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G = D30.23(C2×C4)  order 480 = 25·3·5

8th non-split extension by D30 of C2×C4 acting via C2×C4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D30.C22C4, (C4×Dic3)⋊15D5, D30.23(C2×C4), (C2×C20).265D6, C32(C42⋊D5), (Dic3×C20)⋊25C2, C6.70(C4○D20), C10.D420S3, (C2×C12).193D10, (C2×C30).93C23, C30.54(C22×C4), C30.Q813C2, Dic5.24(C4×S3), Dic3.16(C4×D5), C1517(C42⋊C2), (Dic3×Dic5)⋊17C2, D304C4.10C2, D303C4.15C2, C30.115(C4○D4), C2.3(D60⋊C2), (C2×C60).409C22, (C2×Dic5).103D6, C10.44(D42S3), C10.12(Q83S3), (C2×Dic3).180D10, C2.4(Dic5.D6), (C6×Dic5).54C22, (C2×Dic15).75C22, (C22×D15).29C22, (C10×Dic3).179C22, C2.24(C4×S3×D5), C6.22(C2×C4×D5), C10.55(S3×C2×C4), C52(C4⋊C47S3), (C2×C4).77(S3×D5), C22.48(C2×S3×D5), (C2×D30.C2).2C2, (C3×C10.D4)⋊32C2, (C5×Dic3).35(C2×C4), (C3×Dic5).12(C2×C4), (C2×C6).105(C22×D5), (C2×C10).105(C22×S3), SmallGroup(480,479)

Series: Derived Chief Lower central Upper central

C1C30 — D30.23(C2×C4)
C1C5C15C30C2×C30C6×Dic5C2×D30.C2 — D30.23(C2×C4)
C15C30 — D30.23(C2×C4)
C1C22C2×C4

Generators and relations for D30.23(C2×C4)
 G = < a,b,c,d | a30=b2=d4=1, c2=a15, bab=a-1, cac-1=a11, ad=da, cbc-1=a10b, dbd-1=a15b, cd=dc >

Subgroups: 748 in 152 conjugacy classes, 58 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×8], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4 [×9], C23, D5 [×2], C10 [×3], Dic3 [×2], Dic3 [×2], C12 [×4], D6 [×4], C2×C6, C15, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, Dic5 [×2], Dic5 [×2], C20 [×4], D10 [×4], C2×C10, C4×S3 [×4], C2×Dic3 [×2], C2×Dic3, C2×C12, C2×C12 [×2], C22×S3, D15 [×2], C30 [×3], C42⋊C2, C4×D5 [×4], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C4×Dic3, C4×Dic3, C4⋊Dic3, D6⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C5×Dic3 [×2], C5×Dic3, C3×Dic5 [×2], C3×Dic5, Dic15, C60, D30 [×2], D30 [×2], C2×C30, C4×Dic5, C10.D4, C10.D4, D10⋊C4 [×2], C4×C20, C2×C4×D5, C4⋊C47S3, D30.C2 [×4], C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15, C2×C60, C22×D15, C42⋊D5, Dic3×Dic5, D304C4, C30.Q8, C3×C10.D4, Dic3×C20, D303C4, C2×D30.C2, D30.23(C2×C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], C22×C4, C4○D4 [×2], D10 [×3], C4×S3 [×2], C22×S3, C42⋊C2, C4×D5 [×2], C22×D5, S3×C2×C4, D42S3, Q83S3, S3×D5, C2×C4×D5, C4○D20 [×2], C4⋊C47S3, C2×S3×D5, C42⋊D5, D60⋊C2, C4×S3×D5, Dic5.D6, D30.23(C2×C4)

