direct product, metabelian, nilpotent (class 2), monomial
Aliases: C4○D4×C3×C9, C12.32C62, (C6×C36)⋊15C2, (C2×C12)⋊7C18, (C2×C36)⋊15C6, (D4×C9)⋊11C6, D4⋊2(C3×C18), (C3×D4)⋊5C18, C4.6(C6×C18), (C3×Q8)⋊7C18, (Q8×C9)⋊15C6, Q8⋊4(C3×C18), C22.(C6×C18), (C6×C12).49C6, C36.50(C2×C6), C12.27(C2×C18), C6.15(C2×C62), C62.46(C2×C6), (C2×C6).10C62, (C6×C18).31C22, (C3×C18).60C23, C6.16(C22×C18), C18.30(C22×C6), (C3×C36).81C22, (D4×C32).17C6, (Q8×C32).28C6, (D4×C3×C9)⋊11C2, C2.3(C2×C6×C18), (Q8×C3×C9)⋊11C2, (C2×C4)⋊3(C3×C18), (C2×C6).5(C2×C18), (C2×C18).22(C2×C6), (C2×C12).22(C3×C6), (C3×D4).12(C3×C6), (C3×Q8).24(C3×C6), C3.1(C32×C4○D4), C32.5(C3×C4○D4), (C3×C12).106(C2×C6), (C3×C6).71(C22×C6), (C32×C4○D4).4C3, (C3×C4○D4).6C32, SmallGroup(432,409)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4○D4×C3×C9
G = < a,b,c,d,e | a3=b9=c4=e2=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >
Subgroups: 230 in 200 conjugacy classes, 170 normal (20 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, D4, Q8, C9, C32, C12, C12, C2×C6, C4○D4, C18, C18, C3×C6, C3×C6, C2×C12, C3×D4, C3×Q8, C3×Q8, C3×C9, C36, C2×C18, C3×C12, C3×C12, C62, C3×C4○D4, C3×C4○D4, C3×C18, C3×C18, C2×C36, D4×C9, Q8×C9, C6×C12, D4×C32, Q8×C32, C3×C36, C3×C36, C6×C18, C9×C4○D4, C32×C4○D4, C6×C36, D4×C3×C9, Q8×C3×C9, C4○D4×C3×C9
Quotients: C1, C2, C3, C22, C6, C23, C9, C32, C2×C6, C4○D4, C18, C3×C6, C22×C6, C3×C9, C2×C18, C62, C3×C4○D4, C3×C18, C22×C18, C2×C62, C6×C18, C9×C4○D4, C32×C4○D4, C2×C6×C18, C4○D4×C3×C9
►On 216 points
Generators in S216
(1 57 15)(2 58 16)(3 59 17)(4 60 18)(5 61 10)(6 62 11)(7 63 12)(8 55 13)(9 56 14)(19 206 210)(20 207 211)(21 199 212)(22 200 213)(23 201 214)(24 202 215)(25 203 216)(26 204 208)(27 205 209)(28 54 77)(29 46 78)(30 47 79)(31 48 80)(32 49 81)(33 50 73)(34 51 74)(35 52 75)(36 53 76)(37 174 192)(38 175 193)(39 176 194)(40 177 195)(41 178 196)(42 179 197)(43 180 198)(44 172 190)(45 173 191)(64 82 131)(65 83 132)(66 84 133)(67 85 134)(68 86 135)(69 87 127)(70 88 128)(71 89 129)(72 90 130)(91 104 117)(92 105 109)(93 106 110)(94 107 111)(95 108 112)(96 100 113)(97 101 114)(98 102 115)(99 103 116)(118 136 185)(119 137 186)(120 138 187)(121 139 188)(122 140 189)(123 141 181)(124 142 182)(125 143 183)(126 144 184)(145 158 171)(146 159 163)(147 160 164)(148 161 165)(149 162 166)(150 154 167)(151 155 168)(152 156 169)(153 157 170)
(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)
(1 66 30 107)(2 67 31 108)(3 68 32 100)(4 69 33 101)(5 70 34 102)(6 71 35 103)(7 72 36 104)(8 64 28 105)(9 65 29 106)(10 128 74 98)(11 129 75 99)(12 130 76 91)(13 131 77 92)(14 132 78 93)(15 133 79 94)(16 134 80 95)(17 135 81 96)(18 127 73 97)(19 169 196 142)(20 170 197 143)(21 171 198 144)(22 163 190 136)(23 164 191 137)(24 165 192 138)(25 166 193 139)(26 167 194 140)(27 168 195 141)(37 187 202 148)(38 188 203 149)(39 189 204 150)(40 181 205 151)(41 182 206 152)(42 183 207 153)(43 184 199 145)(44 185 200 146)(45 186 201 147)(46 110 56 83)(47 111 57 84)(48 112 58 85)(49 113 59 86)(50 114 60 87)(51 115 61 88)(52 116 62 89)(53 117 63 90)(54 109 55 82)(118 213 159 172)(119 214 160 173)(120 215 161 174)(121 216 162 175)(122 208 154 176)(123 209 155 177)(124 210 156 178)(125 211 157 179)(126 212 158 180)
