direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C4○D4×C31, D4⋊2C62, Q8⋊2C62, 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
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 71 236 128)(2 72 237 129)(3 73 238 130)(4 74 239 131)(5 75 240 132)(6 76 241 133)(7 77 242 134)(8 78 243 135)(9 79 244 136)(10 80 245 137)(11 81 246 138)(12 82 247 139)(13 83 248 140)(14 84 218 141)(15 85 219 142)(16 86 220 143)(17 87 221 144)(18 88 222 145)(19 89 223 146)(20 90 224 147)(21 91 225 148)(22 92 226 149)(23 93 227 150)(24 63 228 151)(25 64 229 152)(26 65 230 153)(27 66 231 154)(28 67 232 155)(29 68 233 125)(30 69 234 126)(31 70 235 127)(32 213 124 176)(33 214 94 177)(34 215 95 178)(35 216 96 179)(36 217 97 180)(37 187 98 181)(38 188 99 182)(39 189 100 183)(40 190 101 184)(41 191 102 185)(42 192 103 186)(43 193 104 156)(44 194 105 157)(45 195 106 158)(46 196 107 159)(47 197 108 160)(48 198 109 161)(49 199 110 162)(50 200 111 163)(51 201 112 164)(52 202 113 165)(53 203 114 166)(54 204 115 167)(55 205 116 168)(56 206 117 169)(57 207 118 170)(58 208 119 171)(59 209 120 172)(60 210 121 173)(61 211 122 174)(62 212 123 175)
(1 176 236 213)(2 177 237 214)(3 178 238 215)(4 179 239 216)(5 180 240 217)(6 181 241 187)(7 182 242 188)(8 183 243 189)(9 184 244 190)(10 185 245 191)(11 186 246 192)(12 156 247 193)(13 157 248 194)(14 158 218 195)(15 159 219 196)(16 160 220 197)(17 161 221 198)(18 162 222 199)(19 163 223 200)(20 164 224 201)(21 165 225 202)(22 166 226 203)(23 167 227 204)(24 168 228 205)(25 169 229 206)(26 170 230 207)(27 171 231 208)(28 172 232 209)(29 173 233 210)(30 174 234 211)(31 175 235 212)(32 128 124 71)(33 129 94 72)(34 130 95 73)(35 131 96 74)(36 132 97 75)(37 133 98 76)(38 134 99 77)(39 135 100 78)(40 136 101 79)(41 137 102 80)(42 138 103 81)(43 139 104 82)(44 140 105 83)(45 141 106 84)(46 142 107 85)(47 143 108 86)(48 144 109 87)(49 145 110 88)(50 146 111 89)(51 147 112 90)(52 148 113 91)(53 149 114 92)(54 150 115 93)(55 151 116 63)(56 152 117 64)(57 153 118 65)(58 154 119 66)(59 155 120 67)(60 125 121 68)(61 126 122 69)(62 127 123 70)
(32 124)(33 94)(34 95)(35 96)(36 97)(37 98)(38 99)(39 100)(40 101)(41 102)(42 103)(43 104)(44 105)(45 106)(46 107)(47 108)(48 109)(49 110)(50 111)(51 112)(52 113)(53 114)(54 115)(55 116)(56 117)(57 118)(58 119)(59 120)(60 121)(61 122)(62 123)(156 193)(157 194)(158 195)(159 196)(160 197)(161 198)(162 199)(163 200)(164 201)(165 202)(166 203)(167 204)(168 205)(169 206)(170 207)(171 208)(172 209)(173 210)(174 211)(175 212)(176 213)(177 214)(178 215)(179 216)(180 217)(181 187)(182 188)(183 189)(184 190)(185 191)(186 192)
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), (1,71,236,128)(2,72,237,129)(3,73,238,130)(4,74,239,131)(5,75,240,132)(6,76,241,133)(7,77,242,134)(8,78,243,135)(9,79,244,136)(10,80,245,137)(11,81,246,138)(12,82,247,139)(13,83,248,140)(14,84,218,141)(15,85,219,142)(16,86,220,143)(17,87,221,144)(18,88,222,145)(19,89,223,146)(20,90,224,147)(21,91,225,148)(22,92,226,149)(23,93,227,150)(24,63,228,151)(25,64,229,152)(26,65,230,153)(27,66,231,154)(28,67,232,155)(29,68,233,125)(30,69,234,126)(31,70,235,127)(32,213,124,176)(33,214,94,177)(34,215,95,178)(35,216,96,179)(36,217,97,180)(37,187,98,181)(38,188,99,182)(39,189,100,183)(40,190,101,184)(41,191,102,185)(42,192,103,186)(43,193,104,156)(44,194,105,157)(45,195,106,158)(46,196,107,159)(47,197,108,160)(48,198,109,161)(49,199,110,162)(50