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G = C3⋊S3×C52order 450 = 2·32·52

Direct product of C52 and C3⋊S3

Aliases: C3⋊S3×C52, C1528C2, C3⋊(S3×C52), C153(C5×S3), (C5×C15)⋊7S3, (C3×C15)⋊5C10, C322(C5×C10), SmallGroup(450,31)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3×C52
 Chief series C1 — C3 — C32 — C3×C15 — C152 — C3⋊S3×C52
 Lower central C32 — C3⋊S3×C52
 Upper central C1 — C52

Generators and relations for C3⋊S3×C52
G = < a,b,c,d,e | a5=b5=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 224 in 96 conjugacy classes, 56 normal (6 characteristic)
C1, C2, C3, C5, S3, C32, C10, C15, C3⋊S3, C52, C5×S3, C3×C15, C5×C10, C5×C15, C5×C3⋊S3, S3×C52, C152, C3⋊S3×C52
Quotients: C1, C2, C5, S3, C10, C3⋊S3, C52, C5×S3, C5×C10, C5×C3⋊S3, S3×C52, C3⋊S3×C52

Smallest permutation representation of C3⋊S3×C52
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)
(1 148 105 62 14)(2 149 101 63 15)(3 150 102 64 11)(4 146 103 65 12)(5 147 104 61 13)(6 187 140 92 49)(7 188 136 93 50)(8 189 137 94 46)(9 190 138 95 47)(10 186 139 91 48)(16 193 155 58 110)(17 194 151 59 106)(18 195 152 60 107)(19 191 153 56 108)(20 192 154 57 109)(21 198 160 117 69)(22 199 156 118 70)(23 200 157 119 66)(24 196 158 120 67)(25 197 159 116 68)(26 203 111 163 74)(27 204 112 164 75)(28 205 113 165 71)(29 201 114 161 72)(30 202 115 162 73)(31 208 175 122 79)(32 209 171 123 80)(33 210 172 124 76)(34 206 173 125 77)(35 207 174 121 78)(36 169 216 127 84)(37 170 217 128 85)(38 166 218 129 81)(39 167 219 130 82)(40 168 220 126 83)(41 223 180 132 89)(42 224 176 133 90)(43 225 177 134 86)(44 221 178 135 87)(45 222 179 131 88)(51 182 144 96 215)(52 183 145 97 211)(53 184 141 98 212)(54 185 142 99 213)(55 181 143 100 214)
(1 213 31)(2 214 32)(3 215 33)(4 211 34)(5 212 35)(6 113 66)(7 114 67)(8 115 68)(9 111 69)(10 112 70)(11 96 76)(12 97 77)(13 98 78)(14 99 79)(15 100 80)(16 221 81)(17 222 82)(18 223 83)(19 224 84)(20 225 85)(21 190 163)(22 186 164)(23 187 165)(24 188 161)(25 189 162)(26 160 95)(27 156 91)(28 157 92)(29 158 93)(30 159 94)(36 191 176)(37 192 177)(38 193 178)(39 194 179)(40 195 180)(41 126 107)(42 127 108)(43 128 109)(44 129 110)(45 130 106)(46 202 116)(47 203 117)(48 204 118)(49 205 119)(50 201 120)(51 210 150)(52 206 146)(53 207 147)(54 208 148)(55 209 149)(56 90 216)(57 86 217)(58 87 218)(59 88 219)(60 89 220)(61 141 121)(62 142 122)(63 143 123)(64 144 124)(65 145 125)(71 200 140)(72 196 136)(73 197 137)(74 198 138)(75 199 139)(101 181 171)(102 182 172)(103 183 173)(104 184 174)(105 185 175)(131 167 151)(132 168 152)(133 169 153)(134 170 154)(135 166 155)
(1 162 90)(2 163 86)(3 164 87)(4 165 88)(5 161 89)(6 151 77)(7 152 78)(8 153 79)(9 154 80)(10 155 76)(11 112 135)(12 113 131)(13 114 132)(14 115 133)(15 111 134)(16 172 91)(17 173 92)(18 174 93)(19 175 94)(20 171 95)(21 217 214)(22 218 215)(23 219 211)(24 220 212)(25 216 213)(26 225 101)(27 221 102)(28 222 103)(29 223 104)(30 224 105)(31 189 56)(32 190 57)(33 186 58)(34 187 59)(35 188 60)(36 142 116)(37 143 117)(38 144 118)(39 145 119)(40 141 120)(41 147 72)(42 148 73)(43 149 74)(44 150 75)(45 146 71)(46 191 122)(47 192 123)(48 193 124)(49 194 125)(50 195 121)(51 199 129)(52 200 130)(53 196 126)(54 197 127)(55 198 128)(61 201 180)(62 202 176)(63 203 177)(64 204 178)(65 205 179)(66 167 97)(67 168 98)(68 169 99)(69 170 100)(70 166 96)(81 182 156)(82 183 157)(83 184 158)(84 185 159)(85 181 160)(106 206 140)(107 207 136)(108 208 137)(109 209 138)(110 210 139)
(6 167)(7 168)(8 169)(9 170)(10 166)(16 156)(17 157)(18 158)(19 159)(20 160)(21 57)(22 58)(23 59)(24 60)(25 56)(26 225)(27 221)(28 222)(29 223)(30 224)(31 213)(32 214)(33 215)(34 211)(35 212)(36 46)(37 47)(38 48)(39 49)(40 50)(41 72)(42 73)(43 74)(44 75)(45 71)(51 210)(52 206)(53 207)(54 208)(55 209)(66 151)(67 152)(68 153)(69 154)(70 155)(76 96)(77 97)(78 98)(79 99)(80 100)(81 91)(82 92)(83 93)(84 94)(85 95)(86 163)(87 164)(88 165)(89 161)(90 162)(106 200)(107 196)(108 197)(109 198)(110 199)(111 134)(112 135)(113 131)(114 132)(115 133)(116 191)(117 192)(118 193)(119 194)(120 195)(121 141)(122 142)(123 143)(124 144)(125 145)(126 136)(127 137)(128 138)(129 139)(130 140)(171 181)(172 182)(173 183)(174 184)(175 185)(176 202)(177 203)(178 204)(179 205)(180 201)(186 218)(187 219)(188 220)(189 216)(190 217)

