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G = D4.10D30order 480 = 25·3·5

The non-split extension by D4 of D30 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.10D30, Q8.16D30, C30.66C24, C60.90C23, D60.53C22, D30.31C23, C15112- 1+4, Dic30.45C22, Dic15.33C23, C4○D46D15, C55(Q8○D12), (Q8×D15)⋊12C2, (C5×D4).34D6, (C2×C4).24D30, (C5×Q8).54D6, (C3×D4).34D10, (C2×C20).174D6, (C3×Q8).37D10, D42D1512C2, C6.66(C23×D5), (C2×Dic30)⋊17C2, (C2×C12).172D10, C10.66(S3×C23), (C2×C60).90C22, (C2×C30).12C23, D6011C219C2, C2.14(C23×D15), C4.33(C22×D15), C157D4.2C22, C35(D4.10D10), C20.140(C22×S3), (C4×D15).29C22, (D4×C15).39C22, C12.138(C22×D5), (Q8×C15).42C22, C22.4(C22×D15), (C2×Dic15).21C22, (C3×C4○D4)⋊5D5, (C5×C4○D4)⋊9S3, (C15×C4○D4)⋊5C2, (C2×C6).19(C22×D5), (C2×C10).20(C22×S3), SmallGroup(480,1177)

Series: Derived Chief Lower central Upper central

C1C30 — D4.10D30
C1C5C15C30D30C4×D15Q8×D15 — D4.10D30
C15C30 — D4.10D30
C1C2C4○D4

Generators and relations for D4.10D30
 G = < a,b,c,d | a4=b2=1, c30=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=c29 >

Subgroups: 1412 in 292 conjugacy classes, 119 normal (22 characteristic)
C1, C2, C2 [×5], C3, C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], C5, S3 [×2], C6, C6 [×3], C2×C4 [×3], C2×C4 [×12], D4 [×3], D4 [×7], Q8, Q8 [×9], D5 [×2], C10, C10 [×3], Dic3 [×6], C12, C12 [×3], D6 [×2], C2×C6 [×3], C15, C2×Q8 [×5], C4○D4, C4○D4 [×9], Dic5 [×6], C20, C20 [×3], D10 [×2], C2×C10 [×3], Dic6 [×9], C4×S3 [×6], D12, C2×Dic3 [×6], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, D15 [×2], C30, C30 [×3], 2- 1+4, Dic10 [×9], C4×D5 [×6], D20, C2×Dic5 [×6], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, C2×Dic6 [×3], C4○D12 [×3], D42S3 [×6], S3×Q8 [×2], C3×C4○D4, Dic15 [×6], C60, C60 [×3], D30 [×2], C2×C30 [×3], C2×Dic10 [×3], C4○D20 [×3], D42D5 [×6], Q8×D5 [×2], C5×C4○D4, Q8○D12, Dic30 [×9], C4×D15 [×6], D60, C2×Dic15 [×6], C157D4 [×6], C2×C60 [×3], D4×C15 [×3], Q8×C15, D4.10D10, C2×Dic30 [×3], D6011C2 [×3], D42D15 [×6], Q8×D15 [×2], C15×C4○D4, D4.10D30
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], D15, 2- 1+4, C22×D5 [×7], S3×C23, D30 [×7], C23×D5, Q8○D12, C22×D15 [×7], D4.10D10, C23×D15, D4.10D30

