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G = D304Q8order 480 = 25·3·5

4th semidirect product of D30 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D304Q8, Dic3.11D20, C4⋊Dic56S3, C6.16(Q8×D5), C52(D6⋊Q8), C30.50(C2×D4), C2.20(S3×D20), C6.19(C2×D20), (C2×C20).22D6, C10.19(S3×D4), C10.16(S3×Q8), C30.46(C2×Q8), Dic3⋊C419D5, C31(D102Q8), C1517(C22⋊Q8), (C5×Dic3).9D4, D303C4.8C2, (C2×C12).227D10, C2.18(D15⋊Q8), C6.Dic1022C2, (C2×Dic5).37D6, D304C4.14C2, C10.71(C4○D12), C30.121(C4○D4), C6.46(D42D5), (C2×C60).258C22, (C2×C30).119C23, (C2×Dic3).37D10, (C6×Dic5).72C22, C2.17(Dic3.D10), (C2×Dic15).96C22, (C10×Dic3).73C22, (C22×D15).41C22, (C2×C15⋊Q8)⋊9C2, (C2×C4).52(S3×D5), (C3×C4⋊Dic5)⋊17C2, C22.182(C2×S3×D5), (C5×Dic3⋊C4)⋊19C2, (C2×D30.C2).7C2, (C2×C6).131(C22×D5), (C2×C10).131(C22×S3), SmallGroup(480,505)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D304Q8
C1C5C15C30C2×C30C6×Dic5C2×D30.C2 — D304Q8
C15C2×C30 — D304Q8
C1C22C2×C4

Generators and relations for D304Q8
 G = < a,b,c,d | a30=b2=c4=1, d2=c2, bab=cac-1=a-1, dad-1=a11, cbc-1=a13b, dbd-1=a10b, dcd-1=c-1 >

Subgroups: 844 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, Dic10 [×2], C4×D5 [×2], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, Dic3⋊C4, Dic3⋊C4, 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, C4⋊Dic5, C4⋊Dic5, D10⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, D6⋊Q8, D30.C2 [×2], C15⋊Q8 [×2], C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15, C2×C60, C22×D15, D102Q8, D304C4, C6.Dic10, C3×C4⋊Dic5, C5×Dic3⋊C4, D303C4, C2×D30.C2, C2×C15⋊Q8, D304Q8
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, D20 [×2], C22×D5, C4○D12, S3×D4, S3×Q8, S3×D5, C2×D20, D42D5, Q8×D5, D6⋊Q8, C2×S3×D5, D102Q8, D15⋊Q8, S3×D20, Dic3.D10, D304Q8