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

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

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

150 conjugacy classes

 class 1 2 3A 3B 3C 3D 5A ··· 5X 10A ··· 10X 15A ··· 15CR order 1 2 3 3 3 3 5 ··· 5 10 ··· 10 15 ··· 15 size 1 9 2 2 2 2 1 ··· 1 9 ··· 9 2 ··· 2

150 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C5 C10 S3 C5×S3 kernel C3⋊S3×C52 C152 C5×C3⋊S3 C3×C15 C5×C15 C15 # reps 1 1 24 24 4 96

Matrix representation of C3⋊S3×C52 in GL4(𝔽31) generated by

 4 0 0 0 0 4 0 0 0 0 16 0 0 0 0 16
,
 2 0 0 0 0 2 0 0 0 0 16 0 0 0 0 16
,
 30 1 0 0 30 0 0 0 0 0 1 0 0 0 0 1
,
 0 30 0 0 1 30 0 0 0 0 30 30 0 0 1 0
,
 0 30 0 0 30 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(31))| [4,0,0,0,0,4,0,0,0,0,16,0,0,0,0,16],[2,0,0,0,0,2,0,0,0,0,16,0,0,0,0,16],[30,30,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,30,30,0,0,0,0,30,1,0,0,30,0],[0,30,0,0,30,0,0,0,0,0,0,1,0,0,1,0] >;

C3⋊S3×C52 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times C_5^2
% in TeX

G:=Group("C3:S3xC5^2");
// GroupNames label

G:=SmallGroup(450,31);
// by ID

G=gap.SmallGroup(450,31);
# by ID

G:=PCGroup([5,-2,-5,-5,-3,-3,2003,7504]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^5=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