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G = D30.7D4order 480 = 25·3·5

7th non-split extension by D30 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D30.7D4, D6⋊C418D5, C4⋊Dic58S3, C6.44(D4×D5), C10.45(S3×D4), (C2×C20).27D6, C30.56(C2×D4), D6⋊Dic512C2, C54(D6.D4), D303C416C2, C6.12(C4○D20), (C2×C12).229D10, Dic155C424C2, (C2×Dic5).43D6, C30.124(C4○D4), C10.73(C4○D12), C6.48(D42D5), C32(D10.12D4), C2.20(C20⋊D6), (C2×C60).260C22, (C2×C30).128C23, (C22×S3).13D10, C2.18(D60⋊C2), C10.17(Q83S3), (C2×Dic3).109D10, C1512(C22.D4), (C6×Dic5).80C22, C2.19(Dic3.D10), (C10×Dic3).81C22, (C22×D15).43C22, (C2×Dic15).101C22, (C5×D6⋊C4)⋊18C2, (C2×C4).58(S3×D5), (C2×D30.C2)⋊8C2, (C3×C4⋊Dic5)⋊19C2, (C2×C5⋊D12).8C2, C22.188(C2×S3×D5), (S3×C2×C10).26C22, (C2×C6).140(C22×D5), (C2×C10).140(C22×S3), SmallGroup(480,514)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D30.7D4
C1C5C15C30C2×C30C6×Dic5C2×D30.C2 — D30.7D4
C15C2×C30 — D30.7D4
C1C22C2×C4

Generators and relations for D30.7D4
 G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, ac=ca, dad=a11, cbc-1=a15b, dbd=a25b, dcd=a15c-1 >

Subgroups: 908 in 156 conjugacy classes, 46 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×5], C22, C22 [×7], C5, S3 [×3], C6 [×3], C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], D5 [×2], C10 [×3], C10, Dic3 [×2], C12 [×3], D6 [×7], C2×C6, C15, C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×3], C20 [×2], D10 [×4], C2×C10, C2×C10 [×3], C4×S3 [×2], D12 [×2], C2×Dic3, C2×Dic3, C2×C12, C2×C12 [×2], C22×S3, C22×S3, C5×S3, D15 [×2], C30 [×3], C22.D4, C4×D5 [×2], C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×2], C2×C20, C2×C20, C22×D5, C22×C10, Dic3⋊C4, D6⋊C4, D6⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C2×D12, C5×Dic3, C3×Dic5 [×2], Dic15, C60, S3×C10 [×3], D30 [×2], D30 [×2], C2×C30, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C2×C5⋊D4, D6.D4, D30.C2 [×2], C5⋊D12 [×2], C6×Dic5 [×2], C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, C22×D15, D10.12D4, D6⋊Dic5, Dic155C4, C3×C4⋊Dic5, C5×D6⋊C4, D303C4, C2×D30.C2, C2×C5⋊D12, D30.7D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], C22×S3, C22.D4, C22×D5, C4○D12, S3×D4, Q83S3, S3×D5, C4○D20, D4×D5, D42D5, D6.D4, C2×S3×D5, D10.12D4, D60⋊C2, C20⋊D6, Dic3.D10, D30.7D4