Smallest permutation representation of D304Q8
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 45)(2 44)(3 43)(4 42)(5 41)(6 40)(7 39)(8 38)(9 37)(10 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 60)(17 59)(18 58)(19 57)(20 56)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 48)(29 47)(30 46)(61 166)(62 165)(63 164)(64 163)(65 162)(66 161)(67 160)(68 159)(69 158)(70 157)(71 156)(72 155)(73 154)(74 153)(75 152)(76 151)(77 180)(78 179)(79 178)(80 177)(81 176)(82 175)(83 174)(84 173)(85 172)(86 171)(87 170)(88 169)(89 168)(90 167)(91 123)(92 122)(93 121)(94 150)(95 149)(96 148)(97 147)(98 146)(99 145)(100 144)(101 143)(102 142)(103 141)(104 140)(105 139)(106 138)(107 137)(108 136)(109 135)(110 134)(111 133)(112 132)(113 131)(114 130)(115 129)(116 128)(117 127)(118 126)(119 125)(120 124)(181 227)(182 226)(183 225)(184 224)(185 223)(186 222)(187 221)(188 220)(189 219)(190 218)(191 217)(192 216)(193 215)(194 214)(195 213)(196 212)(197 211)(198 240)(199 239)(200 238)(201 237)(202 236)(203 235)(204 234)(205 233)(206 232)(207 231)(208 230)(209 229)(210 228)
(1 116 46 144)(2 115 47 143)(3 114 48 142)(4 113 49 141)(5 112 50 140)(6 111 51 139)(7 110 52 138)(8 109 53 137)(9 108 54 136)(10 107 55 135)(11 106 56 134)(12 105 57 133)(13 104 58 132)(14 103 59 131)(15 102 60 130)(16 101 31 129)(17 100 32 128)(18 99 33 127)(19 98 34 126)(20 97 35 125)(21 96 36 124)(22 95 37 123)(23 94 38 122)(24 93 39 121)(25 92 40 150)(26 91 41 149)(27 120 42 148)(28 119 43 147)(29 118 44 146)(30 117 45 145)(61 182 165 214)(62 181 166 213)(63 210 167 212)(64 209 168 211)(65 208 169 240)(66 207 170 239)(67 206 171 238)(68 205 172 237)(69 204 173 236)(70 203 174 235)(71 202 175 234)(72 201 176 233)(73 200 177 232)(74 199 178 231)(75 198 179 230)(76 197 180 229)(77 196 151 228)(78 195 152 227)(79 194 153 226)(80 193 154 225)(81 192 155 224)(82 191 156 223)(83 190 157 222)(84 189 158 221)(85 188 159 220)(86 187 160 219)(87 186 161 218)(88 185 162 217)(89 184 163 216)(90 183 164 215)
(1 62 46 166)(2 73 47 177)(3 84 48 158)(4 65 49 169)(5 76 50 180)(6 87 51 161)(7 68 52 172)(8 79 53 153)(9 90 54 164)(10 71 55 175)(11 82 56 156)(12 63 57 167)(13 74 58 178)(14 85 59 159)(15 66 60 170)(16 77 31 151)(17 88 32 162)(18 69 33 173)(19 80 34 154)(20 61 35 165)(21 72 36 176)(22 83 37 157)(23 64 38 168)(24 75 39 179)(25 86 40 160)(26 67 41 171)(27 78 42 152)(28 89 43 163)(29 70 44 174)(30 81 45 155)(91 238 149 206)(92 219 150 187)(93 230 121 198)(94 211 122 209)(95 222 123 190)(96 233 124 201)(97 214 125 182)(98 225 126 193)(99 236 127 204)(100 217 128 185)(101 228 129 196)(102 239 130 207)(103 220 131 188)(104 231 132 199)(105 212 133 210)(106 223 134 191)(107 234 135 202)(108 215 136 183)(109 226 137 194)(110 237 138 205)(111 218 139 186)(112 229 140 197)(113 240 141 208)(114 221 142 189)(115 232 143 200)(116 213 144 181)(117 224 145 192)(118 235 146 203)(119 216 147 184)(120 227 148 195)

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

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

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

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○D4D10D10D20C4○D12S3×D4S3×Q8S3×D5D42D5Q8×D5C2×S3×D5D15⋊Q8S3×D20Dic3.D10
kernelD304Q8D304C4C6.Dic10C3×C4⋊Dic5C5×Dic3⋊C4D303C4C2×D30.C2C2×C15⋊Q8C4⋊Dic5C5×Dic3D30Dic3⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12Dic3C10C10C10C2×C4C6C6C22C2C2C2
# reps1111111112222124284112222444

Matrix representation of D304Q8 in GL4(𝔽61) generated by

17100
16100
00060
0011
,
04300
44000
0001
0010
,
22500
565900
00918
00952
,
60000
06000
00500
001111
G:=sub<GL(4,GF(61))| [17,16,0,0,1,1,0,0,0,0,0,1,0,0,60,1],[0,44,0,0,43,0,0,0,0,0,0,1,0,0,1,0],[2,56,0,0,25,59,0,0,0,0,9,9,0,0,18,52],[60,0,0,0,0,60,0,0,0,0,50,11,0,0,0,11] >;

D304Q8 in GAP, Magma, Sage, TeX

D_{30}\rtimes_4Q_8
% in TeX

G:=Group("D30:4Q8");
// GroupNames label

G:=SmallGroup(480,505);
// by ID

G=gap.SmallGroup(480,505);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,219,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=a^-1,d*a*d^-1=a^11,c*b*c^-1=a^13*b,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations

׿
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𝔽