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## G = C32×C5⋊D5order 450 = 2·32·52

### Direct product of C32 and C5⋊D5

Aliases: C32×C5⋊D5, C1526C2, C5⋊(C32×D5), (C5×C15)⋊8C6, (C3×C15)⋊3D5, C152(C3×D5), C524(C3×C6), SmallGroup(450,27)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C32×C5⋊D5
 Chief series C1 — C5 — C52 — C5×C15 — C152 — C32×C5⋊D5
 Lower central C52 — C32×C5⋊D5
 Upper central C1 — C32

Generators and relations for C32×C5⋊D5
G = < a,b,c,d,e | a3=b3=c5=d5=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: 384 in 96 conjugacy classes, 54 normal (6 characteristic)
C1, C2, C3, C5, C6, C32, D5, C15, C3×C6, C52, C3×D5, C3×C15, C5⋊D5, C5×C15, C32×D5, C3×C5⋊D5, C152, C32×C5⋊D5
Quotients: C1, C2, C3, C6, C32, D5, C3×C6, C3×D5, C5⋊D5, C32×D5, C3×C5⋊D5, C32×C5⋊D5

Smallest permutation representation of C32×C5⋊D5
On 225 points
Generators in S225
(1 204 104)(2 205 105)(3 201 101)(4 202 102)(5 203 103)(6 206 106)(7 207 107)(8 208 108)(9 209 109)(10 210 110)(11 211 111)(12 212 112)(13 213 113)(14 214 114)(15 215 115)(16 216 116)(17 217 117)(18 218 118)(19 219 119)(20 220 120)(21 221 121)(22 222 122)(23 223 123)(24 224 124)(25 225 125)(26 151 126)(27 152 127)(28 153 128)(29 154 129)(30 155 130)(31 156 131)(32 157 132)(33 158 133)(34 159 134)(35 160 135)(36 161 136)(37 162 137)(38 163 138)(39 164 139)(40 165 140)(41 166 141)(42 167 142)(43 168 143)(44 169 144)(45 170 145)(46 171 146)(47 172 147)(48 173 148)(49 174 149)(50 175 150)(51 176 76)(52 177 77)(53 178 78)(54 179 79)(55 180 80)(56 181 81)(57 182 82)(58 183 83)(59 184 84)(60 185 85)(61 186 86)(62 187 87)(63 188 88)(64 189 89)(65 190 90)(66 191 91)(67 192 92)(68 193 93)(69 194 94)(70 195 95)(71 196 96)(72 197 97)(73 198 98)(74 199 99)(75 200 100)
(1 54 29)(2 55 30)(3 51 26)(4 52 27)(5 53 28)(6 56 31)(7 57 32)(8 58 33)(9 59 34)(10 60 35)(11 61 36)(12 62 37)(13 63 38)(14 64 39)(15 65 40)(16 66 41)(17 67 42)(18 68 43)(19 69 44)(20 70 45)(21 71 46)(22 72 47)(23 73 48)(24 74 49)(25 75 50)(76 126 101)(77 127 102)(78 128 103)(79 129 104)(80 130 105)(81 131 106)(82 132 107)(83 133 108)(84 134 109)(85 135 110)(86 136 111)(87 137 112)(88 138 113)(89 139 114)(90 140 115)(91 141 116)(92 142 117)(93 143 118)(94 144 119)(95 145 120)(96 146 121)(97 147 122)(98 148 123)(99 149 124)(100 150 125)(151 201 176)(152 202 177)(153 203 178)(154 204 179)(155 205 180)(156 206 181)(157 207 182)(158 208 183)(159 209 184)(160 210 185)(161 211 186)(162 212 187)(163 213 188)(164 214 189)(165 215 190)(166 216 191)(167 217 192)(168 218 193)(169 219 194)(170 220 195)(171 221 196)(172 222 197)(173 223 198)(174 224 199)(175 225 200)
(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 24 19 14 9)(2 25 20 15 10)(3 21 16 11 6)(4 22 17 12 7)(5 23 18 13 8)(26 46 41 36 31)(27 47 42 37 32)(28 48 43 38 33)(29 49 44 39 34)(30 50 45 40 35)(51 71 66 61 56)(52 72 67 62 57)(53 73 68 63 58)(54 74 69 64 59)(55 75 70 65 60)(76 96 91 86 81)(77 97 92 87 82)(78 98 93 88 83)(79 99 94 89 84)(80 100 95 90 85)(101 121 116 111 106)(102 122 117 112 107)(103 123 118 113 108)(104 124 119 114 109)(105 125 120 115 110)(126 146 141 136 131)(127 147 142 137 132)(128 148 143 138 133)(129 149 144 139 134)(130 150 145 140 135)(151 171 166 161 156)(152 172 167 162 157)(153 173 168 163 158)(154 174 169 164 159)(155 175 170 165 160)(176 196 191 186 181)(177 197 192 187 182)(178 198 193 188 183)(179 199 194 189 184)(180 200 195 190 185)(201 221 216 211 206)(202 222 217 212 207)(203 223 218 213 208)(204 224 219 214 209)(205 225 220 215 210)
(1 9)(2 8)(3 7)(4 6)(5 10)(11 22)(12 21)(13 25)(14 24)(15 23)(16 17)(18 20)(26 32)(27 31)(28 35)(29 34)(30 33)(36 47)(37 46)(38 50)(39 49)(40 48)(41 42)(43 45)(51 57)(52 56)(53 60)(54 59)(55 58)(61 72)(62 71)(63 75)(64 74)(65 73)(66 67)(68 70)(76 82)(77 81)(78 85)(79 84)(80 83)(86 97)(87 96)(88 100)(89 99)(90 98)(91 92)(93 95)(101 107)(102 106)(103 110)(104 109)(105 108)(111 122)(112 121)(113 125)(114 124)(115 123)(116 117)(118 120)(126 132)(127 131)(128 135)(129 134)(130 133)(136 147)(137 146)(138 150)(139 149)(140 148)(141 142)(143 145)(151 157)(152 156)(153 160)(154 159)(155 158)(161 172)(162 171)(163 175)(164 174)(165 173)(166 167)(168 170)(176 182)(177 181)(178 185)(179 184)(180 183)(186 197)(187 196)(188 200)(189 199)(190 198)(191 192)(193 195)(201 207)(202 206)(203 210)(204 209)(205 208)(211 222)(212 221)(213 225)(214 224)(215 223)(216 217)(218 220)

