metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C6×Dic5)⋊7C4, C23.37(S3×D5), (C2×Dic5)⋊4Dic3, C30.Q8⋊27C2, C6.D4.6D5, (C22×C10).34D6, (C22×C6).81D10, C15⋊18(C42⋊C2), (Dic3×Dic5)⋊27C2, C30.130(C4○D4), C10.76(C4○D12), C6.50(D4⋊2D5), C22.7(D5×Dic3), C30.134(C22×C4), (C2×C30).170C23, (C2×Dic5).218D6, C30.38D4.5C2, C5⋊4(C23.26D6), (C22×Dic5).4S3, Dic5.22(C2×Dic3), (C2×Dic3).117D10, C2.5(Dic3.D10), C3⋊5(C23.11D10), (C22×C30).32C22, C10.28(C22×Dic3), (C6×Dic5).216C22, (C10×Dic3).99C22, (C2×Dic15).120C22, C6.91(C2×C4×D5), (C2×C6).54(C4×D5), (C2×C6×Dic5).3C2, C2.15(C2×D5×Dic3), C22.74(C2×S3×D5), (C2×C30).109(C2×C4), (C3×Dic5).60(C2×C4), (C2×C10).26(C2×Dic3), (C5×C6.D4).1C2, (C2×C6).182(C22×D5), (C2×C10).182(C22×S3), SmallGroup(480,604)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C6×Dic5)⋊7C4
G = < a,b,c,d | a6=b10=d4=1, c2=b5, ab=ba, ac=ca, dad-1=a-1b5, cbc-1=b-1, bd=db, cd=dc >
Subgroups: 540 in 152 conjugacy classes, 68 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C23, C10, C10, C10, Dic3, C12, C2×C6, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C30, C30, C30, C42⋊C2, C2×Dic5, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C4×Dic3, C4⋊Dic3, C6.D4, C6.D4, C22×C12, C5×Dic3, C3×Dic5, Dic15, C2×C30, C2×C30, C2×C30, C4×Dic5, C10.D4, C23.D5, C5×C22⋊C4, C22×Dic5, C23.26D6, C6×Dic5, C6×Dic5, C10×Dic3, C2×Dic15, C22×C30, C23.11D10, Dic3×Dic5, C30.Q8, C5×C6.D4, C30.38D4, C2×C6×Dic5, (C6×Dic5)⋊7C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, Dic3, D6, C22×C4, C4○D4, D10, C2×Dic3, C22×S3, C42⋊C2, C4×D5, C22×D5, C4○D12, C22×Dic3, S3×D5, C2×C4×D5, D4⋊2D5, C23.26D6, D5×Dic3, C2×S3×D5, C23.11D10, C2×D5×Dic3, Dic3.D10, (C6×Dic5)⋊7C4
►On 240 points
Generators in S240
(1 62 27 51 32 47)(2 63 28 52 33 48)(3 64 29 53 34 49)(4 65 30 54 35 50)(5 66 21 55 36 41)(6 67 22 56 37 42)(7 68 23 57 38 43)(8 69 24 58 39 44)(9 70 25 59 40 45)(10 61 26 60 31 46)(11 209 223 213 234 198)(12 210 224 214 235 199)(13 201 225 215 236 200)(14 202 226 216 237 191)(15 203 227 217 238 192)(16 204 228 218 239 193)(17 205 229 219 240 194)(18 206 230 220 231 195)(19 207 221 211 232 196)(20 208 222 212 233 197)(71 124 85 106 96 120)(72 125 86 107 97 111)(73 126 87 108 98 112)(74 127 88 109 99 113)(75 128 89 110 100 114)(76 129 90 101 91 115)(77 130 81 102 92 116)(78 121 82 103 93 117)(79 122 83 104 94 118)(80 123 84 105 95 119)(131 172 156 161 142 186)(132 173 157 162 143 187)(133 174 158 163 144 188)(134 175 159 164 145 189)(135 176 160 165 146 190)(136 177 151 166 147 181)(137 178 152 167 148 182)(138 179 153 168 149 183)(139 180 154 169 150 184)(140 171 155 170 141 185)
(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)
