direct product, abelian, monomial
Aliases: C152, SmallGroup(225,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C152 |
| C1 — C152 |
| C1 — C152 |
Generators and relations for C152
G = < a,b | a15=b15=1, ab=ba >
Smallest permutation representation of C152
►Regular action on 225 points
Generators in S225
(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)
(1 117 73 217 209 80 149 188 46 124 32 151 101 26 172)(2 118 74 218 210 81 150 189 47 125 33 152 102 27 173)(3 119 75 219 196 82 136 190 48 126 34 153 103 28 174)(4 120 61 220 197 83 137 191 49 127 35 154 104 29 175)(5 106 62 221 198 84 138 192 50 128 36 155 105 30 176)(6 107 63 222 199 85 139 193 51 129 37 156 91 16 177)(7 108 64 223 200 86 140 194 52 130 38 157 92 17 178)(8 109 65 224 201 87 141 195 53 131 39 158 93 18 179)(9 110 66 225 202 88 142 181 54 132 40 159 94 19 180)(10 111 67 211 203 89 143 182 55 133 41 160 95 20 166)(11 112 68 212 204 90 144 183 56 134 42 161 96 21 167)(12 113 69 213 205 76 145 184 57 135 43 162 97 22 168)(13 114 70 214 206 77 146 185 58 121 44 163 98 23 169)(14 115 71 215 207 78 147 186 59 122 45 164 99 24 170)(15 116 72 216 208 79 148 187 60 123 31 165 100 25 171)
G:=sub<Sym(225)| (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), (1,117,73,217,209,80,149,188,46,124,32,151,101,26,172)(2,118,74,218,210,81,150,189,47,125,33,152,102,27,173)(3,119,75,219,196,82,136,190,48,126,34,153,103,28,174)(4,120,61,220,197,83,137,191,49,127,35,154,104,29,175)(5,106,62,221,198,84,138,192,50,128,36,155,105,30,176)(6,107,63,222,199,85,139,193,51,129,37,156,91,16,177)(7,108,64,223,200,86,140,194,52,130,38,157,92,17,178)(8,109,65,224,201,87,141,195,53,131,39,158,93,18,179)(9,110,66,225,202,88,142,181,54,132,40,159,94,19,180)(10,111,67,211,203,89,143,182,55,133,41,160,95,20,166)(11,112,68,212,204,90,144,183,56,134,42,161,96,21,167)(12,113,69,213,205,76,145,184,57,135,43,162,97,22,168)(13,114,70,214,206,77,146,185,58,121,44,163,98,23,169)(14,115,71,215,207,78,147,186,59,122,45,164,99,24,170)(15,116,72,216,208,79,148,187,60,123,31,165,100,25,171)>;
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), (1,117,73,217,209,80,149,188,46,124,32,151,101,26,172)(2,118,74,218,210,81,150,189,47,125,33,152,102,27,173)(3,119,75,219,196,82,136,190,48,126,34,153,103,28,174)(4,120,61,220,197,83,137,191,49,127,35,154,104,29,175)(5,106,62,221,198,84,138,192,50,128,36,155,105,30,176)(6,107,63,222,199,85,139,193,51,129,37,156,91,16,177)(7,108,64,223,200,86,140,194,52,130,38,157,92,17,178)(8,109,65,224,201,87,141,195,53,131,39,158,93,18,179)(9,110,66,225,202,88,142,181,54,132,40,159,94,19,180)(10,111,67,211,203,89,143,182,55,133,41,160,95,20,166)(11,112,68,212,204,90,144,183,56,134,42,161,96,21,167)(12,113,69,213,205,76,145,184,57,135,43,162,97,22,168)(13,114,70,214,206,77,146,185,58,121,44,163,98,23,169)(14,115,71,215,207,78,147,186,59,122,45,164,99,24,170)(15,116,72,216,208,79,148,187,60,123,31,165,100,25,171) );
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)], [(1,117,73,217,209,80,149,188,46,124,32,151,101,26,172),(2,118,74,218,210,81,150,189,47,125,33,152,102,27,173),(3,119,75,219,196,82,136,190,48,126,34,153,103,28,174),(4,120,61,220,197,83,137,191,49,127,35,154,104,29,175),(5,106,62,221,198,84,138,192,50,128,36,155,105,30,176),(6,107,63,222,199,85,139,193,51,129,37,156,91,16,177),(7,108,64,223,200,86,140,194,52,130,38,157,92,17,178),(8,109,65,224,201,87,141,195,53,131,39,158,93,18,179),(9,110,66,225,202,88,142,181,54,132,40,159,94,19,180),(10,111,67,211,203,89,143,182,55,133,41,160,95,20,166),(11,112,68,212,204,90,144,183,56,134,42,161,96,21,167),(12,113,69,213,205,76,145,184,57,135,43,162,97,22,168),(13,114,70,214,206,77,146,185,58,121,44,163,98,23,169),(14,115,71,215,207,78,147,186,59,122,45,164,99,24,170),(15,116,72,216,208,79,148,187,60,123,31,165,100,25,171)]])
C152 is a maximal subgroup of
C15⋊D15
225 conjugacy classes
| class | 1 | 3A | ··· | 3H | 5A | ··· | 5X | 15A | ··· | 15GJ |
| order | 1 | 3 | ··· | 3 | 5 | ··· | 5 | 15 | ··· | 15 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
225 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C5 | C15 |
| kernel | C152 | C5×C15 | C3×C15 | C15 |
| # reps | 1 | 8 | 24 | 192 |
Matrix representation of C152 ►in GL2(𝔽31) generated by
| 18 | 0 |
| 0 | 5 |
| 2 | 0 |
| 0 | 18 |
G:=sub<GL(2,GF(31))| [18,0,0,5],[2,0,0,18] >;
C152 in GAP, Magma, Sage, TeX
C_{15}^2 % in TeX
G:=Group("C15^2"); // GroupNames label
G:=SmallGroup(225,6);
// by ID
G=gap.SmallGroup(225,6);
# by ID
G:=PCGroup([4,-3,-3,-5,-5]);
// Polycyclic
G:=Group<a,b|a^15=b^15=1,a*b=b*a>;
// generators/relations
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