metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C29⋊2M4(2), Dic29.3C4, Dic29.7C22, C29⋊C8⋊2C2, (C2×C58).2C4, C58.6(C2×C4), C22.(C29⋊C4), (C2×Dic29).5C2, C2.6(C2×C29⋊C4), SmallGroup(464,33)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C29⋊M4(2)
G = < a,b,c | a29=b8=c2=1, bab-1=a17, ac=ca, cbc=b5 >
Smallest permutation representation of C29⋊M4(2)
►On 232 points
Generators in S232
(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)
(1 177 59 157 47 207 88 120)(2 189 87 174 48 219 116 137)(3 201 86 162 49 231 115 125)(4 184 85 150 50 214 114 142)(5 196 84 167 51 226 113 130)(6 179 83 155 52 209 112 118)(7 191 82 172 53 221 111 135)(8 203 81 160 54 204 110 123)(9 186 80 148 55 216 109 140)(10 198 79 165 56 228 108 128)(11 181 78 153 57 211 107 145)(12 193 77 170 58 223 106 133)(13 176 76 158 30 206 105 121)(14 188 75 146 31 218 104 138)(15 200 74 163 32 230 103 126)(16 183 73 151 33 213 102 143)(17 195 72 168 34 225 101 131)(18 178 71 156 35 208 100 119)(19 190 70 173 36 220 99 136)(20 202 69 161 37 232 98 124)(21 185 68 149 38 215 97 141)(22 197 67 166 39 227 96 129)(23 180 66 154 40 210 95 117)(24 192 65 171 41 222 94 134)(25 175 64 159 42 205 93 122)(26 187 63 147 43 217 92 139)(27 199 62 164 44 229 91 127)(28 182 61 152 45 212 90 144)(29 194 60 169 46 224 89 132)
(117 154)(118 155)(119 156)(120 157)(121 158)(122 159)(123 160)(124 161)(125 162)(126 163)(127 164)(128 165)(129 166)(130 167)(131 168)(132 169)(133 170)(134 171)(135 172)(136 173)(137 174)(138 146)(139 147)(140 148)(141 149)(142 150)(143 151)(144 152)(145 153)(175 205)(176 206)(177 207)(178 208)(179 209)(180 210)(181 211)(182 212)(183 213)(184 214)(185 215)(186 216)(187 217)(188 218)(189 219)(190 220)(191 221)(192 222)(193 223)(194 224)(195 225)(196 226)(197 227)(198 228)(199 229)(200 230)(201 231)(202 232)(203 204)
G:=sub<Sym(232)| (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), (1,177,59,157,47,207,88,120)(2,189,87,174,48,219,116,137)(3,201,86,162,49,231,115,125)(4,184,85,150,50,214,114,142)(5,196,84,167,51,226,113,130)(6,179,83,155,52,209,112,118)(7,191,82,172,53,221,111,135)(8,203,81,160,54,204,110,123)(9,186,80,148,55,216,109,140)(10,198,79,165,56,228,108,128)(11,181,78,153,57,211,107,145)(12,193,77,170,58,223,106,133)(13,176,76,158,30,206,105,121)(14,188,75,146,31,218,104,138)(15,200,74,163,32,230,103,126)(16,183,73,151,33,213,102,143)(17,195,72,168,34,225,101,131)(18,178,71,156,35,208,100,119)(19,190,70,173,36,220,99,136)(20,202,69,161,37,232,98,124)(21,185,68,149,38,215,97,141)(22,197,67,166,39,227,96,129)(23,180,66,154,40,210,95,117)(24,192,65,171,41,222,94,134)(25,175,64,159,42,205,93,122)(26,187,63,147,43,217,92,139)(27,199,62,164,44,229,91,127)(28,182,61,152,45,212,90,144)(29,194,60,169,46,224,89,132), (117,154)(118,155)(119,156)(120,157)(121,158)(122,159)(123,160)(124,161)(125,162)(126,163)(127,164)(128,165)(129,166)(130,167)(131,168)(132,169)(133,170)(134,171)(135,172)(136,173)(137,174)(138,146)(139,147)(140,148)(141,149)(142,150)(143,151)(144,152)(145,153)(175,205)(176,206)(177,207)(178,208)(179,209)(180,210)(181,211)(182,212)(183,213)(184,214)(185,215)(186,216)(187,217)(188,218)(189,219)(190,220)(191,221)(192,222)(193,223)(194,224)(195,225)(196,226)(197,227)(198,228)(199,229)(200,230)(201,231)(202,232)(203,204)>;
