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G = D307Q8order 480 = 25·3·5

3rd semidirect product of D30 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D307Q8, C60.22D4, (C6×Q8)⋊3D5, (Q8×C30)⋊3C2, (Q8×C10)⋊7S3, (C2×Q8)⋊5D15, C6.46(Q8×D5), C2.9(Q8×D15), C55(D63Q8), C605C421C2, (C2×C4).56D30, C30.99(C2×Q8), C10.46(S3×Q8), C35(D103Q8), C30.389(C2×D4), (C2×C20).155D6, C1537(C22⋊Q8), (C2×C12).251D10, C12.49(C5⋊D4), C4.18(C157D4), C20.47(C3⋊D4), C30.4Q838C2, D303C4.16C2, C30.261(C4○D4), C2.8(Q83D15), (C2×C60).435C22, (C2×C30).314C23, C6.45(Q82D5), C10.45(Q83S3), C22.64(C22×D15), (C2×Dic15).20C22, (C22×D15).89C22, (C2×C4×D15).4C2, C2.21(C2×C157D4), C6.116(C2×C5⋊D4), C10.116(C2×C3⋊D4), (C2×C6).310(C22×D5), (C2×C10).309(C22×S3), SmallGroup(480,911)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D307Q8
C1C5C15C30C2×C30C22×D15C2×C4×D15 — D307Q8
C15C2×C30 — D307Q8
C1C22C2×Q8

Generators and relations for D307Q8
 G = < a,b,c,d | a30=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, cbc-1=a15b, bd=db, dcd-1=c-1 >

Subgroups: 820 in 148 conjugacy classes, 57 normal (33 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×5], Q8 [×2], C23, D5 [×2], C10 [×3], Dic3 [×3], C12 [×2], C12 [×2], D6 [×4], C2×C6, C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×4], C2×C10, C4×S3 [×2], C2×Dic3 [×3], C2×C12, C2×C12 [×2], C3×Q8 [×2], C22×S3, D15 [×2], C30 [×3], C22⋊Q8, C4×D5 [×2], C2×Dic5 [×3], C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], S3×C2×C4, C6×Q8, Dic15 [×3], C60 [×2], C60 [×2], D30 [×2], D30 [×2], C2×C30, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], C2×C4×D5, Q8×C10, D63Q8, C4×D15 [×2], C2×Dic15, C2×Dic15 [×2], C2×C60, C2×C60 [×2], Q8×C15 [×2], C22×D15, D103Q8, C30.4Q8 [×2], C605C4, D303C4 [×2], C2×C4×D15, Q8×C30, D307Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D5, D6 [×3], C2×D4, C2×Q8, C4○D4, D10 [×3], C3⋊D4 [×2], C22×S3, D15, C22⋊Q8, C5⋊D4 [×2], C22×D5, S3×Q8, Q83S3, C2×C3⋊D4, D30 [×3], Q8×D5, Q82D5, C2×C5⋊D4, D63Q8, C157D4 [×2], C22×D15, D103Q8, Q8×D15, Q83D15, C2×C157D4, D307Q8

