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## G = C30.33C24order 480 = 25·3·5

### 33rd non-split extension by C30 of C24 acting via C24/C22=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C30.33C24
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C15⋊D4 — D6.D10 — C30.33C24
 Lower central C15 — C30 — C30.33C24
 Upper central C1 — C2 — Q8

Generators and relations for C30.33C24
G = < a,b,c,d | a60=b2=c2=1, d2=a30, bab=a49, cac=a41, dad-1=a31, cbc=a30b, bd=db, cd=dc >

Subgroups: 1388 in 292 conjugacy classes, 108 normal (24 characteristic)
C1, C2, C2 [×5], C3, C4 [×3], C4 [×7], C22 [×5], C5, S3 [×4], C6, C6, C2×C4 [×15], D4 [×10], Q8, Q8 [×9], D5 [×4], C10, C10, Dic3 [×3], Dic3, C12 [×3], C12 [×3], D6, D6 [×3], C2×C6, C15, C2×Q8 [×5], C4○D4 [×10], Dic5 [×3], Dic5, C20 [×3], C20 [×3], D10, D10 [×3], C2×C10, Dic6 [×3], Dic6 [×3], C4×S3 [×3], C4×S3 [×9], D12 [×6], C3⋊D4 [×4], C2×C12 [×3], C3×Q8, C3×Q8 [×3], C5×S3, C3×D5, D15 [×3], C30, 2- 1+4, Dic10 [×3], Dic10 [×3], C4×D5 [×3], C4×D5 [×9], D20 [×6], C5⋊D4 [×4], C2×C20 [×3], C5×Q8, C5×Q8 [×3], C4○D12 [×6], S3×Q8, S3×Q8 [×3], Q83S3 [×4], C6×Q8, C5×Dic3 [×3], C3×Dic5 [×3], Dic15, C60 [×3], C6×D5, S3×C10, D30 [×3], C4○D20 [×6], Q8×D5, Q8×D5 [×3], Q82D5 [×4], Q8×C10, Q8.15D6, D30.C2 [×6], C15⋊D4, C3⋊D20 [×3], C5⋊D12 [×3], C15⋊Q8 [×3], C3×Dic10 [×3], D5×C12 [×3], C5×Dic6 [×3], S3×C20 [×3], C4×D15 [×3], D60 [×3], Q8×C15, Q8.10D10, D60⋊C2 [×3], D15⋊Q8 [×3], D6.D10 [×3], C12.28D10 [×3], C3×Q8×D5, C5×S3×Q8, Q83D15, C30.33C24
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], 2- 1+4, C22×D5 [×7], S3×C23, S3×D5, C23×D5, Q8.15D6, C2×S3×D5 [×3], Q8.10D10, C22×S3×D5, C30.33C24

