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G = C8×D31order 496 = 24·31

Direct product of C8 and D31

Aliases: C8×D31, C2483C2, D62.2C4, C4.12D62, Dic31.2C4, C124.12C22, C31⋊C86C2, C311(C2×C8), C62.1(C2×C4), C2.1(C4×D31), (C4×D31).3C2, SmallGroup(496,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C31 — C8×D31
 Chief series C1 — C31 — C62 — C124 — C4×D31 — C8×D31
 Lower central C31 — C8×D31
 Upper central C1 — C8

Generators and relations for C8×D31
G = < a,b,c | a8=b31=c2=1, ab=ba, ac=ca, cbc=b-1 >

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

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

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

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

136 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 8A 8B 8C 8D 8E 8F 8G 8H 31A ··· 31O 62A ··· 62O 124A ··· 124AD 248A ··· 248BH order 1 2 2 2 4 4 4 4 8 8 8 8 8 8 8 8 31 ··· 31 62 ··· 62 124 ··· 124 248 ··· 248 size 1 1 31 31 1 1 31 31 1 1 1 1 31 31 31 31 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

136 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 C8 D31 D62 C4×D31 C8×D31 kernel C8×D31 C31⋊C8 C248 C4×D31 Dic31 D62 D31 C8 C4 C2 C1 # reps 1 1 1 1 2 2 8 15 15 30 60

Matrix representation of C8×D31 in GL3(𝔽1489) generated by

 15 0 0 0 1264 0 0 0 1264
,
 1 0 0 0 1211 1 0 1398 943
,
 1 0 0 0 1163 213 0 1032 326
G:=sub<GL(3,GF(1489))| [15,0,0,0,1264,0,0,0,1264],[1,0,0,0,1211,1398,0,1,943],[1,0,0,0,1163,1032,0,213,326] >;

C8×D31 in GAP, Magma, Sage, TeX

C_8\times D_{31}
% in TeX

G:=Group("C8xD31");
// GroupNames label

G:=SmallGroup(496,3);
// by ID

G=gap.SmallGroup(496,3);
# by ID

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

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

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