(1 144 6 149)(2 143 7 148)(3 142 8 147)(4 141 9 146)(5 150 10 145)(11 130 16 125)(12 129 17 124)(13 128 18 123)(14 127 19 122)(15 126 20 121)(21 139 26 134)(22 138 27 133)(23 137 28 132)(24 136 29 131)(25 135 30 140)(31 159 36 154)(32 158 37 153)(33 157 38 152)(34 156 39 151)(35 155 40 160)(41 169 46 164)(42 168 47 163)(43 167 48 162)(44 166 49 161)(45 165 50 170)(51 174 56 179)(52 173 57 178)(53 172 58 177)(54 171 59 176)(55 180 60 175)(61 189 66 184)(62 188 67 183)(63 187 68 182)(64 186 69 181)(65 185 70 190)(71 199 76 194)(72 198 77 193)(73 197 78 192)(74 196 79 191)(75 195 80 200)(81 204 86 209)(82 203 87 208)(83 202 88 207)(84 201 89 206)(85 210 90 205)(91 219 96 214)(92 218 97 213)(93 217 98 212)(94 216 99 211)(95 215 100 220)(101 229 106 224)(102 228 107 223)(103 227 108 222)(104 226 109 221)(105 225 110 230)(111 234 116 239)(112 233 117 238)(113 232 118 237)(114 231 119 236)(115 240 120 235)
(1 111 51 81)(2 112 52 82)(3 113 53 83)(4 114 54 84)(5 115 55 85)(6 116 56 86)(7 117 57 87)(8 118 58 88)(9 119 59 89)(10 120 60 90)(11 188 218 158)(12 189 219 159)(13 190 220 160)(14 181 211 151)(15 182 212 152)(16 183 213 153)(17 184 214 154)(18 185 215 155)(19 186 216 156)(20 187 217 157)(21 101 41 71)(22 102 42 72)(23 103 43 73)(24 104 44 74)(25 105 45 75)(26 106 46 76)(27 107 47 77)(28 108 48 78)(29 109 49 79)(30 110 50 80)(31 124 61 91)(32 125 62 92)(33 126 63 93)(34 127 64 94)(35 128 65 95)(36 129 66 96)(37 130 67 97)(38 121 68 98)(39 122 69 99)(40 123 70 100)(131 221 161 191)(132 222 162 192)(133 223 163 193)(134 224 164 194)(135 225 165 195)(136 226 166 196)(137 227 167 197)(138 228 168 198)(139 229 169 199)(140 230 170 200)(141 231 171 201)(142 232 172 202)(143 233 173 203)(144 234 174 204)(145 235 175 205)(146 236 176 206)(147 237 177 207)(148 238 178 208)(149 239 179 209)(150 240 180 210)
G:=sub<Sym(240)| (1,62,27,51,32,47)(2,63,28,52,33,48)(3,64,29,53,34,49)(4,65,30,54,35,50)(5,66,21,55,36,41)(6,67,22,56,37,42)(7,68,23,57,38,43)(8,69,24,58,39,44)(9,70,25,59,40,45)(10,61,26,60,31,46)(11,209,223,213,234,198)(12,210,224,214,235,199)(13,201,225,215,236,200)(14,202,226,216,237,191)(15,203,227,217,238,192)(16,204,228,218,239,193)(17,205,229,219,240,194)(18,206,230,220,231,195)(19,207,221,211,232,196)(20,208,222,212,233,197)(71,124,85,106,96,120)(72,125,86,107,97,111)(73,126,87,108,98,112)(74,127,88,109,99,113)(75,128,89,110,100,114)(76,129,90,101,91,115)(77,130,81,102,92,116)(78,121,82,103,93,117)(79,122,83,104,94,118)(80,123,84,105,95,119)(131,172,156,161,142,186)(132,173,157,162,143,187)(133,174,158,163,144,188)(134,175,159,164,145,189)(135,176,160,165,146,190)(136,177,151,166,147,181)(137,178,152,167,148,182)(138,179,153,168,149,183)(139,180,154,169,150,184)(140,171,155,170,141,185), (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), (1,144,6,149)(2,143,7,148)(3,142,8,147)(4,141,9,146)(5,150,10,145)(11,130,16,125)(12,129,17,124)(13,128,18,123)(14,127,19,122)(15,126,20,121)(21,139,26,134)(22,138,27,133)(23,137,28,132)(24,136,29,131)(25,135,30,140)(31,159,36,154)(32,158,37,153)(33,157,38,152)(34,156,39,151)(35,155,40,160)(41,169,46,164)(42,168,47,163)(43,167,48,162)(44,166,49,161)(45,165,50,170)(51,174,56,179)(52,173,57,178)(53,172,58,177)(54,171,59,176)(55,180,60,175)(61,189,66,184)(62,188,67,183)(63,187,68,182)(64,186,69,181)(65,185,70,190)(71,199,76,194