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G = D303Q8order 480 = 25·3·5

3rd semidirect product of D30 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D303Q8, C6.15(Q8×D5), (C2×Dic6)⋊5D5, C56(D6⋊Q8), C30.44(C2×Q8), C10.15(S3×Q8), C31(D103Q8), (C2×C20).228D6, (C2×C12).21D10, C10.132(S3×D4), C30.136(C2×D4), C1516(C22⋊Q8), (C5×Dic3).8D4, (C10×Dic6)⋊15C2, C30.66(C4○D4), C10.D419S3, C2.17(D15⋊Q8), Dic155C418C2, C6.Dic1019C2, (C2×Dic5).36D6, D304C4.13C2, D303C4.11C2, C10.12(C4○D12), (C2×C30).114C23, (C2×C60).321C22, C6.14(Q82D5), (C2×Dic3).35D10, Dic3.9(C5⋊D4), C2.15(C12.28D10), (C6×Dic5).68C22, (C2×Dic15).92C22, (C10×Dic3).71C22, (C22×D15).38C22, (C2×C4).50(S3×D5), C2.14(S3×C5⋊D4), C6.33(C2×C5⋊D4), C22.180(C2×S3×D5), (C2×D30.C2).6C2, (C3×C10.D4)⋊22C2, (C2×C6).126(C22×D5), (C2×C10).126(C22×S3), SmallGroup(480,500)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D303Q8
C1C5C15C30C2×C30C6×Dic5C2×D30.C2 — D303Q8
C15C2×C30 — D303Q8
C1C22C2×C4

Generators and relations for D303Q8
 G = < a,b,c,d | a30=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a11, cbc-1=a15b, dbd-1=a10b, dcd-1=c-1 >

Subgroups: 780 in 148 conjugacy classes, 50 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×7], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4 [×7], Q8 [×2], C23, D5 [×2], C10 [×3], Dic3 [×2], Dic3 [×2], C12 [×3], D6 [×4], C2×C6, C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×3], C20 [×4], D10 [×4], C2×C10, Dic6 [×2], C4×S3 [×2], C2×Dic3 [×2], C2×Dic3, C2×C12, C2×C12 [×2], C22×S3, D15 [×2], C30 [×3], C22⋊Q8, C4×D5 [×2], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, Dic3⋊C4 [×2], D6⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4, C5×Dic3 [×2], C5×Dic3, C3×Dic5 [×2], Dic15, C60, D30 [×2], D30 [×2], C2×C30, C10.D4, C10.D4, C4⋊Dic5, D10⋊C4 [×2], C2×C4×D5, Q8×C10, D6⋊Q8, D30.C2 [×2], C6×Dic5 [×2], C5×Dic6 [×2], C10×Dic3 [×2], C2×Dic15, C2×C60, C22×D15, D103Q8, D304C4, Dic155C4, C6.Dic10, C3×C10.D4, D303C4, C2×D30.C2, C10×Dic6, D303Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D5, D6 [×3], C2×D4, C2×Q8, C4○D4, D10 [×3], C22×S3, C22⋊Q8, C5⋊D4 [×2], C22×D5, C4○D12, S3×D4, S3×Q8, S3×D5, Q8×D5, Q82D5, C2×C5⋊D4, D6⋊Q8, C2×S3×D5, D103Q8, D15⋊Q8, C12.28D10, S3×C5⋊D4, D303Q8