(1 107 30 66)(2 108 31 67)(3 100 32 68)(4 101 33 69)(5 102 34 70)(6 103 35 71)(7 104 36 72)(8 105 28 64)(9 106 29 65)(10 98 74 128)(11 99 75 129)(12 91 76 130)(13 92 77 131)(14 93 78 132)(15 94 79 133)(16 95 80 134)(17 96 81 135)(18 97 73 127)(19 169 196 142)(20 170 197 143)(21 171 198 144)(22 163 190 136)(23 164 191 137)(24 165 192 138)(25 166 193 139)(26 167 194 140)(27 168 195 141)(37 187 202 148)(38 188 203 149)(39 189 204 150)(40 181 205 151)(41 182 206 152)(42 183 207 153)(43 184 199 145)(44 185 200 146)(45 186 201 147)(46 83 56 110)(47 84 57 111)(48 85 58 112)(49 86 59 113)(50 87 60 114)(51 88 61 115)(52 89 62 116)(53 90 63 117)(54 82 55 109)(118 213 159 172)(119 214 160 173)(120 215 161 174)(121 216 162 175)(122 208 154 176)(123 209 155 177)(124 210 156 178)(125 211 157 179)(126 212 158 180)
(1 215)(2 216)(3 208)(4 209)(5 210)(6 211)(7 212)(8 213)(9 214)(10 206)(11 207)(12 199)(13 200)(14 201)(15 202)(16 203)(17 204)(18 205)(19 61)(20 62)(21 63)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(28 172)(29 173)(30 174)(31 175)(32 176)(33 177)(34 178)(35 179)(36 180)(37 79)(38 80)(39 81)(40 73)(41 74)(42 75)(43 76)(44 77)(45 78)(46 191)(47 192)(48 193)(49 194)(50 195)(51 196)(52 197)(53 198)(54 190)(64 159)(65 160)(66 161)(67 162)(68 154)(69 155)(70 156)(71 157)(72 158)(82 163)(83 164)(84 165)(85 166)(86 167)(87 168)(88 169)(89 170)(90 171)(91 184)(92 185)(93 186)(94 187)(95 188)(96 189)(97 181)(98 182)(99 183)(100 122)(101 123)(102 124)(103 125)(104 126)(105 118)(106 119)(107 120)(108 121)(109 136)(110 137)(111 138)(112 139)(113 140)(114 141)(115 142)(116 143)(117 144)(127 151)(128 152)(129 153)(130 145)(131 146)(132 147)(133 148)(134 149)(135 150)
G:=sub<Sym(216)| (1,57,15)(2,58,16)(3,59,17)(4,60,18)(5,61,10)(6,62,11)(7,63,12)(8,55,13)(9,56,14)(19,206,210)(20,207,211)(21,199,212)(22,200,213)(23,201,214)(24,202,215)(25,203,216)(26,204,208)(27,205,209)(28,54,77)(29,46,78)(30,47,79)(31,48,80)(32,49,81)(33,50,73)(34,51,74)(35,52,75)(36,53,76)(37,174,192)(38,175,193)(39,176,194)(40,177,195)(41,178,196)(42,179,197)(43,180,198)(44,172,190)(45,173,191)(64,82,131)(65,83,132)(66,84,133)(67,85,134)(68,86,135)(69,87,127)(70,88,128)(71,89,129)(72,90,130)(91,104,117)(92,105,109)(93,106,110)(94,107,111)(95,108,112)(96,100,113)(97,101,114)(98,102,115)(99,103,116)(118,136,185)(119,137,186)(120,138,187)(121,139,188)(122,140,189)(123,141,181)(124,142,182)(125,143,183)(126,144,184)(145,158,171)(146,159,163)(147,160,164)(148,161,165)(149,162,166)(150,154,167)(151,155,168)(152,156,169)(153,157,170), (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), (1,66,30,107)(2,67,31,108)(3,68,32,100)(4,69,33,101)(5,70,34,102)(6,71,35,103)(7,72,36,104)(8,64,28,105)(9,65,29,106)(10,128,74,98)(11,129,75,99)(12,130,76,91)(13,131,77,92)(14,132,78,93)(15,133,79,94)(16,134,80,95)(17,135,81,96)(18,127,73,97)(19,169,196,142)(20,170,197,143)(21,171,198,144)(22,163,190,136)(23,164,191,137)(24,165,192,138)(25,166,193,139)(26,167,194,140)(27