)(72,198,77,193)(73,197,78,192)(74,196,79,191)(75,195,80,200)(81,204,86,209)(82,203,87,208)(83,202,88,207)(84,201,89,206)(85,210,90,205)(91,219,96,214)(92,218,97,213)(93,217,98,212)(94,216,99,211)(95,215,100,220)(101,229,106,224)(102,228,107,223)(103,227,108,222)(104,226,109,221)(105,225,110,230)(111,234,116,239)(112,233,117,238)(113,232,118,237)(114,231,119,236)(115,240,120,235), (1,111,51,81)(2,112,52,82)(3,113,53,83)(4,114,54,84)(5,115,55,85)(6,116,56,86)(7,117,57,87)(8,118,58,88)(9,119,59,89)(10,120,60,90)(11,188,218,158)(12,189,219,159)(13,190,220,160)(14,181,211,151)(15,182,212,152)(16,183,213,153)(17,184,214,154)(18,185,215,155)(19,186,216,156)(20,187,217,157)(21,101,41,71)(22,102,42,72)(23,103,43,73)(24,104,44,74)(25,105,45,75)(26,106,46,76)(27,107,47,77)(28,108,48,78)(29,109,49,79)(30,110,50,80)(31,124,61,91)(32,125,62,92)(33,126,63,93)(34,127,64,94)(35,128,65,95)(36,129,66,96)(37,130,67,97)(38,121,68,98)(39,122,69,99)(40,123,70,100)(131,221,161,191)(132,222,162,192)(133,223,163,193)(134,224,164,194)(135,225,165,195)(136,226,166,196)(137,227,167,197)(138,228,168,198)(139,229,169,199)(140,230,170,200)(141,231,171,201)(142,232,172,202)(143,233,173,203)(144,234,174,204)(145,235,175,205)(146,236,176,206)(147,237,177,207)(148,238,178,208)(149,239,179,209)(150,240,180,210)>;
G:=Group( (1,62,27,51,32,47)(2,63,28,52,33,48)(3,64,29,53,34,49)(4,65,30,54,35,50)(5,66,21,55,36,41)(6,67,22,56,37,42)(7,68,23,57,38,43)(8,69,24,58,39,44)(9,70,25,59,40,45)(10,61,26,60,31,46)(11,209,223,213,234,198)(12,210,224,214,235,199)(13,201,225,215,236,200)(14,202,226,216,237,191)(15,203,227,217,238,192)(16,204,228,218,239,193)(17,205,229,219,240,194)(18,206,230,220,231,195)(19,207,221,211,232,196)(20,208,222,212,233,197)(71,124,85,106,96,120)(72,125,86,107,97,111)(73,126,87,108,98,112)(74,127,88,109,99,113)(75,128,89,110,100,114)(76,129,90,101,91,115)(77,130,81,102,92,116)(78,121,82,103,93,117)(79,122,83,104,94,118)(80,123,84,105,95,119)(131,172,156,161,142,186)(132,173,157,162,143,187)(133,174,158,163,144,188)(134,175,159,164,145,189)(135,176,160,165,146,190)(136,177,151,166,147,181)(137,178,152,167,148,182)(138,179,153,168,149,183)(139,180,154,169,150,184)(140,171,155,170,141,185), (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), (1,144,6,149)(2,143,7,148)(3,142,8,147)(4,141,9,146)(5,150,10,145)(11,130,16,125)(12,129,17,124)(13,128,18,123)(14,127,19,122)(15,126,20,121)(21,139,26,134)(22,138,27,133)(23,137,28,132)(24,136,29,131)(25,135,30,140)(31,159,36,154)(32,158,37,153)(33,157,38,152)(34,156,39,151)(35,155,40,160)(41,169,46,164)(42,168,47,163)(43,167,48,162)(44,166,49,161)(45,165,50,170)(51,174,56,179)(52,173,57,178)(53,172,58,177)(54,171,59,176)(55,180,60,175)(61,189,66,184)(62,188,67,183)(63,187,68,182)(64,186,69,181)(65,185,70,190)(71,199,76,194)(72,198,77,193)(73,197,78,192)(74,196,79,191)(75,195,80,200)(81,204,86,209)(82,203,87,208)(83,202,88,207)(84,201,89,206)(85,210,90,205)(91,219,96,214)(92,218,97,213)(93,217,98,212)(94,216,99,211)(95,215,100