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G = C5×C45order 225 = 32·52

Abelian group of type [5,45]

direct product, abelian, monomial, 5-elementary

Aliases: C5×C45, SmallGroup(225,4)

Series: Derived Chief Lower central Upper central

C1 — C5×C45
C1C3C15C5×C15 — C5×C45
C1 — C5×C45
C1 — C5×C45

Generators and relations for C5×C45
 G = < a,b | a5=b45=1, ab=ba >


Smallest permutation representation of C5×C45
Regular action on 225 points
Generators in S225
(1 116 79 222 163)(2 117 80 223 164)(3 118 81 224 165)(4 119 82 225 166)(5 120 83 181 167)(6 121 84 182 168)(7 122 85 183 169)(8 123 86 184 170)(9 124 87 185 171)(10 125 88 186 172)(11 126 89 187 173)(12 127 90 188 174)(13 128 46 189 175)(14 129 47 190 176)(15 130 48 191 177)(16 131 49 192 178)(17 132 50 193 179)(18 133 51 194 180)(19 134 52 195 136)(20 135 53 196 137)(21 91 54 197 138)(22 92 55 198 139)(23 93 56 199 140)(24 94 57 200 141)(25 95 58 201 142)(26 96 59 202 143)(27 97 60 203 144)(28 98 61 204 145)(29 99 62 205 146)(30 100 63 206 147)(31 101 64 207 148)(32 102 65 208 149)(33 103 66 209 150)(34 104 67 210 151)(35 105 68 211 152)(36 106 69 212 153)(37 107 70 213 154)(38 108 71 214 155)(39 109 72 215 156)(40 110 73 216 157)(41 111 74 217 158)(42 112 75 218 159)(43 113 76 219 160)(44 114 77 220 161)(45 115 78 221 162)
(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)

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

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

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

C5×C45 is a maximal subgroup of   C5⋊D45

225 conjugacy classes

class 1 3A3B5A···5X9A···9F15A···15AV45A···45EN
order1335···59···915···1545···45
size1111···11···11···11···1

225 irreducible representations

dim111111
type+
imageC1C3C5C9C15C45
kernelC5×C45C5×C15C45C52C15C5
# reps1224648144

Matrix representation of C5×C45 in GL2(𝔽181) generated by

420
0125
,
810
082
G:=sub<GL(2,GF(181))| [42,0,0,125],[81,0,0,82] >;

C5×C45 in GAP, Magma, Sage, TeX

C_5\times C_{45}
% in TeX

G:=Group("C5xC45");
// GroupNames label

G:=SmallGroup(225,4);
// by ID

G=gap.SmallGroup(225,4);
# by ID

G:=PCGroup([4,-3,-5,-5,-3,300]);
// Polycyclic

G:=Group<a,b|a^5=b^45=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C5×C45 in TeX

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