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G = D309D4order 480 = 25·3·5

1st semidirect product of D30 and D4 acting via D4/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D309D4, Dic1511D4, C23.10D30, (C2×D60)⋊6C2, C2.9(D4×D15), (C2×C4).7D30, C53(Dic3⋊D4), C22⋊C44D15, (C2×C20).33D6, C6.100(D4×D5), C33(D10⋊D4), C1525(C4⋊D4), C10.102(S3×D4), C30.308(C2×D4), C30.4Q86C2, D303C411C2, C6.97(C4○D20), (C2×C12).205D10, (C22×C6).59D10, (C22×C10).74D6, C30.170(C4○D4), C10.97(C4○D12), (C2×C30).281C23, (C2×C60).174C22, (C22×C30).15C22, C2.11(D6011C2), (C2×Dic15).7C22, (C22×D15).4C22, C22.43(C22×D15), (C2×C4×D15)⋊17C2, (C2×C157D4)⋊2C2, (C5×C22⋊C4)⋊6S3, (C3×C22⋊C4)⋊6D5, (C15×C22⋊C4)⋊8C2, (C2×C6).277(C22×D5), (C2×C10).276(C22×S3), SmallGroup(480,849)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D309D4
C1C5C15C30C2×C30C22×D15C2×C4×D15 — D309D4
C15C2×C30 — D309D4
C1C22C22⋊C4

Generators and relations for D309D4
 G = < a,b,c,d | a30=b2=c4=d2=1, bab=cac-1=dad=a-1, cbc-1=a28b, dbd=a13b, dcd=c-1 >

Subgroups: 1316 in 188 conjugacy classes, 51 normal (47 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×10], C5, S3 [×3], C6 [×3], C6, C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], D5 [×3], C10 [×3], C10, Dic3 [×3], C12 [×2], D6 [×7], C2×C6, C2×C6 [×3], C15, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×3], C20 [×2], D10 [×7], C2×C10, C2×C10 [×3], C4×S3 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C22×S3 [×2], C22×C6, D15 [×3], C30 [×3], C30, C4⋊D4, C4×D5 [×2], D20 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C22×D5 [×2], C22×C10, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4 [×2], Dic15 [×2], Dic15, C60 [×2], D30 [×2], D30 [×5], C2×C30, C2×C30 [×3], C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4 [×2], Dic3⋊D4, C4×D15 [×2], D60 [×2], C2×Dic15 [×2], C157D4 [×4], C2×C60 [×2], C22×D15 [×2], C22×C30, D10⋊D4, C30.4Q8, D303C4, C15×C22⋊C4, C2×C4×D15, C2×D60, C2×C157D4 [×2], D309D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], C22×S3, D15, C4⋊D4, C22×D5, C4○D12, S3×D4 [×2], D30 [×3], C4○D20, D4×D5 [×2], Dic3⋊D4, C22×D15, D10⋊D4, D6011C2, D4×D15 [×2], D309D4