Smallest permutation representation of D30.7D4
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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(44 60)(45 59)(46 58)(47 57)(48 56)(49 55)(50 54)(51 53)(61 65)(62 64)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(91 106)(92 105)(93 104)(94 103)(95 102)(96 101)(97 100)(98 99)(107 120)(108 119)(109 118)(110 117)(111 116)(112 115)(113 114)(121 132)(122 131)(123 130)(124 129)(125 128)(126 127)(133 150)(134 149)(135 148)(136 147)(137 146)(138 145)(139 144)(140 143)(141 142)(151 179)(152 178)(153 177)(154 176)(155 175)(156 174)(157 173)(158 172)(159 171)(160 170)(161 169)(162 168)(163 167)(164 166)(181 191)(182 190)(183 189)(184 188)(185 187)(192 210)(193 209)(194 208)(195 207)(196 206)(197 205)(198 204)(199 203)(200 202)(211 216)(212 215)(213 214)(217 240)(218 239)(219 238)(220 237)(221 236)(222 235)(223 234)(224 233)(225 232)(226 231)(227 230)(228 229)
(1 173 214 60)(2 174 215 31)(3 175 216 32)(4 176 217 33)(5 177 218 34)(6 178 219 35)(7 179 220 36)(8 180 221 37)(9 151 222 38)(10 152 223 39)(11 153 224 40)(12 154 225 41)(13 155 226 42)(14 156 227 43)(15 157 228 44)(16 158 229 45)(17 159 230 46)(18 160 231 47)(19 161 232 48)(20 162 233 49)(21 163 234 50)(22 164 235 51)(23 165 236 52)(24 166 237 53)(25 167 238 54)(26 168 239 55)(27 169 240 56)(28 170 211 57)(29 171 212 58)(30 172 213 59)(61 147 199 119)(62 148 200 120)(63 149 201 91)(64 150 202 92)(65 121 203 93)(66 122 204 94)(67 123 205 95)(68 124 206 96)(69 125 207 97)(70 126 208 98)(71 127 209 99)(72 128 210 100)(73 129 181 101)(74 130 182 102)(75 131 183 103)(76 132 184 104)(77 133 185 105)(78 134 186 106)(79 135 187 107)(80 136 188 108)(81 137 189 109)(82 138 190 110)(83 139 191 111)(84 140 192 112)(85 141 193 113)(86 142 194 114)(87 143 195 115)(88 144 196 116)(89 145 197 117)(90 146 198 118)
(1 86)(2 67)(3 78)(4 89)(5 70)(6 81)(7 62)(8 73)(9 84)(10 65)(11 76)(12 87)(13 68)(14 79)(15 90)(16 71)(17 82)(18 63)(19 74)(20 85)(21 66)(22 77)(23 88)(24 69)(25 80)(26 61)(27 72)(28 83)(29 64)(30 75)(31 138)(32 149)(33 130)(34 141)(35 122)(36 133)(37 144)(38 125)(39 136)(40 147)(41 128)(42 139)(43 150)(44 131)(45 142)(46 123)(47 134)(48 145)(49 126)(50 137)(51 148)(52 129)(53 140)(54 121)(55 132)(56 143)(57 124)(58 135)(59 146)(60 127)(91 175)(92 156)(93 167)(94 178)(95 159)(96 170)(97 151)(98 162)(99 173)(100 154)(101 165)(102 176)(103 157)(104 168)(105 179)(106 160)(107 171)(108 152)(109 163)(110 174)(111 155)(112 166)(113 177)(114 158)(115 169)(116 180)(117 161)(118 172)(119 153)(120 164)(181 221)(182 232)(183 213)(184 224)(185 235)(186 216)(187 227)(188 238)(189 219)(190 230)(191 211)(192 222)(193 233)(194 214)(195 225)(196 236)(197 217)(198 228)(199 239)(200 220)(201 231)(202 212)(203 223)(204 234)(205 215)(206 226)(207 237)(208 218)(209 229)(210 240)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B6A6B6C10A···10F10G10H10I10J12A12B12C12D12E12F15A15B20A20B20C20D20E20F20G20H30A···30F60A···60H
order1222222344444445566610···10101010101212121212121515202020202020202030···3060···60
size1111123030246610102060222222···2121212124420202020444444121212124···44···4

60 irreducible representations

dim1111111122222222222444444444
type++++++++++++++++++++-++
imageC1C2C2C2C2C2C2C2S3D4D5D6D6C4○D4D10D10D10C4○D12C4○D20S3×D4Q83S3S3×D5D4×D5D42D5C2×S3×D5D60⋊C2C20⋊D6Dic3.D10
kernelD30.7D4D6⋊Dic5Dic155C4C3×C4⋊Dic5C5×D6⋊C4D303C4C2×D30.C2C2×C5⋊D12C4⋊Dic5D30D6⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12C22×S3C10C6C10C10C2×C4C6C6C22C2C2C2
# reps1111111112221422248112222444

Matrix representation of D30.7D4 in GL6(𝔽61)

1190000
60430000
0060000
0006000
00005920
000091
,
0430000
4400000
0060000
0015100
0000600
000091
,
25500000
7360000
0052100
00255600
000010
000001
,
30600000
45310000
0061300
0025500
00006020
000001

G:=sub<GL(6,GF(61))| [1,60,0,0,0,0,19,43,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,59,9,0,0,0,0,20,1],[0,44,0,0,0,0,43,0,0,0,0,0,0,0,60,15,0,0,0,0,0,1,0,0,0,0,0,0,60,9,0,0,0,0,0,1],[25,7,0,0,0,0,50,36,0,0,0,0,0,0,5,25,0,0,0,0,21,56,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[30,45,0,0,0,0,60,31,0,0,0,0,0,0,6,2,0,0,0,0,13,55,0,0,0,0,0,0,60,0,0,0,0,0,20,1] >;

D30.7D4 in GAP, Magma, Sage, TeX

D_{30}._7D_4
% in TeX

G:=Group("D30.7D4");
// GroupNames label

G:=SmallGroup(480,514);
// by ID

G=gap.SmallGroup(480,514);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,590,219,142,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^11,c*b*c^-1=a^15*b,d*b*d=a^25*b,d*c*d=a^15*c^-1>;
// generators/relations

׿
×
𝔽