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G = C52×C10order 250 = 2·53

Abelian group of type [5,5,10]

direct product, abelian, monomial, 5-elementary

Aliases: C52×C10, SmallGroup(250,15)

Series: Derived Chief Lower central Upper central

C1 — C52×C10
C1C5C52C53 — C52×C10
C1 — C52×C10
C1 — C52×C10

Generators and relations for C52×C10
 G = < a,b,c | a5=b5=c10=1, ab=ba, ac=ca, bc=cb >

Subgroups: 128, all normal (4 characteristic)
C1, C2, C5 [×31], C10 [×31], C52 [×31], C5×C10 [×31], C53, C52×C10
Quotients: C1, C2, C5 [×31], C10 [×31], C52 [×31], C5×C10 [×31], C53, C52×C10

Smallest permutation representation of C52×C10
Regular action on 250 points
Generators in S250
(1 233 55 95 138)(2 234 56 96 139)(3 235 57 97 140)(4 236 58 98 131)(5 237 59 99 132)(6 238 60 100 133)(7 239 51 91 134)(8 240 52 92 135)(9 231 53 93 136)(10 232 54 94 137)(11 187 229 36 81)(12 188 230 37 82)(13 189 221 38 83)(14 190 222 39 84)(15 181 223 40 85)(16 182 224 31 86)(17 183 225 32 87)(18 184 226 33 88)(19 185 227 34 89)(20 186 228 35 90)(21 66 111 41 209)(22 67 112 42 210)(23 68 113 43 201)(24 69 114 44 202)(25 70 115 45 203)(26 61 116 46 204)(27 62 117 47 205)(28 63 118 48 206)(29 64 119 49 207)(30 65 120 50 208)(71 249 164 219 149)(72 250 165 220 150)(73 241 166 211 141)(74 242 167 212 142)(75 243 168 213 143)(76 244 169 214 144)(77 245 170 215 145)(78 246 161 216 146)(79 247 162 217 147)(80 248 163 218 148)(101 159 199 127 171)(102 160 200 128 172)(103 151 191 129 173)(104 152 192 130 174)(105 153 193 121 175)(106 154 194 122 176)(107 155 195 123 177)(108 156 196 124 178)(109 157 197 125 179)(110 158 198 126 180)
(1 193 25 75 15)(2 194 26 76 16)(3 195 27 77 17)(4 196 28 78 18)(5 197 29 79 19)(6 198 30 80 20)(7 199 21 71 11)(8 200 22 72 12)(9 191 23 73 13)(10 192 24 74 14)(31 96 106 46 214)(32 97 107 47 215)(33 98 108 48 216)(34 99 109 49 217)(35 100 110 50 218)(36 91 101 41 219)(37 92 102 42 220)(38 93 103 43 211)(39 94 104 44 212)(40 95 105 45 213)(51 171 111 164 229)(52 172 112 165 230)(53 173 113 166 221)(54 174 114 167 222)(55 175 115 168 223)(56 176 116 169 224)(57 177 117 170 225)(58 178 118 161 226)(59 179 119 162 227)(60 180 120 163 228)(61 244 182 234 122)(62 245 183 235 123)(63 246 184 236 124)(64 247 185 237 125)(65 248 186 238 126)(66 249 187 239 127)(67 250 188 240 128)(68 241 189 231 129)(69 242 190 232 130)(70 243 181 233 121)(81 134 159 209 149)(82 135 160 210 150)(83 136 151 201 141)(84 137 152 202 142)(85 138 153 203 143)(86 139 154 204 144)(87 140 155 205 145)(88 131 156 206 146)(89 132 157 207 147)(90 133 158 208 148)
(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)(231 232 233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248 249 250)

