176)(154,175)(155,174)(187,235)(188,234)(189,233)(190,232)(191,231)(192,230)(193,229)(194,228)(195,227)(196,226)(197,225)(198,224)(199,223)(200,222)(201,221)(202,220)(203,219)(204,218)(205,248)(206,247)(207,246)(208,245)(209,244)(210,243)(211,242)(212,241)(213,240)(214,239)(215,238)(216,237)(217,236)>;

G:=Group( (1,236,108,176,35,187,72,154)(2,237,109,177,36,188,73,155)(3,238,110,178,37,189,74,125)(4,239,111,179,38,190,75,126)(5,240,112,180,39,191,76,127)(6,241,113,181,40,192,77,128)(7,242,114,182,41,193,78,129)(8,243,115,183,42,194,79,130)(9,244,116,184,43,195,80,131)(10,245,117,185,44,196,81,132)(11,246,118,186,45,197,82,133)(12,247,119,156,46,198,83,134)(13,248,120,157,47,199,84,135)(14,218,121,158,48,200,85,136)(15,219,122,159,49,201,86,137)(16,220,123,160,50,202,87,138)(17,221,124,161,51,203,88,139)(18,222,94,162,52,204,89,140)(19,223,95,163,53,205,90,141)(20,224,96,164,54,206,91,142)(21,225,97,165,55,207,92,143)(22,226,98,166,56,208,93,144)(23,227,99,167,57,209,63,145)(24,228,100,168,58,210,64,146)(25,229,101,169,59,211,65,147)(26,230,102,170,60,212,66,148)(27,231,103,171,61,213,67,149)(28,232,104,172,62,214,68,150)(29,233,105,173,32,215,69,151)(30,234,106,174,33,216,70,152)(31,235,107,175,34,217,71,153), (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)(218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248), (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)(32,37)(33,36)(34,35)(38,62)(39,61)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(94,121)(95,120)(96,119)(97,118)(98,117)(99,116)(100,115)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)(122,124)(125,173)(126,172)(127,171)(128,170)(129,169)(130,168)(131,167)(132,166)(133,165)(134,164)(135,163)(136,162)(137,161)(138,160)(139,159)(140,158)(141,157)(142,156)(143,186)(144,185)(145,184)(146,183)(147,182)(148,181)(149,180)(150,179)(151,178)(152,177)(153,176)(154,175)(155,174)(187,235)(188,234)(189,233)(190,232)(191,231)(192,230)(193,229)(194,228)(195,227)(196,226)(197,225)(198,224)(199,223)(200,222)(201,221)(202,220)(203,219)(204,218)(205,248)(206,247)(207,246)(208,245)(209,244)(210,243)(211,242)(212,241)(213,240)(214,239)(215,238)(216,237)(217,236) );

G=PermutationGroup([(1,236,108,176,35,187,72,154),(2,237,109,177,36,188,73,155),(3,238,110,178,37,189,74,125),(4,239,111,179,38,190,75,126),(5,240,112,180,39,191,76,127),(6,241,113,181,40,192,77,128),(7,242,114,182,41,193,78,129),(8,243,115,183,42,194,79,130),(9,244,116,184,43,195,80,131),(10,245,117,185,44,196,81,132),(11,246,118,186,45,197,82,133),(12,247,119,156,46,198,83,134),(13,248,120,157,47,199,84,135),(14,218,121,158,48,200,85,136),(15,219,122,159,49,201,86,137),(16,220,123,160,50,202,87,138),(17,221,124,161,51,203,88,139),(18,222,94,162,52,204,89,140),(19,223,95,163,53,205,90,141),(20,224,96,164,54,206,91,142),(21,225,97,165,55,207,92,143),(22,226,98,166,56,208,93,144),(23,227,99,167,57,209,63,145),(24,228,100,168,58,210,64,146),(25,229,101,169,59,211,65,147),(26,230,102,170,60,212,66,148),(27,231,103,171,61,213,67,149),(28,232,104,172,62,214,68,150),(29,233,105,173,32,215,69,151),(30,234,106,174,33,216,70,152),(31,235,107,175,34,217,71,153)], [(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),(218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248)], [(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),(32,37),(33,36),(34,35),(38,62),(39,61),(40,60),(41,59),(42,58),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(81,93),(82,92),(83,91),(84,90),(85,89),(86,88),(94,121),(95,120),(96,119),(97,118),(98,117),(99,116),(100,115),(101,114),(102,113),(103,112),(104,111),(105,110),(106,109),(107,108),(122,124),(125,173),(126,172),(127,171),(128,170),(129,169),(130,168),(131,167),(132,166),(133,165),(134,164),(135,163),(136,162),(137,161),(138,160),(139,159),(140,158),(141,157),(142,156),(143,186),(144,185),(145,184),(146,183),(147,182),(148,181),(149,180),(150,179),(151,178),(152,177),(153,176),(154,175),(155,174),(187,235),(188,234),(189,233),(190,232),(191,231),(192,230),(193,229),(194,228),(195,227),(196,226),(197,225),(198,224),(199,223),(200,222),(201,221),(202,220),(203,219),(204,218),(205,248),(206,247),(207,246),(208,245),(209,244),(210,243),(211,242),(212,241),(213,240),(214,239),(215,238),(216,237),(217,236)])

130 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D31A···31O62A···62O124A···124AD248A···248BH
order122444888831···3162···62124···124248···248
size116211622262622···22···22···22···2

130 irreducible representations

dim11111122222
type++++++
imageC1C2C2C2C4C4M4(2)D31D62C4×D31C8⋊D31
kernelC8⋊D31C31⋊C8C248C4×D31Dic31D62C31C8C4C2C1
# reps111122215153060

Matrix representation of C8⋊D31 in GL2(𝔽1489) generated by

14801223
13679
,
1491
1461470
,
982130
680507
G:=sub<GL(2,GF(1489))| [1480,1367,1223,9],[149,146,1,1470],[982,680,130,507] >;

C8⋊D31 in GAP, Magma, Sage, TeX

C_8\rtimes D_{31}
% in TeX

G:=Group("C8:D31");
// GroupNames label

G:=SmallGroup(496,4);
// by ID

G=gap.SmallGroup(496,4);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-31,101,26,42,12004]);
// Polycyclic

G:=Group<a,b,c|a^8=b^31=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C8⋊D31 in TeX

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