Smallest permutation representation of D307Q8
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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 60)(14 59)(15 58)(16 57)(17 56)(18 55)(19 54)(20 53)(21 52)(22 51)(23 50)(24 49)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(61 114)(62 113)(63 112)(64 111)(65 110)(66 109)(67 108)(68 107)(69 106)(70 105)(71 104)(72 103)(73 102)(74 101)(75 100)(76 99)(77 98)(78 97)(79 96)(80 95)(81 94)(82 93)(83 92)(84 91)(85 120)(86 119)(87 118)(88 117)(89 116)(90 115)(121 153)(122 152)(123 151)(124 180)(125 179)(126 178)(127 177)(128 176)(129 175)(130 174)(131 173)(132 172)(133 171)(134 170)(135 169)(136 168)(137 167)(138 166)(139 165)(140 164)(141 163)(142 162)(143 161)(144 160)(145 159)(146 158)(147 157)(148 156)(149 155)(150 154)(181 237)(182 236)(183 235)(184 234)(185 233)(186 232)(187 231)(188 230)(189 229)(190 228)(191 227)(192 226)(193 225)(194 224)(195 223)(196 222)(197 221)(198 220)(199 219)(200 218)(201 217)(202 216)(203 215)(204 214)(205 213)(206 212)(207 211)(208 240)(209 239)(210 238)
(1 90 43 101)(2 61 44 102)(3 62 45 103)(4 63 46 104)(5 64 47 105)(6 65 48 106)(7 66 49 107)(8 67 50 108)(9 68 51 109)(10 69 52 110)(11 70 53 111)(12 71 54 112)(13 72 55 113)(14 73 56 114)(15 74 57 115)(16 75 58 116)(17 76 59 117)(18 77 60 118)(19 78 31 119)(20 79 32 120)(21 80 33 91)(22 81 34 92)(23 82 35 93)(24 83 36 94)(25 84 37 95)(26 85 38 96)(27 86 39 97)(28 87 40 98)(29 88 41 99)(30 89 42 100)(121 233 178 195)(122 234 179 196)(123 235 180 197)(124 236 151 198)(125 237 152 199)(126 238 153 200)(127 239 154 201)(128 240 155 202)(129 211 156 203)(130 212 157 204)(131 213 158 205)(132 214 159 206)(133 215 160 207)(134 216 161 208)(135 217 162 209)(136 218 163 210)(137 219 164 181)(138 220 165 182)(139 221 166 183)(140 222 167 184)(141 223 168 185)(142 224 169 186)(143 225 170 187)(144 226 171 188)(145 227 172 189)(146 228 173 190)(147 229 174 191)(148 230 175 192)(149 231 176 193)(150 232 177 194)
(1 151 43 124)(2 152 44 125)(3 153 45 126)(4 154 46 127)(5 155 47 128)(6 156 48 129)(7 157 49 130)(8 158 50 131)(9 159 51 132)(10 160 52 133)(11 161 53 134)(12 162 54 135)(13 163 55 136)(14 164 56 137)(15 165 57 138)(16 166 58 139)(17 167 59 140)(18 168 60 141)(19 169 31 142)(20 170 32 143)(21 171 33 144)(22 172 34 145)(23 173 35 146)(24 174 36 147)(25 175 37 148)(26 176 38 149)(27 177 39 150)(28 178 40 121)(29 179 41 122)(30 180 42 123)(61 237 102 199)(62 238 103 200)(63 239 104 201)(64 240 105 202)(65 211 106 203)(66 212 107 204)(67 213 108 205)(68 214 109 206)(69 215 110 207)(70 216 111 208)(71 217 112 209)(72 218 113 210)(73 219 114 181)(74 220 115 182)(75 221 116 183)(76 222 117 184)(77 223 118 185)(78 224 119 186)(79 225 120 187)(80 226 91 188)(81 227 92 189)(82 228 93 190)(83 229 94 191)(84 230 95 192)(85 231 96 193)(86 232 97 194)(87 233 98 195)(88 234 99 196)(89 235 100 197)(90 236 101 198)

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C10A···10F12A···12F15A15B15C15D20A···20L30A···30L60A···60X
order1222223444444445566610···1012···121515151520···2030···3060···60
size111130302224430306060222222···24···422224···42···24···4

84 irreducible representations

dim111111222222222222444444
type++++++++-+++++-+-+-+
imageC1C2C2C2C2C2S3D4Q8D5D6C4○D4D10C3⋊D4D15C5⋊D4D30C157D4S3×Q8Q83S3Q8×D5Q82D5Q8×D15Q83D15
kernelD307Q8C30.4Q8C605C4D303C4C2×C4×D15Q8×C30Q8×C10C60D30C6×Q8C2×C20C30C2×C12C20C2×Q8C12C2×C4C4C10C10C6C6C2C2
# reps12121112223264481216112244

Matrix representation of D307Q8 in GL6(𝔽61)

6000000
0600000
0052500
00135300
0000600
0000060
,
6000000
5510000
0001700
0018000
000010
0000060
,
27520000
47340000
001000
000100
000001
0000600
,
100000
010000
001000
000100
0000110
0000050

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,5,13,0,0,0,0,25,53,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,55,0,0,0,0,0,1,0,0,0,0,0,0,0,18,0,0,0,0,17,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60],[27,47,0,0,0,0,52,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,50] >;

D307Q8 in GAP, Magma, Sage, TeX

D_{30}\rtimes_7Q_8
% in TeX

G:=Group("D30:7Q8");
// GroupNames label

G:=SmallGroup(480,911);
// by ID

G=gap.SmallGroup(480,911);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,100,2693,18822]);
// 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,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