G:=sub<Sym(250)| (1,233,55,95,138)(2,234,56,96,139)(3,235,57,97,140)(4,236,58,98,131)(5,237,59,99,132)(6,238,60,100,133)(7,239,51,91,134)(8,240,52,92,135)(9,231,53,93,136)(10,232,54,94,137)(11,187,229,36,81)(12,188,230,37,82)(13,189,221,38,83)(14,190,222,39,84)(15,181,223,40,85)(16,182,224,31,86)(17,183,225,32,87)(18,184,226,33,88)(19,185,227,34,89)(20,186,228,35,90)(21,66,111,41,209)(22,67,112,42,210)(23,68,113,43,201)(24,69,114,44,202)(25,70,115,45,203)(26,61,116,46,204)(27,62,117,47,205)(28,63,118,48,206)(29,64,119,49,207)(30,65,120,50,208)(71,249,164,219,149)(72,250,165,220,150)(73,241,166,211,141)(74,242,167,212,142)(75,243,168,213,143)(76,244,169,214,144)(77,245,170,215,145)(78,246,161,216,146)(79,247,162,217,147)(80,248,163,218,148)(101,159,199,127,171)(102,160,200,128,172)(103,151,191,129,173)(104,152,192,130,174)(105,153,193,121,175)(106,154,194,122,176)(107,155,195,123,177)(108,156,196,124,178)(109,157,197,125,179)(110,158,198,126,180), (1,193,25,75,15)(2,194,26,76,16)(3,195,27,77,17)(4,196,28,78,18)(5,197,29,79,19)(6,198,30,80,20)(7,199,21,71,11)(8,200,22,72,12)(9,191,23,73,13)(10,192,24,74,14)(31,96,106,46,214)(32,97,107,47,215)(33,98,108,48,216)(34,99,109,49,217)(35,100,110,50,218)(36,91,101,41,219)(37,92,102,42,220)(38,93,103,43,211)(39,94,104,44,212)(40,95,105,45,213)(51,171,111,164,229)(52,172,112,165,230)(53,173,113,166,221)(54,174,114,167,222)(55,175,115,168,223)(56,176,116,169,224)(57,177,117,170,225)(58,178,118,161,226)(59,179,119,162,227)(60,180,120,163,228)(61,244,182,234,122)(62,245,183,235,123)(63,246,184,236,124)(64,247,185,237,125)(65,248,186,238,126)(66,249,187,239,127)(67,250,188,240,128)(68,241,189,231,129)(69,242,190,232,130)(70,243,181,233,121)(81,134,159,209,149)(82,135,160,210,150)(83,136,151,201,141)(84,137,152,202,142)(85,138,153,203,143)(86,139,154,204,144)(87,140,155,205,145)(88,131,156,206,146)(89,132,157,207,147)(90,133,158,208,148), (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)(231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250)>;

G:=Group( (1,233,55,95,138)(2,234,56,96,139)(3,235,57,97,140)(4,236,58,98,131)(5,237,59,99,132)(6,238,60,100,133)(7,239,51,91,134)(8,240,52,92,135)(9,231,53,93,136)(10,232,54,94,137)(11,187,229,36,81)(12,188,230,37,82)(13,189,221,38,83)(14,190,222,39,84)(15,181,223,40,85)(16,182,224,31,86)(17,183,225,32,87)(18,184,226,33,88)(19,185,227,34,89)(20,186,228,35,90)(21,66,111,41,209)(22,67,112,42,210)(23,68,113,43,201)(24,69,114,44,202)(25,70,115,45,203)(26,61,116,46,204)(27,62,117,47,205)(28,63,118,48,206)(29,64,119,49,207)(30,65,120,50,208)(71,249,164,219,149)(72,250,165,220,150)(73,241,166,211,141)(74,242,167,212,142)(75,243,168,213,143)(76,244,169,214,144)(77,245,170,215,145)(78,246,161,216,146)(79,247,162,217,147)(80,248,163,218,148)(101,159,199,127,171)(102,160,200,128,172)(103,151,191,129,173)(104,152,192,130,174)(105,153,193,121,175)(106,154,194,122,176)(107,155,195,123,177)(108,156,196,124,178)(109,157,197,125,179)(110,158,198,126,180), (1,193,25,75,15)(2,194,26,76,16)(3,195,27,77,17)(4,196,28,78,18)(5,197,29,79,19)(6,198,30,80,20)(7,199,21,71,11)(8,200,22,72,12)(9,191,23,73,13)(10,192,24,74,14)(31,96,106,46,214)(32,97,107,47,215)(33,98,108,48,216)(34,99,109,49,217)(35,100,110,50,218)(36,91,101,41,219)(37,92,102,42,220)(38,93,103,43,211)(39,94,104,44,212)(40,95,105,45,213)(51,171,111,164,229)(52,172,112,165,230)(53,173,113,166,221)(54,174,114,167,222)(55,175,115,168,223)(56,176,116,169,224)(57,177,117,170,225)(58,178,118,161,226)(59,179,119,162,227)(60,180,120,163,228)(61,244,182,234,122)(62,245,183,235,123)(63,246,184,236,124)(64,247,185,237,125)(65,248,186,238,126)(66,249,187,239,127)(67,250,188,240,128)(68,241,189,231,129)(69,242,190,232,130)(70,243,181,233,121)(81,134,159,209,149)(82,135,160,210,150)(83,136,151,201,141)(84,137,152,202,142)(85,138,153,203,143)(86,139,154,204,144)(87,140,155,205,145)(88,131,156,206,146)(89,132,157,207,147)(90,133,158,208,148), (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)(231,232,233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248,249,250) );

