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.C2⋊2C4, (C4×Dic3)⋊15D5, D30.23(C2×C4), (C2×C20).265D6, C3⋊2(C42⋊D5), (Dic3×C20)⋊25C2, C6.70(C4○D20), C10.D4⋊20S3, (C2×C12).193D10, (C2×C30).93C23, C30.54(C22×C4), C30.Q8⋊13C2, Dic5.24(C4×S3), Dic3.16(C4×D5), C15⋊17(C42⋊C2), (Dic3×Dic5)⋊17C2, D30⋊4C4.10C2, D30⋊3C4.15C2, C30.115(C4○D4), C2.3(D60⋊C2), (C2×C60).409C22, (C2×Dic5).103D6, C10.44(D4⋊2S3), C10.12(Q8⋊3S3), (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), C5⋊2(C4⋊C4⋊7S3), (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
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, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, D5, C10, Dic3, Dic3, C12, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, Dic5, C20, D10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, D15, C30, C42⋊C2, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C5×Dic3, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, D30, D30, C2×C30, C4×Dic5, C10.D4, C10.D4, D10⋊C4, C4×C20, C2×C4×D5, C4⋊C4⋊7S3, D30.C2, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, C22×D15, C42⋊D5, Dic3×Dic5, D30⋊4C4, C30.Q8, C3×C10.D4, Dic3×C20, D30⋊3C4, C2×D30.C2, D30.23(C2×C4)
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, C4○D4, D10, C4×S3, C22×S3, C42⋊C2, C4×D5, C22×D5, S3×C2×C4, D4⋊2S3, Q8⋊3S3, S3×D5, C2×C4×D5, C4○D20, C4⋊C4⋊7S3, C2×S3×D5, C42⋊D5, D60⋊C2, C4×S3×D5, Dic5.D6, 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 40)(32 39)(33 38)(34 37)(35 36)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(91 105)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)(121 140)(122 139)(123 138)(124 137)(125 136)(126 135)(127 134)(128 133)(129 132)(130 131)(141 150)(142 149)(143 148)(144 147)(145 146)(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)
(1 146 16 131)(2 127 17 142)(3 138 18 123)(4 149 19 134)(5 130 20 145)(6 141 21 126)(7 122 22 137)(8 133 23 148)(9 144 24 129)(10 125 25 140)(11 136 26 121)(12 147 27 132)(13 128 28 143)(14 139 29 124)(15 150 30 135)(31 151 46 166)(32 162 47 177)(33 173 48 158)(34 154 49 169)(35 165 50 180)(36 176 51 161)(37 157 52 172)(38 168 53 153)(39 179 54 164)(40 160 55 175)(41 171 56 156)(42 152 57 167)(43 163 58 178)(44 174 59 159)(45 155 60 170)(61 191 76 206)(62 202 77 187)(63 183 78 198)(64 194 79 209)(65 205 80 190)(66 186 81 201)(67 197 82 182)(68 208 83 193)(69 189 84 204)(70 200 85 185)(71 181 86 196)(72 192 87 207)(73 203 88 188)(74 184 89 199)(75 195 90 210)(91 211 106 226)(92 222 107 237)(93 233 108 218)(94 214 109 229)(95 225 110 240)(96 236 111 221)(97 217 112 232)(98 228 113 213)(99 239 114 224)(100 220 115 235)(101 231 116 216)(102 212 117 227)(103 223 118 238)(104 234 119 219)(105 215 120 230)
(1 106 51 61)(2 107 52 62)(3 108 53 63)(4 109 54 64)(5 110 55 65)(6 111 56 66)(7 112 57 67)(8 113 58 68)(9 114 59 69)(10 115 60 70)(11 116 31 71)(12 117 32 72)(13 118 33 73)(14 119 34 74)(15 120 35 75)(16 91 36 76)(17 92 37 77)(18 93 38 78)(19 94 39 79)(20 95 40 80)(21 96 41 81)(22 97 42 82)(23 98 43 83)(24 99 44 84)(25 100 45 85)(26 101 46 86)(27 102 47 87)(28 103 48 88)(29 104 49 89)(30 105 50 90)(121 231 166 196)(122 232 167 197)(123 233 168 198)(124 234 169 199)(125 235 170 200)(126 236 171 201)(127 237 172 202)(128 238 173 203)(129 239 174 204)(130 240 175 205)(131 211 176 206)(132 212 177 207)(133 213 178 208)(134 214 179 209)(135 215 180 210)(136 216 151 181)(137 217 152 182)(138 218 153 183)(139 219 154 184)(140 220 155 185)(141 221 156 186)(142 222 157 187)(143 223 158 188)(144 224 159 189)(145 225 160 190)(146 226 161 191)(147 227 162 192)(148 228 163 193)(149 229 164 194)(150 230 165 195)
