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G = C152order 225 = 32·52

Abelian group of type [15,15]

direct product, abelian, monomial

Aliases: C152, SmallGroup(225,6)

Series: Derived Chief Lower central Upper central

C1 — C152
C1C5C52C5×C15 — 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 46 77 135 34 136 188 69 102 208 173 21 214 151 118)(2 47 78 121 35 137 189 70 103 209 174 22 215 152 119)(3 48 79 122 36 138 190 71 104 210 175 23 216 153 120)(4 49 80 123 37 139 191 72 105 196 176 24 217 154 106)(5 50 81 124 38 140 192 73 91 197 177 25 218 155 107)(6 51 82 125 39 141 193 74 92 198 178 26 219 156 108)(7 52 83 126 40 142 194 75 93 199 179 27 220 157 109)(8 53 84 127 41 143 195 61 94 200 180 28 221 158 110)(9 54 85 128 42 144 181 62 95 201 166 29 222 159 111)(10 55 86 129 43 145 182 63 96 202 167 30 223 160 112)(11 56 87 130 44 146 183 64 97 203 168 16 224 161 113)(12 57 88 131 45 147 184 65 98 204 169 17 225 162 114)(13 58 89 132 31 148 185 66 99 205 170 18 211 163 115)(14 59 90 133 32 149 186 67 100 206 171 19 212 164 116)(15 60 76 134 33 150 187 68 101 207 172 20 213 165 117)

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

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

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

C152 is a maximal subgroup of   C15⋊D15

225 conjugacy classes

class 1 3A···3H5A···5X15A···15GJ
order13···35···515···15
size11···11···11···1

225 irreducible representations

dim1111
type+
imageC1C3C5C15
kernelC152C5×C15C3×C15C15
# reps1824192

Matrix representation of C152 in GL2(𝔽31) generated by

180
05
,
20
018
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

Export

Subgroup lattice of C152 in TeX

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