metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C29⋊2C8, C58.2C4, C4.2D29, C2.Dic29, C116.2C2, SmallGroup(232,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C29 — C29⋊2C8 |
Generators and relations for C29⋊2C8
G = < a,b | a29=b8=1, bab-1=a-1 >
Smallest permutation representation of C29⋊2C8
►Regular action 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)
(1 204 94 146 33 175 69 117)(2 232 95 174 34 203 70 145)(3 231 96 173 35 202 71 144)(4 230 97 172 36 201 72 143)(5 229 98 171 37 200 73 142)(6 228 99 170 38 199 74 141)(7 227 100 169 39 198 75 140)(8 226 101 168 40 197 76 139)(9 225 102 167 41 196 77 138)(10 224 103 166 42 195 78 137)(11 223 104 165 43 194 79 136)(12 222 105 164 44 193 80 135)(13 221 106 163 45 192 81 134)(14 220 107 162 46 191 82 133)(15 219 108 161 47 190 83 132)(16 218 109 160 48 189 84 131)(17 217 110 159 49 188 85 130)(18 216 111 158 50 187 86 129)(19 215 112 157 51 186 87 128)(20 214 113 156 52 185 59 127)(21 213 114 155 53 184 60 126)(22 212 115 154 54 183 61 125)(23 211 116 153 55 182 62 124)(24 210 88 152 56 181 63 123)(25 209 89 151 57 180 64 122)(26 208 90 150 58 179 65 121)(27 207 91 149 30 178 66 120)(28 206 92 148 31 177 67 119)(29 205 93 147 32 176 68 118)
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), (1,204,94,146,33,175,69,117)(2,232,95,174,34,203,70,145)(3,231,96,173,35,202,71,144)(4,230,97,172,36,201,72,143)(5,229,98,171,37,200,73,142)(6,228,99,170,38,199,74,141)(7,227,100,169,39,198,75,140)(8,226,101,168,40,197,76,139)(9,225,102,167,41,196,77,138)(10,224,103,166,42,195,78,137)(11,223,104,165,43,194,79,136)(12,222,105,164,44,193,80,135)(13,221,106,163,45,192,81,134)(14,220,107,162,46,191,82,133)(15,219,108,161,47,190,83,132)(16,218,109,160,48,189,84,131)(17,217,110,159,49,188,85,130)(18,216,111,158,50,187,86,129)(19,215,112,157,51,186,87,128)(20,214,113,156,52,185,59,127)(21,213,114,155,53,184,60,126)(22,212,115,154,54,183,61,125)(23,211,116,153,55,182,62,124)(24,210,88,152,56,181,63,123)(25,209,89,151,57,180,64,122)(26,208,90,150,58,179,65,121)(27,207,91,149,30,178,66,120)(28,206,92,148,31,177,67,119)(29,205,93,147,32,176,68,118)>;
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), (1,204,94,146,33,175,69,117)(2,232,95,174,34,203,70,145)(3,231,96,173,35,202,71,144)(4,230,97,172,36,201,72,143)(5,229,98,171,37,200,73,142)(6,228,99,170,38,199,74,141)(7,227,100,169,39,198,75,140)(8,226,101,168,40,197,76,139)(9,225,102,167,41,196,77,138)(10,224,103,166,42,195,78,137)(11,223,104,165,43,194,79,136)(12,222,105,164,44,193,80,135)(13,221,106,163,45,192,81,134)(14,220,107,162,46,191,82,133)(15,219,108,161,47,190,83,132)(16,218,109,160,48,189,84,131)(17,217,110,159,49,188,85,130)(18,216,111,158,50,187,86,129)(19,215,112,157,51,186,87,128)(20,214,113,156,52,185,59,127)(21,213,114,155,53,184,60,126)(22,212,115,154,54,183,61,125)(23,211,116,153,55,182,62,124)(24,210,88,152,56,181,63,123)(25,209,89,151,57,180,64,122)(26,208,90,150,58,179,65,121)(27,207,91,149,30,178,66,120)(28,206,92,148,31,177,67,119)(29,205,93,147,32,176,68,118) );
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)], [(1,204,94,146,33,175,69,117),(2,232,95,174,34,203,70,145),(3,231,96,173,35,202,71,144),(4,230,97,172,36,201,72,143),(5,229,98,171,37,200,73,142),(6,228,99,170,38,199,74,141),(7,227,100,169,39,198,75,140),(8,226,101,168,40,197,76,139),(9,225,102,167,41,196,77,138),(10,224,103,166,42,195,78,137),(11,223,104,165,43,194,79,136),(12,222,105,164,44,193,80,135),(13,221,106,163,45,192,81,134),(14,220,107,162,46,191,82,133),(15,219,108,161,47,190,83,132),(16,218,109,160,48,189,84,131),(17,217,110,159,49,188,85,130),(18,216,111,158,50,187,86,129),(19,215,112,157,51,186,87,128),(20,214,113,156,52,185,59,127),(21,213,114,155,53,184,60,126),(22,212,115,154,54,183,61,125),(23,211,116,153,55,182,62,124),(24,210,88,152,56,181,63,123),(25,209,89,151,57,180,64,122),(26,208,90,150,58,179,65,121),(27,207,91,149,30,178,66,120),(28,206,92,148,31,177,67,119),(29,205,93,147,32,176,68,118)]])
C29⋊2C8 is a maximal subgroup of
C29⋊C16 C8×D29 C8⋊D29 C4.Dic29 D4⋊D29 D4.D29 Q8⋊D29 C29⋊Q16
C29⋊2C8 is a maximal quotient of
C29⋊2C16
64 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 29A | ··· | 29N | 58A | ··· | 58N | 116A | ··· | 116AB |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 1 | 1 | 29 | 29 | 29 | 29 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | - | |||
| image | C1 | C2 | C4 | C8 | D29 | Dic29 | C29⋊2C8 |
| kernel | C29⋊2C8 | C116 | C58 | C29 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 14 | 14 | 28 |
Matrix representation of C29⋊2C8 ►in GL3(𝔽233) generated by
| 1 | 0 | 0 |
| 0 | 232 | 1 |
| 0 | 112 | 120 |
| 12 | 0 | 0 |
| 0 | 184 | 106 |
| 0 | 140 | 49 |
G:=sub<GL(3,GF(233))| [1,0,0,0,232,112,0,1,120],[12,0,0,0,184,140,0,106,49] >;
C29⋊2C8 in GAP, Magma, Sage, TeX
C_{29}\rtimes_2C_8 % in TeX
G:=Group("C29:2C8"); // GroupNames label
G:=SmallGroup(232,1);
// by ID
G=gap.SmallGroup(232,1);
# by ID
G:=PCGroup([4,-2,-2,-2,-29,8,21,3587]);
// Polycyclic
G:=Group<a,b|a^29=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
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