metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C116.1C4, D58.3C4, C29⋊1M4(2), Dic29.5C22, C29⋊C8⋊1C2, C4.(C29⋊C4), C58.2(C2×C4), (C4×D29).3C2, C2.4(C2×C29⋊C4), SmallGroup(464,29)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C116.C4
G = < a,b | a116=1, b4=a58, bab-1=a75 >
Smallest permutation representation of C116.C4
►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)
(1 194 88 223 59 136 30 165)(2 177 29 182 60 119 87 124)(3 160 86 141 61 218 28 199)(4 143 27 216 62 201 85 158)(5 126 84 175 63 184 26 117)(6 225 25 134 64 167 83 192)(7 208 82 209 65 150 24 151)(8 191 23 168 66 133 81 226)(9 174 80 127 67 232 22 185)(10 157 21 202 68 215 79 144)(11 140 78 161 69 198 20 219)(12 123 19 120 70 181 77 178)(13 222 76 195 71 164 18 137)(14 205 17 154 72 147 75 212)(15 188 74 229 73 130 16 171)(31 148 58 153 89 206 116 211)(32 131 115 228 90 189 57 170)(33 230 56 187 91 172 114 129)(34 213 113 146 92 155 55 204)(35 196 54 221 93 138 112 163)(36 179 111 180 94 121 53 122)(37 162 52 139 95 220 110 197)(38 145 109 214 96 203 51 156)(39 128 50 173 97 186 108 231)(40 227 107 132 98 169 49 190)(41 210 48 207 99 152 106 149)(42 193 105 166 100 135 47 224)(43 176 46 125 101 118 104 183)(44 159 103 200 102 217 45 142)
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), (1,194,88,223,59,136,30,165)(2,177,29,182,60,119,87,124)(3,160,86,141,61,218,28,199)(4,143,27,216,62,201,85,158)(5,126,84,175,63,184,26,117)(6,225,25,134,64,167,83,192)(7,208,82,209,65,150,24,151)(8,191,23,168,66,133,81,226)(9,174,80,127,67,232,22,185)(10,157,21,202,68,215,79,144)(11,140,78,161,69,198,20,219)(12,123,19,120,70,181,77,178)(13,222,76,195,71,164,18,137)(14,205,17,154,72,147,75,212)(15,188,74,229,73,130,16,171)(31,148,58,153,89,206,116,211)(32,131,115,228,90,189,57,170)(33,230,56,187,91,172,114,129)(34,213,113,146,92,155,55,204)(35,196,54,221,93,138,112,163)(36,179,111,180,94,121,53,122)(37,162,52,139,95,220,110,197)(38,145,109,214,96,203,51,156)(39,128,50,173,97,186,108,231)(40,227,107,132,98,169,49,190)(41,210,48,207,99,152,106,149)(42,193,105,166,100,135,47,224)(43,176,46,125,101,118,104,183)(44,159,103,200,102,217,45,142)>;
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), (1,194,88,223,59,136,30,165)(2,177,29,182,60,119,87,124)(3,160,86,141,61,218,28,199)(4,143,27,216,62,201,85,158)(5,126,84,175,63,184,26,117)(6,225,25,134,64,167,83,192)(7,208,82,209,65,150,24,151)(8,191,23,168,66,133,81,226)(9,174,80,127,67,232,22,185)(10,157,21,202,68,215,79,144)(11,140,78,161,69,198,20,219)(12,123,19,120,70,181,77,178)(13,222,76,195,71,164,18,137)(14,205,17,154,72,147,75,212)(15,188,74,229,73,130,16,171)(31,148,58,153,89,206,116,211)(32,131,115,228,90,189,57,170)(33,230,56,187,91,172,114,129)(34,213,113,146,92,155,55,204)(35,196,54,221,93,138,112,163)(36,179,111,180,94,121,53,122)(37,162,52,139,95,220,110,197)(38,145,109,214,96,203,51,156)(39,128,50,173,97,186,108,231)(40,227,107,132,98,169,49,190)(41,210,48,207,99,152,106,149)(42,193,105,166,100,135,47,224)(43,176,46,125,101,118,104,183)(44,159,103,200,102,217,45,142) );
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)], [(1,194,88,223,59,136,30,165),(2,177,29,182,60,119,87,124),(3,160,86,141,61,218,28,199),(4,143,27,216,62,201,85,158),(5,126,84,175,63,184,26,117),(6,225,25,134,64,167,83,192),(7,208,82,209,65,150,24,151),(8,191,23,168,66,133,81,226),(9,174,80,127,67,232,22,185),(10,157,21,202,68,215,79,144),(11,140,78,161,69,198,20,219),(12,123,19,120,70,181,77,178),(13,222,76,195,71,164,18,137),(14,205,17,154,72,147,75,212),(15,188,74,229,73,130,16,171),(31,148,58,153,89,206,116,211),(32,131,115,228,90,189,57,170),(33,230,56,187,91,172,114,129),(34,213,113,146,92,155,55,204),(35,196,54,221,93,138,112,163),(36,179,111,180,94,121,53,122),(37,162,52,139,95,220,110,197),(38,145,109,214,96,203,51,156),(39,128,50,173,97,186,108,231),(40,227,107,132,98,169,49,190),(41,210,48,207,99,152,106,149),(42,193,105,166,100,135,47,224),(43,176,46,125,101,118,104,183),(44,159,103,200,102,217,45,142)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 29A | ··· | 29G | 58A | ··· | 58G | 116A | ··· | 116N |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 58 | 2 | 29 | 29 | 58 | 58 | 58 | 58 | 4 | ··· | 4 | 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 | C116.C4 |
| kernel | C116.C4 | C29⋊C8 | C4×D29 | C116 | D58 | C29 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 7 | 7 | 14 |
Matrix representation of C116.C4 ►in GL6(𝔽233)
| 89 | 0 | 0 | 0 | 0 | 0 |
| 150 | 144 | 0 | 0 | 0 | 0 |
| 0 | 0 | 195 | 29 | 38 | 28 |
| 0 | 0 | 205 | 75 | 115 | 151 |
| 0 | 0 | 82 | 157 | 156 | 67 |
| 0 | 0 | 166 | 161 | 213 | 2 |
| 136 | 112 | 0 | 0 | 0 | 0 |
| 169 | 97 | 0 | 0 | 0 | 0 |
| 0 | 0 | 193 | 216 | 116 | 229 |
| 0 | 0 | 160 | 93 | 186 | 65 |
| 0 | 0 | 12 | 158 | 64 | 211 |
| 0 | 0 | 126 | 221 | 22 | 116 |
G:=sub<GL(6,GF(233))| [89,150,0,0,0,0,0,144,0,0,0,0,0,0,195,205,82,166,0,0,29,75,157,161,0,0,38,115,156,213,0,0,28,151,67,2],[136,169,0,0,0,0,112,97,0,0,0,0,0,0,193,160,12,126,0,0,216,93,158,221,0,0,116,186,64,22,0,0,229,65,211,116] >;
C116.C4 in GAP, Magma, Sage, TeX
C_{116}.C_4 % in TeX
G:=Group("C116.C4"); // GroupNames label
G:=SmallGroup(464,29);
// by ID
G=gap.SmallGroup(464,29);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-29,20,101,46,42,4804,2814]);
// Polycyclic
G:=Group<a,b|a^116=1,b^4=a^58,b*a*b^-1=a^75>;
// generators/relations
Export