,200,111,163)(51,201,112,164)(52,202,113,165)(53,203,114,166)(54,204,115,167)(55,205,116,168)(56,206,117,169)(57,207,118,170)(58,208,119,171)(59,209,120,172)(60,210,121,173)(61,211,122,174)(62,212,123,175), (1,176,236,213)(2,177,237,214)(3,178,238,215)(4,179,239,216)(5,180,240,217)(6,181,241,187)(7,182,242,188)(8,183,243,189)(9,184,244,190)(10,185,245,191)(11,186,246,192)(12,156,247,193)(13,157,248,194)(14,158,218,195)(15,159,219,196)(16,160,220,197)(17,161,221,198)(18,162,222,199)(19,163,223,200)(20,164,224,201)(21,165,225,202)(22,166,226,203)(23,167,227,204)(24,168,228,205)(25,169,229,206)(26,170,230,207)(27,171,231,208)(28,172,232,209)(29,173,233,210)(30,174,234,211)(31,175,235,212)(32,128,124,71)(33,129,94,72)(34,130,95,73)(35,131,96,74)(36,132,97,75)(37,133,98,76)(38,134,99,77)(39,135,100,78)(40,136,101,79)(41,137,102,80)(42,138,103,81)(43,139,104,82)(44,140,105,83)(45,141,106,84)(46,142,107,85)(47,143,108,86)(48,144,109,87)(49,145,110,88)(50,146,111,89)(51,147,112,90)(52,148,113,91)(53,149,114,92)(54,150,115,93)(55,151,116,63)(56,152,117,64)(57,153,118,65)(58,154,119,66)(59,155,120,67)(60,125,121,68)(61,126,122,69)(62,127,123,70), (32,124)(33,94)(34,95)(35,96)(36,97)(37,98)(38,99)(39,100)(40,101)(41,102)(42,103)(43,104)(44,105)(45,106)(46,107)(47,108)(48,109)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,121)(61,122)(62,123)(156,193)(157,194)(158,195)(159,196)(160,197)(161,198)(162,199)(163,200)(164,201)(165,202)(166,203)(167,204)(168,205)(169,206)(170,207)(171,208)(172,209)(173,210)(174,211)(175,212)(176,213)(177,214)(178,215)(179,216)(180,217)(181,187)(182,188)(183,189)(184,190)(185,191)(186,192)>;
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), (1,71,236,128)(2,72,237,129)(3,73,238,130)(4,74,239,131)(5,75,240,132)(6,76,241,133)(7,77,242,134)(8,78,243,135)(9,79,244,136)(10,80,245,137)(11,81,246,138)(12,82,247,139)(13,83,248,140)(14,84,218,141)(15,85,219,142)(16,86,220,143)(17,87,221,144)(18,88,222,145)(19,89,223,146)(20,90,224,147)(21,91,225,148)(22,92,226,149)(23,93,227,150)(24,63,228,151)(25,64,229,152)(26,65,230,153)(27,66,231,154)(28,67,232,155)(29,68,233,125)(30,69,234,126)(31,70,235,127)(32,213,124,176)(33,214,94,177)(34,215,95,178)(35,216,96,179)(36,217,97,180)(37,187,98,181)(38,188,99,182)(39,189,100,183)(40,190,101,184)(41,191,102,185)(42,192,103,186)(43,193,104,156)(44,194,105,157)(45,195,106,158)(46,196,107,159)(47,197,108,160)(48,198,109,161)(49,199,110,162)(50,200,111,163)(51,201,112,164)(52,202,113,165)(53,203,114,166)(54,204,115,167)(55,205,116,168)(56,206,117,169)(57,207,118,170)(58,208,119,171)(59,209,120,172)(60,210,121,173)(61,211,122,174)(62,212,123,175), (1,176,236,213)(2,177,237,214)(3,178,238,215)(4,179,239,216)(5,180,240,217)(6,181,241,187)(7,182,242,188)(8,183,243,189)(9,184,244,190)(10,185,245,191)(11,186,246,192)(12,156,247,193)(13,157,248,194)(14,158,218,195)(15,159,219,196)(16,160,220,197)(17,161,221,198)(18,162,222,199)(19,163,223,200)(20,164,224,201)(21,165,225,202)(22,166,226,203)(23,167,227,204)(24,168,228,205)(25,169,229,206)(26,170,230,207)(27,171,231,208)(28,172,232,209)(29,173,233,210)(30,174,234,211)(31,175,235,212)(32,128,124,71)(33,129,94,72)(34,130,95,73)(35,131,96,74)(36,132,97,75)(37,133,98,76)(38,134,99,77)(39,135,100,78)(40,136,101,79)(41,137,102,80)(42,138,103,81)(43,139,104,82)(44,140,105,83)(45,141,106,84)(46,142,107,85)(47,143,108,86)(48,144,109,87)(49,145,110,88)(50,146