Smallest permutation representation of C30.33C24
On 240 points
Generators in S240
```(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)
(1 184)(2 233)(3 222)(4 211)(5 200)(6 189)(7 238)(8 227)(9 216)(10 205)(11 194)(12 183)(13 232)(14 221)(15 210)(16 199)(17 188)(18 237)(19 226)(20 215)(21 204)(22 193)(23 182)(24 231)(25 220)(26 209)(27 198)(28 187)(29 236)(30 225)(31 214)(32 203)(33 192)(34 181)(35 230)(36 219)(37 208)(38 197)(39 186)(40 235)(41 224)(42 213)(43 202)(44 191)(45 240)(46 229)(47 218)(48 207)(49 196)(50 185)(51 234)(52 223)(53 212)(54 201)(55 190)(56 239)(57 228)(58 217)(59 206)(60 195)(61 151)(62 140)(63 129)(64 178)(65 167)(66 156)(67 145)(68 134)(69 123)(70 172)(71 161)(72 150)(73 139)(74 128)(75 177)(76 166)(77 155)(78 144)(79 133)(80 122)(81 171)(82 160)(83 149)(84 138)(85 127)(86 176)(87 165)(88 154)(89 143)(90 132)(91 121)(92 170)(93 159)(94 148)(95 137)(96 126)(97 175)(98 164)(99 153)(100 142)(101 131)(102 180)(103 169)(104 158)(105 147)(106 136)(107 125)(108 174)(109 163)(110 152)(111 141)(112 130)(113 179)(114 168)(115 157)(116 146)(117 135)(118 124)(119 173)(120 162)
(1 229)(2 210)(3 191)(4 232)(5 213)(6 194)(7 235)(8 216)(9 197)(10 238)(11 219)(12 200)(13 181)(14 222)(15 203)(16 184)(17 225)(18 206)(19 187)(20 228)(21 209)(22 190)(23 231)(24 212)(25 193)(26 234)(27 215)(28 196)(29 237)(30 218)(31 199)(32 240)(33 221)(34 202)(35 183)(36 224)(37 205)(38 186)(39 227)(40 208)(41 189)(42 230)(43 211)(44 192)(45 233)(46 214)(47 195)(48 236)(49 217)(50 198)(51 239)(52 220)(53 201)(54 182)(55 223)(56 204)(57 185)(58 226)(59 207)(60 188)(61 170)(62 151)(63 132)(64 173)(65 154)(66 135)(67 176)(68 157)(69 138)(70 179)(71 160)(72 141)(73 122)(74 163)(75 144)(76 125)(77 166)(78 147)(79 128)(80 169)(81 150)(82 131)(83 172)(84 153)(85 134)(86 175)(87 156)(88 137)(89 178)(90 159)(91 140)(92 121)(93 162)(94 143)(95 124)(96 165)(97 146)(98 127)(99 168)(100 149)(101 130)(102 171)(103 152)(104 133)(105 174)(106 155)(107 136)(108 177)(109 158)(110 139)(111 180)(112 161)(113 142)(114 123)(115 164)(116 145)(117 126)(118 167)(119 148)(120 129)
(1 114 31 84)(2 85 32 115)(3 116 33 86)(4 87 34 117)(5 118 35 88)(6 89 36 119)(7 120 37 90)(8 91 38 61)(9 62 39 92)(10 93 40 63)(11 64 41 94)(12 95 42 65)(13 66 43 96)(14 97 44 67)(15 68 45 98)(16 99 46 69)(17 70 47 100)(18 101 48 71)(19 72 49 102)(20 103 50 73)(21 74 51 104)(22 105 52 75)(23 76 53 106)(24 107 54 77)(25 78 55 108)(26 109 56 79)(27 80 57 110)(28 111 58 81)(29 82 59 112)(30 113 60 83)(121 197 151 227)(122 228 152 198)(123 199 153 229)(124 230 154 200)(125 201 155 231)(126 232 156 202)(127 203 157 233)(128 234 158 204)(129 205 159 235)(130 236 160 206)(131 207 161 237)(132 238 162 208)(133 209 163 239)(134 240 164 210)(135 211 165 181)(136 182 166 212)(137 213 167 183)(138 184 168 214)(139 215 169 185)(140 186 170 216)(141 217 171 187)(142 188 172 218)(143 219 173 189)(144 190 174 220)(145 221 175 191)(146 192 176 222)(147 223 177 193)(148 194 178 224)(149 225 179 195)(150 196 180 226)```

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 6A 6B 6C 10A 10B 10C 10D 10E 10F 12A 12B 12C 12D 12E 12F 15A 15B 20A ··· 20F 20G ··· 20L 30A 30B 60A ··· 60F order 1 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 10 10 10 10 10 10 12 12 12 12 12 12 15 15 20 ··· 20 20 ··· 20 30 30 60 ··· 60 size 1 1 6 10 30 30 30 2 2 2 2 6 6 6 10 10 10 30 2 2 2 10 10 2 2 6 6 6 6 4 4 4 20 20 20 4 4 4 ··· 4 12 ··· 12 4 4 8 ··· 8

57 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + - + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 D10 D10 D10 2- 1+4 S3×D5 Q8.15D6 C2×S3×D5 Q8.10D10 C30.33C24 kernel C30.33C24 D60⋊C2 D15⋊Q8 D6.D10 C12.28D10 C3×Q8×D5 C5×S3×Q8 Q8⋊3D15 Q8×D5 S3×Q8 Dic10 C4×D5 C5×Q8 Dic6 C4×S3 C3×Q8 C15 Q8 C5 C4 C3 C1 # reps 1 3 3 3 3 1 1 1 1 2 3 3 1 6 6 2 1 2 2 6 4 2

Matrix representation of C30.33C24 in GL8(𝔽61)

 1 43 55 35 0 0 0 0 18 43 40 55 0 0 0 0 50 38 36 36 0 0 0 0 38 50 25 59 0 0 0 0 0 0 0 0 0 60 0 59 0 0 0 0 1 0 2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0
,
 45 60 0 39 0 0 0 0 16 16 39 31 0 0 0 0 54 58 27 60 0 0 0 0 58 0 1 34 0 0 0 0 0 0 0 0 25 17 50 0 0 0 0 0 44 36 0 11 0 0 0 0 36 0 36 17 0 0 0 0 0 25 44 25
,
 34 60 30 22 0 0 0 0 1 16 30 30 0 0 0 0 23 58 45 1 0 0 0 0 16 23 60 27 0 0 0 0 0 0 0 0 44 25 27 0 0 0 0 0 36 17 0 34 0 0 0 0 17 0 17 25 0 0 0 0 0 44 36 44
,
 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 59 0 0 0 0 0 0 60 0 59 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1

`G:=sub<GL(8,GF(61))| [1,18,50,38,0,0,0,0,43,43,38,50,0,0,0,0,55,40,36,25,0,0,0,0,35,55,36,59,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,2,0,60,0,0,0,0,59,0,1,0],[45,16,54,58,0,0,0,0,60,16,58,0,0,0,0,0,0,39,27,1,0,0,0,0,39,31,60,34,0,0,0,0,0,0,0,0,25,44,36,0,0,0,0,0,17,36,0,25,0,0,0,0,50,0,36,44,0,0,0,0,0,11,17,25],[34,1,23,16,0,0,0,0,60,16,58,23,0,0,0,0,30,30,45,60,0,0,0,0,22,30,1,27,0,0,0,0,0,0,0,0,44,36,17,0,0,0,0,0,25,17,0,44,0,0,0,0,27,0,17,36,0,0,0,0,0,34,25,44],[60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,1,0,0,0,0,0,0,60,0,1,0,0,0,0,59,0,1,0,0,0,0,0,0,59,0,1] >;`

C30.33C24 in GAP, Magma, Sage, TeX

`C_{30}._{33}C_2^4`
`% in TeX`

`G:=Group("C30.33C2^4");`
`// GroupNames label`

`G:=SmallGroup(480,1105);`
`// by ID`

`G=gap.SmallGroup(480,1105);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,100,185,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^60=b^2=c^2=1,d^2=a^30,b*a*b=a^49,c*a*c=a^41,d*a*d^-1=a^31,c*b*c=a^30*b,b*d=d*b,c*d=d*c>;`
`// generators/relations`

׿
×
𝔽