,220)(101,229,106,224)(102,228,107,223)(103,227,108,222)(104,226,109,221)(105,225,110,230)(111,234,116,239)(112,233,117,238)(113,232,118,237)(114,231,119,236)(115,240,120,235), (1,111,51,81)(2,112,52,82)(3,113,53,83)(4,114,54,84)(5,115,55,85)(6,116,56,86)(7,117,57,87)(8,118,58,88)(9,119,59,89)(10,120,60,90)(11,188,218,158)(12,189,219,159)(13,190,220,160)(14,181,211,151)(15,182,212,152)(16,183,213,153)(17,184,214,154)(18,185,215,155)(19,186,216,156)(20,187,217,157)(21,101,41,71)(22,102,42,72)(23,103,43,73)(24,104,44,74)(25,105,45,75)(26,106,46,76)(27,107,47,77)(28,108,48,78)(29,109,49,79)(30,110,50,80)(31,124,61,91)(32,125,62,92)(33,126,63,93)(34,127,64,94)(35,128,65,95)(36,129,66,96)(37,130,67,97)(38,121,68,98)(39,122,69,99)(40,123,70,100)(131,221,161,191)(132,222,162,192)(133,223,163,193)(134,224,164,194)(135,225,165,195)(136,226,166,196)(137,227,167,197)(138,228,168,198)(139,229,169,199)(140,230,170,200)(141,231,171,201)(142,232,172,202)(143,233,173,203)(144,234,174,204)(145,235,175,205)(146,236,176,206)(147,237,177,207)(148,238,178,208)(149,239,179,209)(150,240,180,210) );
G=PermutationGroup([[(1,62,27,51,32,47),(2,63,28,52,33,48),(3,64,29,53,34,49),(4,65,30,54,35,50),(5,66,21,55,36,41),(6,67,22,56,37,42),(7,68,23,57,38,43),(8,69,24,58,39,44),(9,70,25,59,40,45),(10,61,26,60,31,46),(11,209,223,213,234,198),(12,210,224,214,235,199),(13,201,225,215,236,200),(14,202,226,216,237,191),(15,203,227,217,238,192),(16,204,228,218,239,193),(17,205,229,219,240,194),(18,206,230,220,231,195),(19,207,221,211,232,196),(20,208,222,212,233,197),(71,124,85,106,96,120),(72,125,86,107,97,111),(73,126,87,108,98,112),(74,127,88,109,99,113),(75,128,89,110,100,114),(76,129,90,101,91,115),(77,130,81,102,92,116),(78,121,82,103,93,117),(79,122,83,104,94,118),(80,123,84,105,95,119),(131,172,156,161,142,186),(132,173,157,162,143,187),(133,174,158,163,144,188),(134,175,159,164,145,189),(135,176,160,165,146,190),(136,177,151,166,147,181),(137,178,152,167,148,182),(138,179,153,168,149,183),(139,180,154,169,150,184),(140,171,155,170,141,185)], [(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)], [(1,144,6,149),(2,143,7,148),(3,142,8,147),(4,141,9,146),(5,150,10,145),(11,130,16,125),(12,129,17,124),(13,128,18,123),(14,127,19,122),(15,126,20,121),(21,139,26,134),(22,138,27,133),(23,137,28,132),(24,136,29,131),(25,135,30,140),(31,159,36,154),(32,158,37,153),(33,157,38,152),(34,156,39,151),(35,155,40,160),(41,169,46,164),(42,168,47,163),(43,167,48,162),(44,166,49,161),(45,165,50,170),(51,174,56,179),(52,173,57,178),(53,172,58,177),(54,171,59,176),(55,180,60,175),(61,189,66,184),(62,188,67,183),(63,187,68,182),(64,186,69,181),(65,185,70,190),(71,199,76,194),(72,198,77,193),(73,197,78,192),(74,196,79,191),(75,195,80,200),(81,204,86,209),(82,203,87,208),(83,202,88,207),(84,201,89,206),(85,210,90,205),(91,219,96,214),(92,218,97,213),(93,217,98,212),(94,216,99,211),(95,215,100,220),(101,229,106,224),(102,228,107,223),(103,227,108,222),(104,226,109,221),(105,225,110,230),(111