,111,89)(51,147,112,90)(52,148,113,91)(53,149,114,92)(54,150,115,93)(55,151,116,63)(56,152,117,64)(57,153,118,65)(58,154,119,66)(59,155,120,67)(60,125,121,68)(61,126,122,69)(62,127,123,70), (32,124)(33,94)(34,95)(35,96)(36,97)(37,98)(38,99)(39,100)(40,101)(41,102)(42,103)(43,104)(44,105)(45,106)(46,107)(47,108)(48,109)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,121)(61,122)(62,123)(156,193)(157,194)(158,195)(159,196)(160,197)(161,198)(162,199)(163,200)(164,201)(165,202)(166,203)(167,204)(168,205)(169,206)(170,207)(171,208)(172,209)(173,210)(174,211)(175,212)(176,213)(177,214)(178,215)(179,216)(180,217)(181,187)(182,188)(183,189)(184,190)(185,191)(186,192) );
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)], [(1,71,236,128),(2,72,237,129),(3,73,238,130),(4,74,239,131),(5,75,240,132),(6,76,241,133),(7,77,242,134),(8,78,243,135),(9,79,244,136),(10,80,245,137),(11,81,246,138),(12,82,247,139),(13,83,248,140),(14,84,218,141),(15,85,219,142),(16,86,220,143),(17,87,221,144),(18,88,222,145),(19,89,223,146),(20,90,224,147),(21,91,225,148),(22,92,226,149),(23,93,227,150),(24,63,228,151),(25,64,229,152),(26,65,230,153),(27,66,231,154),(28,67,232,155),(29,68,233,125),(30,69,234,126),(31,70,235,127),(32,213,124,176),(33,214,94,177),(34,215,95,178),(35,216,96,179),(36,217,97,180),(37,187,98,181),(38,188,99,182),(39,189,100,183),(40,190,101,184),(41,191,102,185),(42,192,103,186),(43,193,104,156),(44,194,105,157),(45,195,106,158),(46,196,107,159),(47,197,108,160),(48,198,109,161),(49,199,110,162),(50,200,111,163),(51,201,112,164),(52,202,113,165),(53,203,114,166),(54,204,115,167),(55,205,116,168),(56,206,117,169),(57,207,118,170),(58,208,119,171),(59,209,120,172),(60,210,121,173),(61,211,122,174),(62,212,123,175)], [(1,176,236,213),(2,177,237,214),(3,178,238,215),(4,179,239,216),(5,180,240,217),(6,181,241,187),(7,182,242,188),(8,183,243,189),(9,184,244,190),(10,185,245,191),(11,186,246,192),(12,156,247,193),(13,157,248,194),(14,158,218,195),(15,159,219,196),(16,160,220,197),(17,161,221,198),(18,162,222,199),(19,163,223,200),(20,164,224,201),(21,165,225,202),(22,166,226,203),(23,167,227,204),(24,168,228,205),(25,169,229,206),(26,170,230,207),(27,171,231,208),(28,172,232,209),(29,173,233,210),(30,174,234,211),(31,175,235,212),(32,128,124,71),(33,129,94,72),(34,130,95,73),(35,131,96,74),(36,132,97,75),(37,133,98,76),(38,134,99,77),(39,135,100,78),(40,136,101,79),(41,137,102,80),(42,138,103,81),(43,139,104,82),(44,140,105,83),(45,141,106,84),(46,142,107,85),(47,143,108,86),(48,144,109,87),(49,145,110,88),(50,146,111,89),(51,147,112,90),(52,148,113,91),(53,149,114,92),(54,150,115,93),(55,151,116,63),(56,152,117,64),(57,153,118,65),(58,154,119,66),(59,155,120,67),(60,125,121,68),(61,126,122,69),(62,127,123,70)], [(32,124),(33,94),(34,95),(35,96),(36,97),(37,98),(38,99),(39,100),(40,101),(41,102),(42,103),(43,104),(44,105),(45,106),(46,107),(47,108),(48,109),(49,110),(50,111),(51,112),(52,113),(53,114),(54,115),(55,116),(56,117),(57,118),(58,119),(59,120),(60,121),(61,122),(62,123),(156,193),(157,194),(158,195),(159,196),(160,197),(161,198),(162,199),(163,200),(164,201),(165,202),(166,203),(167,204),(168,205),(169,206),(170,207),(171,208),(172,209),(173,210),(174,211),(175,212),(176,213),(177,214),(178,215),(179,216),(180,217),(181,187),(182,188),(183,189),(184,190),(185,191),(186,192)]])
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
Export