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), (1,177,59,157,47,207,88,120)(2,189,87,174,48,219,116,137)(3,201,86,162,49,231,115,125)(4,184,85,150,50,214,114,142)(5,196,84,167,51,226,113,130)(6,179,83,155,52,209,112,118)(7,191,82,172,53,221,111,135)(8,203,81,160,54,204,110,123)(9,186,80,148,55,216,109,140)(10,198,79,165,56,228,108,128)(11,181,78,153,57,211,107,145)(12,193,77,170,58,223,106,133)(13,176,76,158,30,206,105,121)(14,188,75,146,31,218,104,138)(15,200,74,163,32,230,103,126)(16,183,73,151,33,213,102,143)(17,195,72,168,34,225,101,131)(18,178,71,156,35,208,100,119)(19,190,70,173,36,220,99,136)(20,202,69,161,37,232,98,124)(21,185,68,149,38,215,97,141)(22,197,67,166,39,227,96,129)(23,180,66,154,40,210,95,117)(24,192,65,171,41,222,94,134)(25,175,64,159,42,205,93,122)(26,187,63,147,43,217,92,139)(27,199,62,164,44,229,91,127)(28,182,61,152,45,212,90,144)(29,194,60,169,46,224,89,132), (117,154)(118,155)(119,156)(120,157)(121,158)(122,159)(123,160)(124,161)(125,162)(126,163)(127,164)(128,165)(129,166)(130,167)(131,168)(132,169)(133,170)(134,171)(135,172)(136,173)(137,174)(138,146)(139,147)(140,148)(141,149)(142,150)(143,151)(144,152)(145,153)(175,205)(176,206)(177,207)(178,208)(179,209)(180,210)(181,211)(182,212)(183,213)(184,214)(185,215)(186,216)(187,217)(188,218)(189,219)(190,220)(191,221)(192,222)(193,223)(194,224)(195,225)(196,226)(197,227)(198,228)(199,229)(200,230)(201,231)(202,232)(203,204) );
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)], [(1,177,59,157,47,207,88,120),(2,189,87,174,48,219,116,137),(3,201,86,162,49,231,115,125),(4,184,85,150,50,214,114,142),(5,196,84,167,51,226,113,130),(6,179,83,155,52,209,112,118),(7,191,82,172,53,221,111,135),(8,203,81,160,54,204,110,123),(9,186,80,148,55,216,109,140),(10,198,79,165,56,228,108,128),(11,181,78,153,57,211,107,145),(12,193,77,170,58,223,106,133),(13,176,76,158,30,206,105,121),(14,188,75,146,31,218,104,138),(15,200,74,163,32,230,103,126),(16,183,73,151,33,213,102,143),(17,195,72,168,34,225,101,131),(18,178,71,156,35,208,100,119),(19,190,70,173,36,220,99,136),(20,202,69,161,37,232,98,124),(21,185,68,149,38,215,97,141),(22,197,67,166,39,227,96,129),(23,180,66,154,40,210,95,117),(24,192,65,171,41,222,94,134),(25,175,64,159,42,205,93,122),(26,187,63,147,43,217,92,139),(27,199,62,164,44,229,91,127),(28,182,61,152,45,212,90,144),(29,194,60,169,46,224,89,132)], [(117,154),(118,155),(119,156),(120,157),(121,158),(122,159),(123,160),(124,161),(125,162),(126,163),(127,164),(128,165),(129,166),(130,167),(131,168),(132,169),(133,170),(134,171),(135,172),(136,173),(137,174),(138,146),(139,147),(140,148),(141,149),(142,150),(143,151),(144,152),(145,153),(175,205),(176,206),(177,207),(178,208),(179,209),(180,210),(181,211),(182,212),(183,213),(184,214),(185,215),(186,216),(187,217),(188,218),(189,219),(190,220),(191,221),(192,222),(193,223),(194,224),(195,225),(196,226),(197,227),(198,228),(199,229),(200,230),(201,231),(202,232),(203,204)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 29A | ··· | 29G | 58A | ··· | 58U |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 29 | ··· | 29 | 58 | ··· | 58 |
| size | 1 | 1 | 2 | 29 | 29 | 58 | 58 | 58 | 58 | 58 | 4 | ··· | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C4 | C4 | M4(2) | C29⋊C4 | C2×C29⋊C4 | C29⋊M4(2) |
| kernel | C29⋊M4(2) | C29⋊C8 | C2×Dic29 | Dic29 | C2×C58 | C29 | C22 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 7 | 7 | 14 |
Matrix representation of C29⋊M4(2) ►in GL4(𝔽233) generated by
| 0 | 1 | 0 | 0 |
| 232 | 118 | 0 | 0 |
| 0 | 0 | 158 | 109 |
| 0 | 0 | 124 | 205 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 198 | 33 | 0 | 0 |
| 97 | 35 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 232 | 0 |
| 0 | 0 | 0 | 232 |
G:=sub<GL(4,GF(233))| [0,232,0,0,1,118,0,0,0,0,158,124,0,0,109,205],[0,0,198,97,0,0,33,35,1,0,0,0,0,1,0,0],[1,0,0,0,0,1,0,0,0,0,232,0,0,0,0,232] >;
C29⋊M4(2) in GAP, Magma, Sage, TeX
C_{29}\rtimes M_4(2) % in TeX
G:=Group("C29:M4(2)"); // GroupNames label
G:=SmallGroup(464,33);
// by ID
G=gap.SmallGroup(464,33);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-29,20,101,42,4804,2814]);
// Polycyclic
G:=Group<a,b,c|a^29=b^8=c^2=1,b*a*b^-1=a^17,a*c=c*a,c*b*c=b^5>;
// generators/relations
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