,168,195,141)(37,187,202,148)(38,188,203,149)(39,189,204,150)(40,181,205,151)(41,182,206,152)(42,183,207,153)(43,184,199,145)(44,185,200,146)(45,186,201,147)(46,110,56,83)(47,111,57,84)(48,112,58,85)(49,113,59,86)(50,114,60,87)(51,115,61,88)(52,116,62,89)(53,117,63,90)(54,109,55,82)(118,213,159,172)(119,214,160,173)(120,215,161,174)(121,216,162,175)(122,208,154,176)(123,209,155,177)(124,210,156,178)(125,211,157,179)(126,212,158,180), (1,107,30,66)(2,108,31,67)(3,100,32,68)(4,101,33,69)(5,102,34,70)(6,103,35,71)(7,104,36,72)(8,105,28,64)(9,106,29,65)(10,98,74,128)(11,99,75,129)(12,91,76,130)(13,92,77,131)(14,93,78,132)(15,94,79,133)(16,95,80,134)(17,96,81,135)(18,97,73,127)(19,169,196,142)(20,170,197,143)(21,171,198,144)(22,163,190,136)(23,164,191,137)(24,165,192,138)(25,166,193,139)(26,167,194,140)(27,168,195,141)(37,187,202,148)(38,188,203,149)(39,189,204,150)(40,181,205,151)(41,182,206,152)(42,183,207,153)(43,184,199,145)(44,185,200,146)(45,186,201,147)(46,83,56,110)(47,84,57,111)(48,85,58,112)(49,86,59,113)(50,87,60,114)(51,88,61,115)(52,89,62,116)(53,90,63,117)(54,82,55,109)(118,213,159,172)(119,214,160,173)(120,215,161,174)(121,216,162,175)(122,208,154,176)(123,209,155,177)(124,210,156,178)(125,211,157,179)(126,212,158,180), (1,215)(2,216)(3,208)(4,209)(5,210)(6,211)(7,212)(8,213)(9,214)(10,206)(11,207)(12,199)(13,200)(14,201)(15,202)(16,203)(17,204)(18,205)(19,61)(20,62)(21,63)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,172)(29,173)(30,174)(31,175)(32,176)(33,177)(34,178)(35,179)(36,180)(37,79)(38,80)(39,81)(40,73)(41,74)(42,75)(43,76)(44,77)(45,78)(46,191)(47,192)(48,193)(49,194)(50,195)(51,196)(52,197)(53,198)(54,190)(64,159)(65,160)(66,161)(67,162)(68,154)(69,155)(70,156)(71,157)(72,158)(82,163)(83,164)(84,165)(85,166)(86,167)(87,168)(88,169)(89,170)(90,171)(91,184)(92,185)(93,186)(94,187)(95,188)(96,189)(97,181)(98,182)(99,183)(100,122)(101,123)(102,124)(103,125)(104,126)(105,118)(106,119)(107,120)(108,121)(109,136)(110,137)(111,138)(112,139)(113,140)(114,141)(115,142)(116,143)(117,144)(127,151)(128,152)(129,153)(130,145)(131,146)(132,147)(133,148)(134,149)(135,150)>;
G:=Group( (1,57,15)(2,58,16)(3,59,17)(4,60,18)(5,61,10)(6,62,11)(7,63,12)(8,55,13)(9,56,14)(19,206,210)(20,207,211)(21,199,212)(22,200,213)(23,201,214)(24,202,215)(25,203,216)(26,204,208)(27,205,209)(28,54,77)(29,46,78)(30,47,79)(31,48,80)(32,49,81)(33,50,73)(34,51,74)(35,52,75)(36,53,76)(37,174,192)(38,175,193)(39,176,194)(40,177,195)(41,178,196)(42,179,197)(43,180,198)(44,172,190)(45,173,191)(64,82,131)(65,83,132)(66,84,133)(67,85,134)(68,86,135)(69,87,127)(70,88,128)(71,89,129)(72,90,130)(91,104,117)(92,105,109)(93,106,110)(94,107,111)(95,108,112)(96,100,113)(97,101,114)(98,102,115)(99,103,116)(118,136,185)(119,137,186)(120,138,187)(121,139,188)(122,140,189)(123,141,181)(124,142,182)(125,143,183)(126,144,184)(145,158,171)(146,159,163)(147,160,164)(148,161,165)(149,162,166)(150,154,167)(151,155,168)(152,156,169)(153,157,170), (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), (1,66,30,107)(2,67,31,108)(3,68,32,100)(4,69,33,101)(5,70,34,102)(6,71