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G = D30.28D4order 480 = 25·3·5

1st non-split extension by D30 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D30.28D4, C23.9D30, C2.8(D4×D15), C6.99(D4×D5), (C2×C4).6D30, C22⋊C43D15, C605C410C2, (C2×C20).32D6, D303C45C2, (C2×C12).33D10, C10.101(S3×D4), C30.307(C2×D4), C55(C23.9D6), C6.96(C4○D20), C30.38D44C2, C30.4Q810C2, (C22×C6).58D10, (C22×C10).73D6, C30.169(C4○D4), C10.96(C4○D12), C2.8(D42D15), C6.92(D42D5), C35(D10.12D4), (C2×C30).280C23, (C2×C60).173C22, C10.92(D42S3), C1525(C22.D4), (C22×C30).14C22, C2.10(D6011C2), C22.42(C22×D15), (C22×D15).80C22, (C2×Dic15).158C22, (C2×C4×D15)⋊16C2, (C5×C22⋊C4)⋊5S3, (C3×C22⋊C4)⋊5D5, (C15×C22⋊C4)⋊7C2, (C2×C157D4).3C2, (C2×C6).276(C22×D5), (C2×C10).275(C22×S3), SmallGroup(480,848)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D30.28D4
C1C5C15C30C2×C30C22×D15C2×C4×D15 — D30.28D4
C15C2×C30 — D30.28D4
C1C22C22⋊C4

Generators and relations for D30.28D4
 G = < a,b,c,d | a30=b2=c4=1, d2=a15, bab=a-1, ac=ca, ad=da, cbc-1=a15b, bd=db, dcd-1=a15c-1 >

Subgroups: 932 in 156 conjugacy classes, 49 normal (47 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×5], C22, C22 [×7], C5, S3 [×2], C6 [×3], C6, C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, D5 [×2], C10 [×3], C10, Dic3 [×3], C12 [×2], D6 [×4], C2×C6, C2×C6 [×3], C15, C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×3], C20 [×2], D10 [×4], C2×C10, C2×C10 [×3], C4×S3 [×2], C2×Dic3 [×3], C3⋊D4 [×2], C2×C12 [×2], C22×S3, C22×C6, D15 [×2], C30 [×3], C30, C22.D4, C4×D5 [×2], C2×Dic5 [×3], C5⋊D4 [×2], C2×C20 [×2], C22×D5, C22×C10, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, C2×C3⋊D4, Dic15 [×3], C60 [×2], D30 [×2], D30 [×2], C2×C30, C2×C30 [×3], C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C2×C5⋊D4, C23.9D6, C4×D15 [×2], C2×Dic15 [×3], C157D4 [×2], C2×C60 [×2], C22×D15, C22×C30, D10.12D4, C30.4Q8, C605C4, D303C4, C30.38D4, C15×C22⋊C4, C2×C4×D15, C2×C157D4, D30.28D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], C22×S3, D15, C22.D4, C22×D5, C4○D12, S3×D4, D42S3, D30 [×3], C4○D20, D4×D5, D42D5, C23.9D6, C22×D15, D10.12D4, D6011C2, D4×D15, D42D15, D30.28D4

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

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B6A6B6C6D6E10A···10F10G10H10I10J12A12B12C12D15A15B15C15D20A···20H30A···30L30M···30T60A···60P
order122222234444444556666610···1010101010121212121515151520···2030···3030···3060···60
size11114303022243030606022222442···24444444422224···42···24···44···4

84 irreducible representations

dim1111111122222222222222444444
type+++++++++++++++++++-+-+-
imageC1C2C2C2C2C2C2C2S3D4D5D6D6C4○D4D10D10D15C4○D12D30D30C4○D20D6011C2S3×D4D42S3D4×D5D42D5D4×D15D42D15
kernelD30.28D4C30.4Q8C605C4D303C4C30.38D4C15×C22⋊C4C2×C4×D15C2×C157D4C5×C22⋊C4D30C3×C22⋊C4C2×C20C22×C10C30C2×C12C22×C6C22⋊C4C10C2×C4C23C6C2C10C10C6C6C2C2
# reps11111111122214424484816112244

Matrix representation of D30.28D4 in GL8(𝔽61)

600000000
060000000
0053200000
004970000
0000176000
00001000
000000600
000000060
,
600000000
11000000
0053200000
00680000
0000472400
0000301400
000000600
00000001
,
5039000000
011000000
00100000
00010000
00001000
00000100
000000011
000000500
,
110000000
011000000
006000000
000600000
000060000
000006000
000000500
000000011

G:=sub<GL(8,GF(61))| [60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,53,49,0,0,0,0,0,0,20,7,0,0,0,0,0,0,0,0,17,1,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[60,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,53,6,0,0,0,0,0,0,20,8,0,0,0,0,0,0,0,0,47,30,0,0,0,0,0,0,24,14,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1],[50,0,0,0,0,0,0,0,39,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,11,0],[11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,11] >;

D30.28D4 in GAP, Magma, Sage, TeX

D_{30}._{28}D_4
% in TeX

G:=Group("D30.28D4");
// GroupNames label

G:=SmallGroup(480,848);
// by ID

G=gap.SmallGroup(480,848);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,254,219,2693,18822]);
// Polycyclic

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

׿
×
𝔽