Smallest permutation representation of D4.10D30
On 240 points
Generators in S240
(1 216 31 186)(2 187 32 217)(3 218 33 188)(4 189 34 219)(5 220 35 190)(6 191 36 221)(7 222 37 192)(8 193 38 223)(9 224 39 194)(10 195 40 225)(11 226 41 196)(12 197 42 227)(13 228 43 198)(14 199 44 229)(15 230 45 200)(16 201 46 231)(17 232 47 202)(18 203 48 233)(19 234 49 204)(20 205 50 235)(21 236 51 206)(22 207 52 237)(23 238 53 208)(24 209 54 239)(25 240 55 210)(26 211 56 181)(27 182 57 212)(28 213 58 183)(29 184 59 214)(30 215 60 185)(61 140 91 170)(62 171 92 141)(63 142 93 172)(64 173 94 143)(65 144 95 174)(66 175 96 145)(67 146 97 176)(68 177 98 147)(69 148 99 178)(70 179 100 149)(71 150 101 180)(72 121 102 151)(73 152 103 122)(74 123 104 153)(75 154 105 124)(76 125 106 155)(77 156 107 126)(78 127 108 157)(79 158 109 128)(80 129 110 159)(81 160 111 130)(82 131 112 161)(83 162 113 132)(84 133 114 163)(85 164 115 134)(86 135 116 165)(87 166 117 136)(88 137 118 167)(89 168 119 138)(90 139 120 169)
(1 78)(2 109)(3 80)(4 111)(5 82)(6 113)(7 84)(8 115)(9 86)(10 117)(11 88)(12 119)(13 90)(14 61)(15 92)(16 63)(17 94)(18 65)(19 96)(20 67)(21 98)(22 69)(23 100)(24 71)(25 102)(26 73)(27 104)(28 75)(29 106)(30 77)(31 108)(32 79)(33 110)(34 81)(35 112)(36 83)(37 114)(38 85)(39 116)(40 87)(41 118)(42 89)(43 120)(44 91)(45 62)(46 93)(47 64)(48 95)(49 66)(50 97)(51 68)(52 99)(53 70)(54 101)(55 72)(56 103)(57 74)(58 105)(59 76)(60 107)(121 240)(122 211)(123 182)(124 213)(125 184)(126 215)(127 186)(128 217)(129 188)(130 219)(131 190)(132 221)(133 192)(134 223)(135 194)(136 225)(137 196)(138 227)(139 198)(140 229)(141 200)(142 231)(143 202)(144 233)(145 204)(146 235)(147 206)(148 237)(149 208)(150 239)(151 210)(152 181)(153 212)(154 183)(155 214)(156 185)(157 216)(158 187)(159 218)(160 189)(161 220)(162 191)(163 222)(164 193)(165 224)(166 195)(167 226)(168 197)(169 228)(170 199)(171 230)(172 201)(173 232)(174 203)(175 234)(176 205)(177 236)(178 207)(179 238)(180 209)
(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)
(1 141 31 171)(2 170 32 140)(3 139 33 169)(4 168 34 138)(5 137 35 167)(6 166 36 136)(7 135 37 165)(8 164 38 134)(9 133 39 163)(10 162 40 132)(11 131 41 161)(12 160 42 130)(13 129 43 159)(14 158 44 128)(15 127 45 157)(16 156 46 126)(17 125 47 155)(18 154 48 124)(19 123 49 153)(20 152 50 122)(21 121 51 151)(22 150 52 180)(23 179 53 149)(24 148 54 178)(25 177 55 147)(26 146 56 176)(27 175 57 145)(28 144 58 174)(29 173 59 143)(30 142 60 172)(61 187 91 217)(62 216 92 186)(63 185 93 215)(64 214 94 184)(65 183 95 213)(66 212 96 182)(67 181 97 211)(68 210 98 240)(69 239 99 209)(70 208 100 238)(71 237 101 207)(72 206 102 236)(73 235 103 205)(74 204 104 234)(75 233 105 203)(76 202 106 232)(77 231 107 201)(78 200 108 230)(79 229 109 199)(80 198 110 228)(81 227 111 197)(82 196 112 226)(83 225 113 195)(84 194 114 224)(85 223 115 193)(86 192 116 222)(87 221 117 191)(88 190 118 220)(89 219 119 189)(90 188 120 218)

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

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

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

87 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E···4J5A5B6A6B6C6D10A10B10C···10H12A12B12C12D12E15A15B15C15D20A20B20C20D20E···20J30A30B30C30D30E···30P60A···60H60I···60T
order1222222344444···4556666101010···101212121212151515152020202020···203030303030···3060···6060···60
size1122230302222230···30222444224···422444222222224···422224···42···24···4

87 irreducible representations

dim1111112222222222224444
type++++++++++++++++++----
imageC1C2C2C2C2C2S3D5D6D6D6D10D10D10D15D30D30D302- 1+4Q8○D12D4.10D10D4.10D30
kernelD4.10D30C2×Dic30D6011C2D42D15Q8×D15C15×C4○D4C5×C4○D4C3×C4○D4C2×C20C5×D4C5×Q8C2×C12C3×D4C3×Q8C4○D4C2×C4D4Q8C15C5C3C1
# reps133621123316624121241248

Matrix representation of D4.10D30 in GL4(𝔽61) generated by

6001414
060470
03510
262601
,
383525
2623258
4331326
49125558
,
2553110
8111152
1731588
29473128
,
59571348
2324558
26502123
24503840
G:=sub<GL(4,GF(61))| [60,0,0,26,0,60,35,26,14,47,1,0,14,0,0,1],[38,26,43,49,35,23,31,12,2,2,3,55,5,58,26,58],[25,8,17,29,53,11,31,47,1,11,58,31,10,52,8,28],[59,23,26,24,57,2,50,50,13,45,21,38,48,58,23,40] >;

D4.10D30 in GAP, Magma, Sage, TeX

D_4._{10}D_{30}
% in TeX

G:=Group("D4.10D30");
// GroupNames label

G:=SmallGroup(480,1177);
// by ID

G=gap.SmallGroup(480,1177);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,219,100,675,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=1,c^30=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^29>;
// generators/relations

׿
×
𝔽