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)(31,40)(32,39)(33,38)(34,37)(35,36)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(91,105)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)(121,140)(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)(141,150)(142,149)(143,148)(144,147)(145,146)(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), (1,146,16,131)(2,127,17,142)(3,138,18,123)(4,149,19,134)(5,130,20,145)(6,141,21,126)(7,122,22,137)(8,133,23,148)(9,144,24,129)(10,125,25,140)(11,136,26,121)(12,147,27,132)(13,128,28,143)(14,139,29,124)(15,150,30,135)(31,151,46,166)(32,162,47,177)(33,173,48,158)(34,154,49,169)(35,165,50,180)(36,176,51,161)(37,157,52,172)(38,168,53,153)(39,179,54,164)(40,160,55,175)(41,171,56,156)(42,152,57,167)(43,163,58,178)(44,174,59,159)(45,155,60,170)(61,191,76,206)(62,202,77,187)(63,183,78,198)(64,194,79,209)(65,205,80,190)(66,186,81,201)(67,197,82,182)(68,208,83,193)(69,189,84,204)(70,200,85,185)(71,181,86,196)(72,192,87,207)(73,203,88,188)(74,184,89,199)(75,195,90,210)(91,211,106,226)(92,222,107,237)(93,233,108,218)(94,214,109,229)(95,225,110,240)(96,236,111,221)(97,217,112,232)(98,228,113,213)(99,239,114,224)(100,220,115,235)(101,231,116,216)(102,212,117,227)(103,223,118,238)(104,234,119,219)(105,215,120,230), (1,106,51,61)(2,107,52,62)(3,108,53,63)(4,109,54,64)(5,110,55,65)(6,111,56,66)(7,112,57,67)(8,113,58,68)(9,114,59,69)(10,115,60,70)(11,116,31,71)(12,117,32,72)(13,118,33,73)(14,119,34,74)(15,120,35,75)(16,91,36,76)(17,92,37,77)(18,93,38,78)(19,94,39,79)(20,95,40,80)(21,96,41,81)(22,97,42,82)(23,98,43,83)(24,99,44,84)(25,100,45,85)(26,101,46,86)(27,102,47,87)(28,103,48,88)(29,104,49,89)(30,105,50,90)(121,231,166,196)(122,232,167,197)(123,233,168,198)(124,234,169,199)(125,235,170,200)(126,236,171,201)(127,237,172,202)(128,238,173,203)(129,239,174,204)(130,240,175,205)(131,211,176,206)(132,212,177,207)(133,213,178,208)(134,214,179,209)(135,215,180,210)(136,216,151,181)(137,217,152,182)(138,218,153,183)(139,219,154,184)(140,220,155,185)(141,221,156,186)(142,222,157,187)(143,223,158,188)(144,224,159,189)(145,225,160,190)(146,226,161,191)(147,227,162,192)(148,228,163,193)(149,229,164,194)(150,230,165,195)>;
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)(31,40)(32,39)(33,38)(34,37)(35,36)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(91,105)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)(121,140)(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)(141,150)(142,149)(143,148)(144,147)(145,146)(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), (1,146,16,131)(2,127,17,142)(3,138,18,123)(4,149,19,134)(5,130,20,145)(6,141,21,126)(7,122,22,137)(8,133,23,148)(9,144,24,129)(10,125,25,140)(11,136,26,121)(12,147,27,132)(13,128,28,143)(14,139,29,124)(15,150,30,135)(31,151,46,166)(32,162,47,177)(33,173,48,158)(34,154,49,169)(35,165,50,180)(36,176,51,161)(37,157,52,172)(38,168,53,153)(39,179,54,164)(40,160,55,175)(41,171,56,156)(42,152,57,167)(43,163,58,178)(44,174,59,159)(45,155,60,170)(61,191,76,206)(62,202,77,187)(63,183,78,198)(64,194,79,209)(65,205,80,190)(66,186,81,201)(67,197,82,182)(68,208,83,193)(69,189,84,204)(70,200,85,185)(71,181,86,196)(72,192,87,207)(73,203,88,188)(74,184,89,199)(75,195,90,210)(91,211,106,226)(92,222,107,237)(93,233,108,218)(94,214,109,229)(95,225,110,240)(96,236,111,221)(97,217,112,232)(98,228,113,213)(99,239,114,224)(100,220,115,235)(101,231,116,216)(102,212,117,227)(103,223,118,238)(104,234,119,219)(105,215,120,230), (1,106,51,61)(2,107,52,62)(3,108,53,63)(4,109,54,64)(5,110,55,65)(6,111,56,66)(7,112,57,67)(8,113,58,68)(9,114,59,69)(10,115,60,70)(11,116,31,71)(12,117,32,72)(13,118,33,73)(14,119,34,74)(15,120,35,75)(16,91,36,76)(17,92,37,77)(18,93,38,78)(19,94,39,79)(20,95,40,80)(21,96,41,81)(22,97,42,82)(23,98,43,83)(24,99,44,84)(25,100,45,85)(26,101,46,86)(27,102,47,87)(28,103,48,88)(29,104,49,89)(30,105,50,90)(121,231,166,196)(122,232,167,197)(123,233,168,198)(124,234,169,199)(125,235,170,200)(126,236,171,201)(127,237,172,202)(128,238,173,203)(129,239,174,204)(130,240,175,205)(131,211,176,206)(132,212,177,207)(133,213,178,208)(134,214,179,209)(135,215,180,210)(136,216,151,181)(137,217,152,182)(138,218,153,183)(139,219,154,184)(140,220,155,185)(141,221,156,186)(142,222,157,187)(143,223,158,188)(144,224,159,189)(145,225,160,190)(146,226,161,191)(147,227,162,192)(148,228,163,193)(149,229,164,194)(150,230,165,195) );