Smallest permutation representation of D309D4
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 87)(2 86)(3 85)(4 84)(5 83)(6 82)(7 81)(8 80)(9 79)(10 78)(11 77)(12 76)(13 75)(14 74)(15 73)(16 72)(17 71)(18 70)(19 69)(20 68)(21 67)(22 66)(23 65)(24 64)(25 63)(26 62)(27 61)(28 90)(29 89)(30 88)(31 185)(32 184)(33 183)(34 182)(35 181)(36 210)(37 209)(38 208)(39 207)(40 206)(41 205)(42 204)(43 203)(44 202)(45 201)(46 200)(47 199)(48 198)(49 197)(50 196)(51 195)(52 194)(53 193)(54 192)(55 191)(56 190)(57 189)(58 188)(59 187)(60 186)(91 174)(92 173)(93 172)(94 171)(95 170)(96 169)(97 168)(98 167)(99 166)(100 165)(101 164)(102 163)(103 162)(104 161)(105 160)(106 159)(107 158)(108 157)(109 156)(110 155)(111 154)(112 153)(113 152)(114 151)(115 180)(116 179)(117 178)(118 177)(119 176)(120 175)(121 240)(122 239)(123 238)(124 237)(125 236)(126 235)(127 234)(128 233)(129 232)(130 231)(131 230)(132 229)(133 228)(134 227)(135 226)(136 225)(137 224)(138 223)(139 222)(140 221)(141 220)(142 219)(143 218)(144 217)(145 216)(146 215)(147 214)(148 213)(149 212)(150 211)
(1 156 73 95)(2 155 74 94)(3 154 75 93)(4 153 76 92)(5 152 77 91)(6 151 78 120)(7 180 79 119)(8 179 80 118)(9 178 81 117)(10 177 82 116)(11 176 83 115)(12 175 84 114)(13 174 85 113)(14 173 86 112)(15 172 87 111)(16 171 88 110)(17 170 89 109)(18 169 90 108)(19 168 61 107)(20 167 62 106)(21 166 63 105)(22 165 64 104)(23 164 65 103)(24 163 66 102)(25 162 67 101)(26 161 68 100)(27 160 69 99)(28 159 70 98)(29 158 71 97)(30 157 72 96)(31 218 200 130)(32 217 201 129)(33 216 202 128)(34 215 203 127)(35 214 204 126)(36 213 205 125)(37 212 206 124)(38 211 207 123)(39 240 208 122)(40 239 209 121)(41 238 210 150)(42 237 181 149)(43 236 182 148)(44 235 183 147)(45 234 184 146)(46 233 185 145)(47 232 186 144)(48 231 187 143)(49 230 188 142)(50 229 189 141)(51 228 190 140)(52 227 191 139)(53 226 192 138)(54 225 193 137)(55 224 194 136)(56 223 195 135)(57 222 196 134)(58 221 197 133)(59 220 198 132)(60 219 199 131)
(1 240)(2 239)(3 238)(4 237)(5 236)(6 235)(7 234)(8 233)(9 232)(10 231)(11 230)(12 229)(13 228)(14 227)(15 226)(16 225)(17 224)(18 223)(19 222)(20 221)(21 220)(22 219)(23 218)(24 217)(25 216)(26 215)(27 214)(28 213)(29 212)(30 211)(31 164)(32 163)(33 162)(34 161)(35 160)(36 159)(37 158)(38 157)(39 156)(40 155)(41 154)(42 153)(43 152)(44 151)(45 180)(46 179)(47 178)(48 177)(49 176)(50 175)(51 174)(52 173)(53 172)(54 171)(55 170)(56 169)(57 168)(58 167)(59 166)(60 165)(61 134)(62 133)(63 132)(64 131)(65 130)(66 129)(67 128)(68 127)(69 126)(70 125)(71 124)(72 123)(73 122)(74 121)(75 150)(76 149)(77 148)(78 147)(79 146)(80 145)(81 144)(82 143)(83 142)(84 141)(85 140)(86 139)(87 138)(88 137)(89 136)(90 135)(91 182)(92 181)(93 210)(94 209)(95 208)(96 207)(97 206)(98 205)(99 204)(100 203)(101 202)(102 201)(103 200)(104 199)(105 198)(106 197)(107 196)(108 195)(109 194)(110 193)(111 192)(112 191)(113 190)(114 189)(115 188)(116 187)(117 186)(118 185)(119 184)(120 183)

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F5A5B6A6B6C6D6E10A···10F10G10H10I10J12A12B12C12D15A15B15C15D20A···20H30A···30L30M···30T60A···60P
order122222223444444556666610···1010101010121212121515151520···2030···3030···3060···60
size11114303060222430306022222442···24444444422224···42···24···44···4

84 irreducible representations

dim1111111222222222222222444
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2S3D4D4D5D6D6C4○D4D10D10D15C4○D12D30D30C4○D20D6011C2S3×D4D4×D5D4×D15
kernelD309D4C30.4Q8D303C4C15×C22⋊C4C2×C4×D15C2×D60C2×C157D4C5×C22⋊C4Dic15D30C3×C22⋊C4C2×C20C22×C10C30C2×C12C22×C6C22⋊C4C10C2×C4C23C6C2C10C6C2
# reps11111121222212424484816248

Matrix representation of D309D4 in GL6(𝔽61)

45530000
8230000
0060000
0006000
0000600
0000060
,
18600000
18430000
0060000
0046100
000010
0000060
,
56380000
950000
0011000
00435000
000010
000001
,
56380000
950000
00112700
00435000
000001
000010

G:=sub<GL(6,GF(61))| [45,8,0,0,0,0,53,23,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[18,18,0,0,0,0,60,43,0,0,0,0,0,0,60,46,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60],[56,9,0,0,0,0,38,5,0,0,0,0,0,0,11,43,0,0,0,0,0,50,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[56,9,0,0,0,0,38,5,0,0,0,0,0,0,11,43,0,0,0,0,27,50,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

D309D4 in GAP, Magma, Sage, TeX

D_{30}\rtimes_9D_4
% in TeX

G:=Group("D30:9D4");
// GroupNames label

G:=SmallGroup(480,849);
// by ID

G=gap.SmallGroup(480,849);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,590,219,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^28*b,d*b*d=a^13*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