G=PermutationGroup([(1,233,55,95,138),(2,234,56,96,139),(3,235,57,97,140),(4,236,58,98,131),(5,237,59,99,132),(6,238,60,100,133),(7,239,51,91,134),(8,240,52,92,135),(9,231,53,93,136),(10,232,54,94,137),(11,187,229,36,81),(12,188,230,37,82),(13,189,221,38,83),(14,190,222,39,84),(15,181,223,40,85),(16,182,224,31,86),(17,183,225,32,87),(18,184,226,33,88),(19,185,227,34,89),(20,186,228,35,90),(21,66,111,41,209),(22,67,112,42,210),(23,68,113,43,201),(24,69,114,44,202),(25,70,115,45,203),(26,61,116,46,204),(27,62,117,47,205),(28,63,118,48,206),(29,64,119,49,207),(30,65,120,50,208),(71,249,164,219,149),(72,250,165,220,150),(73,241,166,211,141),(74,242,167,212,142),(75,243,168,213,143),(76,244,169,214,144),(77,245,170,215,145),(78,246,161,216,146),(79,247,162,217,147),(80,248,163,218,148),(101,159,199,127,171),(102,160,200,128,172),(103,151,191,129,173),(104,152,192,130,174),(105,153,193,121,175),(106,154,194,122,176),(107,155,195,123,177),(108,156,196,124,178),(109,157,197,125,179),(110,158,198,126,180)], [(1,193,25,75,15),(2,194,26,76,16),(3,195,27,77,17),(4,196,28,78,18),(5,197,29,79,19),(6,198,30,80,20),(7,199,21,71,11),(8,200,22,72,12),(9,191,23,73,13),(10,192,24,74,14),(31,96,106,46,214),(32,97,107,47,215),(33,98,108,48,216),(34,99,109,49,217),(35,100,110,50,218),(36,91,101,41,219),(37,92,102,42,220),(38,93,103,43,211),(39,94,104,44,212),(40,95,105,45,213),(51,171,111,164,229),(52,172,112,165,230),(53,173,113,166,221),(54,174,114,167,222),(55,175,115,168,223),(56,176,116,169,224),(57,177,117,170,225),(58,178,118,161,226),(59,179,119,162,227),(60,180,120,163,228),(61,244,182,234,122),(62,245,183,235,123),(63,246,184,236,124),(64,247,185,237,125),(65,248,186,238,126),(66,249,187,239,127),(67,250,188,240,128),(68,241,189,231,129),(69,242,190,232,130),(70,243,181,233,121),(81,134,159,209,149),(82,135,160,210,150),(83,136,151,201,141),(84,137,152,202,142),(85,138,153,203,143),(86,139,154,204,144),(87,140,155,205,145),(88,131,156,206,146),(89,132,157,207,147),(90,133,158,208,148)], [(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),(231,232,233,234,235,236,237,238,239,240),(241,242,243,244,245,246,247,248,249,250)])

C52×C10 is a maximal subgroup of   C5312C4

250 conjugacy classes

class 1  2 5A···5DT10A···10DT
order125···510···10
size111···11···1

250 irreducible representations

dim1111
type++
imageC1C2C5C10
kernelC52×C10C53C5×C10C52
# reps11124124

Matrix representation of C52×C10 in GL3(𝔽11) generated by

900
010
001
,
100
030
005
,
100
090
007
G:=sub<GL(3,GF(11))| [9,0,0,0,1,0,0,0,1],[1,0,0,0,3,0,0,0,5],[1,0,0,0,9,0,0,0,7] >;

C52×C10 in GAP, Magma, Sage, TeX

C_5^2\times C_{10}
% in TeX

G:=Group("C5^2xC10");
// GroupNames label

G:=SmallGroup(250,15);
// by ID

G=gap.SmallGroup(250,15);
# by ID

G:=PCGroup([4,-2,-5,-5,-5]);
// Polycyclic

G:=Group<a,b,c|a^5=b^5=c^10=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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