Smallest permutation representation of D303Q8
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)
(1 209)(2 208)(3 207)(4 206)(5 205)(6 204)(7 203)(8 202)(9 201)(10 200)(11 199)(12 198)(13 197)(14 196)(15 195)(16 194)(17 193)(18 192)(19 191)(20 190)(21 189)(22 188)(23 187)(24 186)(25 185)(26 184)(27 183)(28 182)(29 181)(30 210)(31 226)(32 225)(33 224)(34 223)(35 222)(36 221)(37 220)(38 219)(39 218)(40 217)(41 216)(42 215)(43 214)(44 213)(45 212)(46 211)(47 240)(48 239)(49 238)(50 237)(51 236)(52 235)(53 234)(54 233)(55 232)(56 231)(57 230)(58 229)(59 228)(60 227)(61 96)(62 95)(63 94)(64 93)(65 92)(66 91)(67 120)(68 119)(69 118)(70 117)(71 116)(72 115)(73 114)(74 113)(75 112)(76 111)(77 110)(78 109)(79 108)(80 107)(81 106)(82 105)(83 104)(84 103)(85 102)(86 101)(87 100)(88 99)(89 98)(90 97)(121 179)(122 178)(123 177)(124 176)(125 175)(126 174)(127 173)(128 172)(129 171)(130 170)(131 169)(132 168)(133 167)(134 166)(135 165)(136 164)(137 163)(138 162)(139 161)(140 160)(141 159)(142 158)(143 157)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)(150 180)
(1 117 210 86)(2 118 181 87)(3 119 182 88)(4 120 183 89)(5 91 184 90)(6 92 185 61)(7 93 186 62)(8 94 187 63)(9 95 188 64)(10 96 189 65)(11 97 190 66)(12 98 191 67)(13 99 192 68)(14 100 193 69)(15 101 194 70)(16 102 195 71)(17 103 196 72)(18 104 197 73)(19 105 198 74)(20 106 199 75)(21 107 200 76)(22 108 201 77)(23 109 202 78)(24 110 203 79)(25 111 204 80)(26 112 205 81)(27 113 206 82)(28 114 207 83)(29 115 208 84)(30 116 209 85)(31 137 216 168)(32 138 217 169)(33 139 218 170)(34 140 219 171)(35 141 220 172)(36 142 221 173)(37 143 222 174)(38 144 223 175)(39 145 224 176)(40 146 225 177)(41 147 226 178)(42 148 227 179)(43 149 228 180)(44 150 229 151)(45 121 230 152)(46 122 231 153)(47 123 232 154)(48 124 233 155)(49 125 234 156)(50 126 235 157)(51 127 236 158)(52 128 237 159)(53 129 238 160)(54 130 239 161)(55 131 240 162)(56 132 211 163)(57 133 212 164)(58 134 213 165)(59 135 214 166)(60 136 215 167)
(1 151 210 150)(2 162 181 131)(3 173 182 142)(4 154 183 123)(5 165 184 134)(6 176 185 145)(7 157 186 126)(8 168 187 137)(9 179 188 148)(10 160 189 129)(11 171 190 140)(12 152 191 121)(13 163 192 132)(14 174 193 143)(15 155 194 124)(16 166 195 135)(17 177 196 146)(18 158 197 127)(19 169 198 138)(20 180 199 149)(21 161 200 130)(22 172 201 141)(23 153 202 122)(24 164 203 133)(25 175 204 144)(26 156 205 125)(27 167 206 136)(28 178 207 147)(29 159 208 128)(30 170 209 139)(31 94 216 63)(32 105 217 74)(33 116 218 85)(34 97 219 66)(35 108 220 77)(36 119 221 88)(37 100 222 69)(38 111 223 80)(39 92 224 61)(40 103 225 72)(41 114 226 83)(42 95 227 64)(43 106 228 75)(44 117 229 86)(45 98 230 67)(46 109 231 78)(47 120 232 89)(48 101 233 70)(49 112 234 81)(50 93 235 62)(51 104 236 73)(52 115 237 84)(53 96 238 65)(54 107 239 76)(55 118 240 87)(56 99 211 68)(57 110 212 79)(58 91 213 90)(59 102 214 71)(60 113 215 82)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C10A···10F12A12B12C12D12E12F15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222223444444445566610···1012121212121215152020202020···2030···3060···60
size1111303024661010122060222222···2442020202044444412···124···44···4

60 irreducible representations

dim1111111122222222222444444444
type++++++++++-++++++-+-+++
imageC1C2C2C2C2C2C2C2S3D4Q8D5D6D6C4○D4D10D10C5⋊D4C4○D12S3×D4S3×Q8S3×D5Q8×D5Q82D5C2×S3×D5D15⋊Q8C12.28D10S3×C5⋊D4
kernelD303Q8D304C4Dic155C4C6.Dic10C3×C10.D4D303C4C2×D30.C2C10×Dic6C10.D4C5×Dic3D30C2×Dic6C2×Dic5C2×C20C30C2×Dic3C2×C12Dic3C10C10C10C2×C4C6C6C22C2C2C2
# reps1111111112222124284112222444

Matrix representation of D303Q8 in GL4(𝔽61) generated by

11900
604300
00160
0010
,
04300
44000
00600
00601
,
306000
453100
003846
001523
,
1000
0100
00011
00110
G:=sub<GL(4,GF(61))| [1,60,0,0,19,43,0,0,0,0,1,1,0,0,60,0],[0,44,0,0,43,0,0,0,0,0,60,60,0,0,0,1],[30,45,0,0,60,31,0,0,0,0,38,15,0,0,46,23],[1,0,0,0,0,1,0,0,0,0,0,11,0,0,11,0] >;

D303Q8 in GAP, Magma, Sage, TeX

D_{30}\rtimes_3Q_8
% in TeX

G:=Group("D30:3Q8");
// GroupNames label

G:=SmallGroup(480,500);
// by ID

G=gap.SmallGroup(480,500);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,c*b*c^-1=a^15*b,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