Smallest permutation representation of D30.23(C2×C4)
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 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(51 60)(52 59)(53 58)(54 57)(55 56)(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)(91 115)(92 114)(93 113)(94 112)(95 111)(96 110)(97 109)(98 108)(99 107)(100 106)(101 105)(102 104)(116 120)(117 119)(121 150)(122 149)(123 148)(124 147)(125 146)(126 145)(127 144)(128 143)(129 142)(130 141)(131 140)(132 139)(133 138)(134 137)(135 136)(151 170)(152 169)(153 168)(154 167)(155 166)(156 165)(157 164)(158 163)(159 162)(160 161)(171 180)(172 179)(173 178)(174 177)(175 176)(181 185)(182 184)(186 210)(187 209)(188 208)(189 207)(190 206)(191 205)(192 204)(193 203)(194 202)(195 201)(196 200)(197 199)(211 225)(212 224)(213 223)(214 222)(215 221)(216 220)(217 219)(226 240)(227 239)(228 238)(229 237)(230 236)(231 235)(232 234)
(1 121 16 136)(2 132 17 147)(3 143 18 128)(4 124 19 139)(5 135 20 150)(6 146 21 131)(7 127 22 142)(8 138 23 123)(9 149 24 134)(10 130 25 145)(11 141 26 126)(12 122 27 137)(13 133 28 148)(14 144 29 129)(15 125 30 140)(31 156 46 171)(32 167 47 152)(33 178 48 163)(34 159 49 174)(35 170 50 155)(36 151 51 166)(37 162 52 177)(38 173 53 158)(39 154 54 169)(40 165 55 180)(41 176 56 161)(42 157 57 172)(43 168 58 153)(44 179 59 164)(45 160 60 175)(61 201 76 186)(62 182 77 197)(63 193 78 208)(64 204 79 189)(65 185 80 200)(66 196 81 181)(67 207 82 192)(68 188 83 203)(69 199 84 184)(70 210 85 195)(71 191 86 206)(72 202 87 187)(73 183 88 198)(74 194 89 209)(75 205 90 190)(91 231 106 216)(92 212 107 227)(93 223 108 238)(94 234 109 219)(95 215 110 230)(96 226 111 211)(97 237 112 222)(98 218 113 233)(99 229 114 214)(100 240 115 225)(101 221 116 236)(102 232 117 217)(103 213 118 228)(104 224 119 239)(105 235 120 220)
(1 96 56 86)(2 97 57 87)(3 98 58 88)(4 99 59 89)(5 100 60 90)(6 101 31 61)(7 102 32 62)(8 103 33 63)(9 104 34 64)(10 105 35 65)(11 106 36 66)(12 107 37 67)(13 108 38 68)(14 109 39 69)(15 110 40 70)(16 111 41 71)(17 112 42 72)(18 113 43 73)(19 114 44 74)(20 115 45 75)(21 116 46 76)(22 117 47 77)(23 118 48 78)(24 119 49 79)(25 120 50 80)(26 91 51 81)(27 92 52 82)(28 93 53 83)(29 94 54 84)(30 95 55 85)(121 226 161 206)(122 227 162 207)(123 228 163 208)(124 229 164 209)(125 230 165 210)(126 231 166 181)(127 232 167 182)(128 233 168 183)(129 234 169 184)(130 235 170 185)(131 236 171 186)(132 237 172 187)(133 238 173 188)(134 239 174 189)(135 240 175 190)(136 211 176 191)(137 212 177 192)(138 213 178 193)(139 214 179 194)(140 215 180 195)(141 216 151 196)(142 217 152 197)(143 218 153 198)(144 219 154 199)(145 220 155 200)(146 221 156 201)(147 222 157 202)(148 223 158 203)(149 224 159 204)(150 225 160 205)

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

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

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

78 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I4J4K4L4M4N5A5B6A6B6C10A···10F12A12B12C12D12E12F15A15B20A···20H20I···20X30A···30F60A···60H
order1222223444444444444445566610···10121212121212151520···2020···2030···3060···60
size11113030222333366101010103030222222···24420202020442···26···64···44···4

78 irreducible representations

dim11111111122222222224444444
type++++++++++++++-++++
imageC1C2C2C2C2C2C2C2C4S3D5D6D6C4○D4D10D10C4×S3C4×D5C4○D20D42S3Q83S3S3×D5C2×S3×D5D60⋊C2C4×S3×D5Dic5.D6
kernelD30.23(C2×C4)Dic3×Dic5D304C4C30.Q8C3×C10.D4Dic3×C20D303C4C2×D30.C2D30.C2C10.D4C4×Dic3C2×Dic5C2×C20C30C2×Dic3C2×C12Dic5Dic3C6C10C10C2×C4C22C2C2C2
# reps111111118122144248161122444

Matrix representation of D30.23(C2×C4) in GL6(𝔽61)

17600000
100000
0060000
0006000
000001
00006060
,
17600000
44440000
0060000
0023100
000001
000010
,
100000
010000
0011000
0001100
000010
00006060
,
1100000
0110000
0011600
00386000
0000600
0000060

G:=sub<GL(6,GF(61))| [17,1,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,1,60],[17,44,0,0,0,0,60,44,0,0,0,0,0,0,60,23,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,60,0,0,0,0,0,60],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,38,0,0,0,0,16,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;

D30.23(C2×C4) in GAP, Magma, Sage, TeX

D_{30}._{23}(C_2\times C_4)
% in TeX

G:=Group("D30.23(C2xC4)");
// GroupNames label

G:=SmallGroup(480,479);
// by ID

G=gap.SmallGroup(480,479);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,422,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=b^2=d^4=1,c^2=a^15,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^10*b,d*b*d^-1=a^15*b,c*d=d*c>;
// generators/relations

׿
×
𝔽