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

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

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

126 conjugacy classes

 class 1 2 3A ··· 3H 5A ··· 5L 6A ··· 6H 15A ··· 15CR order 1 2 3 ··· 3 5 ··· 5 6 ··· 6 15 ··· 15 size 1 25 1 ··· 1 2 ··· 2 25 ··· 25 2 ··· 2

126 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C3 C6 D5 C3×D5 kernel C32×C5⋊D5 C152 C3×C5⋊D5 C5×C15 C3×C15 C15 # reps 1 1 8 8 12 96

Matrix representation of C32×C5⋊D5 in GL5(𝔽31)

 25 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 25 0 0 0 0 0 25
,
 1 0 0 0 0 0 0 1 0 0 0 30 12 0 0 0 0 0 13 14 0 0 0 30 30
,
 1 0 0 0 0 0 30 12 0 0 0 19 19 0 0 0 0 0 18 18 0 0 0 12 0
,
 30 0 0 0 0 0 30 12 0 0 0 0 1 0 0 0 0 0 0 13 0 0 0 12 0

G:=sub<GL(5,GF(31))| [25,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,25,0,0,0,0,0,25],[1,0,0,0,0,0,0,30,0,0,0,1,12,0,0,0,0,0,13,30,0,0,0,14,30],[1,0,0,0,0,0,30,19,0,0,0,12,19,0,0,0,0,0,18,12,0,0,0,18,0],[30,0,0,0,0,0,30,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,13,0] >;

C32×C5⋊D5 in GAP, Magma, Sage, TeX

C_3^2\times C_5\rtimes D_5
% in TeX

G:=Group("C3^2xC5:D5");
// GroupNames label

G:=SmallGroup(450,27);
// by ID

G=gap.SmallGroup(450,27);
# by ID

G:=PCGroup([5,-2,-3,-3,-5,-5,1443,9004]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^5=d^5=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