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## G = C4○D4×C31order 496 = 24·31

### Direct product of C31 and C4○D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C4○D4×C31, D42C62, Q82C62, C62.13C23, C124.21C22, (C2×C4)⋊3C62, (C2×C124)⋊7C2, (D4×C31)⋊5C2, C4.5(C2×C62), (Q8×C31)⋊5C2, C22.(C2×C62), C2.3(C22×C62), (C2×C62).2C22, SmallGroup(496,40)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4○D4×C31
 Chief series C1 — C2 — C62 — C2×C62 — D4×C31 — C4○D4×C31
 Lower central C1 — C2 — C4○D4×C31
 Upper central C1 — C124 — C4○D4×C31

Generators and relations for C4○D4×C31
G = < a,b,c,d | a31=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Smallest permutation representation of C4○D4×C31
On 248 points
Generators in S248
(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 59 171 188)(2 60 172 189)(3 61 173 190)(4 62 174 191)(5 32 175 192)(6 33 176 193)(7 34 177 194)(8 35 178 195)(9 36 179 196)(10 37 180 197)(11 38 181 198)(12 39 182 199)(13 40 183 200)(14 41 184 201)(15 42 185 202)(16 43 186 203)(17 44 156 204)(18 45 157 205)(19 46 158 206)(20 47 159 207)(21 48 160 208)(22 49 161 209)(23 50 162 210)(24 51 163 211)(25 52 164 212)(26 53 165 213)(27 54 166 214)(28 55 167 215)(29 56 168 216)(30 57 169 217)(31 58 170 187)(63 245 113 141)(64 246 114 142)(65 247 115 143)(66 248 116 144)(67 218 117 145)(68 219 118 146)(69 220 119 147)(70 221 120 148)(71 222 121 149)(72 223 122 150)(73 224 123 151)(74 225 124 152)(75 226 94 153)(76 227 95 154)(77 228 96 155)(78 229 97 125)(79 230 98 126)(80 231 99 127)(81 232 100 128)(82 233 101 129)(83 234 102 130)(84 235 103 131)(85 236 104 132)(86 237 105 133)(87 238 106 134)(88 239 107 135)(89 240 108 136)(90 241 109 137)(91 242 110 138)(92 243 111 139)(93 244 112 140)
(1 98 171 79)(2 99 172 80)(3 100 173 81)(4 101 174 82)(5 102 175 83)(6 103 176 84)(7 104 177 85)(8 105 178 86)(9 106 179 87)(10 107 180 88)(11 108 181 89)(12 109 182 90)(13 110 183 91)(14 111 184 92)(15 112 185 93)(16 113 186 63)(17 114 156 64)(18 115 157 65)(19 116 158 66)(20 117 159 67)(21 118 160 68)(22 119 161 69)(23 120 162 70)(24 121 163 71)(25 122 164 72)(26 123 165 73)(27 124 166 74)(28 94 167 75)(29 95 168 76)(30 96 169 77)(31 97 170 78)(32 130 192 234)(33 131 193 235)(34 132 194 236)(35 133 195 237)(36 134 196 238)(37 135 197 239)(38 136 198 240)(39 137 199 241)(40 138 200 242)(41 139 201 243)(42 140 202 244)(43 141 203 245)(44 142 204 246)(45 143 205 247)(46 144 206 248)(47 145 207 218)(48 146 208 219)(49 147 209 220)(50 148 210 221)(51 149 211 222)(52 150 212 223)(53 151 213 224)(54 152 214 225)(55 153 215 226)(56 154 216 227)(57 155 217 228)(58 125 187 229)(59 126 188 230)(60 127 189 231)(61 128 190 232)(62 129 191 233)
(63 113)(64 114)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 121)(72 122)(73 123)(74 124)(75 94)(76 95)(77 96)(78 97)(79 98)(80 99)(81 100)(82 101)(83 102)(84 103)(85 104)(86 105)(87 106)(88 107)(89 108)(90 109)(91 110)(92 111)(93 112)(125 229)(126 230)(127 231)(128 232)(129 233)(130 234)(131 235)(132 236)(133 237)(134 238)(135 239)(136 240)(137 241)(138 242)(139 243)(140 244)(141 245)(142 246)(143 247)(144 248)(145 218)(146 219)(147 220)(148 221)(149 222)(150 223)(151 224)(152 225)(153 226)(154 227)(155 228)

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

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

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

310 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 31A ··· 31AD 62A ··· 62AD 62AE ··· 62DP 124A ··· 124BH 124BI ··· 124ET order 1 2 2 2 2 4 4 4 4 4 31 ··· 31 62 ··· 62 62 ··· 62 124 ··· 124 124 ··· 124 size 1 1 2 2 2 1 1 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

310 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C31 C62 C62 C62 C4○D4 C4○D4×C31 kernel C4○D4×C31 C2×C124 D4×C31 Q8×C31 C4○D4 C2×C4 D4 Q8 C31 C1 # reps 1 3 3 1 30 90 90 30 2 60

Matrix representation of C4○D4×C31 in GL2(𝔽373) generated by

 75 0 0 75
,
 269 0 0 269
,
 271 2 206 102
,
 1 0 102 372
G:=sub<GL(2,GF(373))| [75,0,0,75],[269,0,0,269],[271,206,2,102],[1,102,0,372] >;

C4○D4×C31 in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_{31}
% in TeX

G:=Group("C4oD4xC31");
// GroupNames label

G:=SmallGroup(496,40);
// by ID

G=gap.SmallGroup(496,40);
# by ID

G:=PCGroup([5,-2,-2,-2,-31,-2,2501,942]);
// Polycyclic

G:=Group<a,b,c,d|a^31=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations

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