,234,116,239),(112,233,117,238),(113,232,118,237),(114,231,119,236),(115,240,120,235)], [(1,111,51,81),(2,112,52,82),(3,113,53,83),(4,114,54,84),(5,115,55,85),(6,116,56,86),(7,117,57,87),(8,118,58,88),(9,119,59,89),(10,120,60,90),(11,188,218,158),(12,189,219,159),(13,190,220,160),(14,181,211,151),(15,182,212,152),(16,183,213,153),(17,184,214,154),(18,185,215,155),(19,186,216,156),(20,187,217,157),(21,101,41,71),(22,102,42,72),(23,103,43,73),(24,104,44,74),(25,105,45,75),(26,106,46,76),(27,107,47,77),(28,108,48,78),(29,109,49,79),(30,110,50,80),(31,124,61,91),(32,125,62,92),(33,126,63,93),(34,127,64,94),(35,128,65,95),(36,129,66,96),(37,130,67,97),(38,121,68,98),(39,122,69,99),(40,123,70,100),(131,221,161,191),(132,222,162,192),(133,223,163,193),(134,224,164,194),(135,225,165,195),(136,226,166,196),(137,227,167,197),(138,228,168,198),(139,229,169,199),(140,230,170,200),(141,231,171,201),(142,232,172,202),(143,233,173,203),(144,234,174,204),(145,235,175,205),(146,236,176,206),(147,237,177,207),(148,238,178,208),(149,239,179,209),(150,240,180,210)]])
72 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 | ··· | 6G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | ··· | 12H | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30N |
| 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 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | ··· | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D5 | Dic3 | D6 | D6 | C4○D4 | D10 | D10 | C4×D5 | C4○D12 | S3×D5 | D4⋊2D5 | D5×Dic3 | C2×S3×D5 | Dic3.D10 |
| kernel | (C6×Dic5)⋊7C4 | Dic3×Dic5 | C30.Q8 | C5×C6.D4 | C30.38D4 | C2×C6×Dic5 | C6×Dic5 | C22×Dic5 | C6.D4 | C2×Dic5 | C2×Dic5 | C22×C10 | C30 | C2×Dic3 | C22×C6 | C2×C6 | C10 | C23 | C6 | C22 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 2 | 4 | 2 | 1 | 4 | 4 | 2 | 8 | 8 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of (C6×Dic5)⋊7C4 ►in GL6(𝔽61)
| 48 | 0 | 0 | 0 | 0 | 0 |
| 17 | 47 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 0 | 0 | 0 |
| 0 | 0 | 0 | 48 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 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 | 43 |
| 50 | 0 | 0 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 25 |
| 0 | 0 | 0 | 0 | 36 | 57 |
| 11 | 46 | 0 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [48,17,0,0,0,0,0,47,0,0,0,0,0,0,47,0,0,0,0,0,0,48,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,60,43],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,4,36,0,0,0,0,25,57],[11,0,0,0,0,0,46,50,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;
(C6×Dic5)⋊7C4 in GAP, Magma, Sage, TeX
(C_6\times {\rm Dic}_5)\rtimes_7C_4 % in TeX
G:=Group("(C6xDic5):7C4"); // GroupNames label
G:=SmallGroup(480,604);
// by ID
G=gap.SmallGroup(480,604);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,219,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^10=d^4=1,c^2=b^5,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^5,c*b*c^-1=b^-1,b*d=d*b,c*d=d*c>;
// generators/relations