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),(31,40),(32,39),(33,38),(34,37),(35,36),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(61,75),(62,74),(63,73),(64,72),(65,71),(66,70),(67,69),(76,90),(77,89),(78,88),(79,87),(80,86),(81,85),(82,84),(91,105),(92,104),(93,103),(94,102),(95,101),(96,100),(97,99),(106,120),(107,119),(108,118),(109,117),(110,116),(111,115),(112,114),(121,140),(122,139),(123,138),(124,137),(125,136),(126,135),(127,134),(128,133),(129,132),(130,131),(141,150),(142,149),(143,148),(144,147),(145,146),(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)], [(1,146,16,131),(2,127,17,142),(3,138,18,123),(4,149,19,134),(5,130,20,145),(6,141,21,126),(7,122,22,137),(8,133,23,148),(9,144,24,129),(10,125,25,140),(11,136,26,121),(12,147,27,132),(13,128,28,143),(14,139,29,124),(15,150,30,135),(31,151,46,166),(32,162,47,177),(33,173,48,158),(34,154,49,169),(35,165,50,180),(36,176,51,161),(37,157,52,172),(38,168,53,153),(39,179,54,164),(40,160,55,175),(41,171,56,156),(42,152,57,167),(43,163,58,178),(44,174,59,159),(45,155,60,170),(61,191,76,206),(62,202,77,187),(63,183,78,198),(64,194,79,209),(65,205,80,190),(66,186,81,201),(67,197,82,182),(68,208,83,193),(69,189,84,204),(70,200,85,185),(71,181,86,196),(72,192,87,207),(73,203,88,188),(74,184,89,199),(75,195,90,210),(91,211,106,226),(92,222,107,237),(93,233,108,218),(94,214,109,229),(95,225,110,240),(96,236,111,221),(97,217,112,232),(98,228,113,213),(99,239,114,224),(100,220,115,235),(101,231,116,216),(102,212,117,227),(103,223,118,238),(104,234,119,219),(105,215,120,230)], [(1,106,51,61),(2,107,52,62),(3,108,53,63),(4,109,54,64),(5,110,55,65),(6,111,56,66),(7,112,57,67),(8,113,58,68),(9,114,59,69),(10,115,60,70),(11,116,31,71),(12,117,32,72),(13,118,33,73),(14,119,34,74),(15,120,35,75),(16,91,36,76),(17,92,37,77),(18,93,38,78),(19,94,39,79),(20,95,40,80),(21,96,41,81),(22,97,42,82),(23,98,43,83),(24,99,44,84),(25,100,45,85),(26,101,46,86),(27,102,47,87),(28,103,48,88),(29,104,49,89),(30,105,50,90),(121,231,166,196),(122,232,167,197),(123,233,168,198),(124,234,169,199),(125,235,170,200),(126,236,171,201),(127,237,172,202),(128,238,173,203),(129,239,174,204),(130,240,175,205),(131,211,176,206),(132,212,177,207),(133,213,178,208),(134,214,179,209),(135,215,180,210),(136,216,151,181),(137,217,152,182),(138,218,153,183),(139,219,154,184),(140,220,155,185),(141,221,156,186),(142,222,157,187),(143,223,158,188),(144,224,159,189),(145,225,160,190),(146,226,161,191),(147,227,162,192),(148,228,163,193),(149,229,164,194),(150,230,165,195)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20X | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 30 | 30 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 10 | 10 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D5 | D6 | D6 | C4○D4 | D10 | D10 | C4×S3 | C4×D5 | C4○D20 | D4⋊2S3 | Q8⋊3S3 | S3×D5 | C2×S3×D5 | D60⋊C2 | C4×S3×D5 | Dic5.D6 |
| kernel | D30.23(C2×C4) | Dic3×Dic5 | D30⋊4C4 | C30.Q8 | C3×C10.D4 | Dic3×C20 | D30⋊3C4 | C2×D30.C2 | D30.C2 | C10.D4 | C4×Dic3 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | Dic5 | Dic3 | C6 | C10 | C10 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 4 | 8 | 16 | 1 | 1 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of D30.23(C2×C4) ►in GL6(𝔽61)
| 17 | 60 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 60 | 60 |
| 17 | 60 | 0 | 0 | 0 | 0 |
| 44 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 23 | 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 | 0 |
| 0 | 0 | 0 | 0 | 60 | 60 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 38 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
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