,35,103)(7,72,36,104)(8,64,28,105)(9,65,29,106)(10,128,74,98)(11,129,75,99)(12,130,76,91)(13,131,77,92)(14,132,78,93)(15,133,79,94)(16,134,80,95)(17,135,81,96)(18,127,73,97)(19,169,196,142)(20,170,197,143)(21,171,198,144)(22,163,190,136)(23,164,191,137)(24,165,192,138)(25,166,193,139)(26,167,194,140)(27,168,195,141)(37,187,202,148)(38,188,203,149)(39,189,204,150)(40,181,205,151)(41,182,206,152)(42,183,207,153)(43,184,199,145)(44,185,200,146)(45,186,201,147)(46,110,56,83)(47,111,57,84)(48,112,58,85)(49,113,59,86)(50,114,60,87)(51,115,61,88)(52,116,62,89)(53,117,63,90)(54,109,55,82)(118,213,159,172)(119,214,160,173)(120,215,161,174)(121,216,162,175)(122,208,154,176)(123,209,155,177)(124,210,156,178)(125,211,157,179)(126,212,158,180), (1,107,30,66)(2,108,31,67)(3,100,32,68)(4,101,33,69)(5,102,34,70)(6,103,35,71)(7,104,36,72)(8,105,28,64)(9,106,29,65)(10,98,74,128)(11,99,75,129)(12,91,76,130)(13,92,77,131)(14,93,78,132)(15,94,79,133)(16,95,80,134)(17,96,81,135)(18,97,73,127)(19,169,196,142)(20,170,197,143)(21,171,198,144)(22,163,190,136)(23,164,191,137)(24,165,192,138)(25,166,193,139)(26,167,194,140)(27,168,195,141)(37,187,202,148)(38,188,203,149)(39,189,204,150)(40,181,205,151)(41,182,206,152)(42,183,207,153)(43,184,199,145)(44,185,200,146)(45,186,201,147)(46,83,56,110)(47,84,57,111)(48,85,58,112)(49,86,59,113)(50,87,60,114)(51,88,61,115)(52,89,62,116)(53,90,63,117)(54,82,55,109)(118,213,159,172)(119,214,160,173)(120,215,161,174)(121,216,162,175)(122,208,154,176)(123,209,155,177)(124,210,156,178)(125,211,157,179)(126,212,158,180), (1,215)(2,216)(3,208)(4,209)(5,210)(6,211)(7,212)(8,213)(9,214)(10,206)(11,207)(12,199)(13,200)(14,201)(15,202)(16,203)(17,204)(18,205)(19,61)(20,62)(21,63)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,172)(29,173)(30,174)(31,175)(32,176)(33,177)(34,178)(35,179)(36,180)(37,79)(38,80)(39,81)(40,73)(41,74)(42,75)(43,76)(44,77)(45,78)(46,191)(47,192)(48,193)(49,194)(50,195)(51,196)(52,197)(53,198)(54,190)(64,159)(65,160)(66,161)(67,162)(68,154)(69,155)(70,156)(71,157)(72,158)(82,163)(83,164)(84,165)(85,166)(86,167)(87,168)(88,169)(89,170)(90,171)(91,184)(92,185)(93,186)(94,187)(95,188)(96,189)(97,181)(98,182)(99,183)(100,122)(101,123)(102,124)(103,125)(104,126)(105,118)(106,119)(107,120)(108,121)(109,136)(110,137)(111,138)(112,139)(113,140)(114,141)(115,142)(116,143)(117,144)(127,151)(128,152)(129,153)(130,145)(131,146)(132,147)(133,148)(134,149)(135,150) );
G=PermutationGroup([[(1,57,15),(2,58,16),(3,59,17),(4,60,18),(5,61,10),(6,62,11),(7,63,12),(8,55,13),(9,56,14),(19,206,210),(20,207,211),(21,199,212),(22,200,213),(23,201,214),(24,202,215),(25,203,216),(26,204,208),(27,205,209),(28,54,77),(29,46,78),(30,47,79),(31,48,80),(32,49,81),(33,50,73),(34,51,74),(35,52,75),(36,53,76),(37,174,192),(38,175,193),(39,176,194),(40,177,195),(41,178,196),(42,179,197),(43,180,198),(44,172,190),(45,173,191),(64,82,131),(65,83,132),(66,84,133),(67,85,134),(68,86,135),(69,87,127),(70,88,128),(71,89,129),(72,90,130),(91,104,117),(92,105,109),(93,106,110),(94,107,111),(95,108,112),(96,100,113),(97,101,114),(98,102,115),(99,103,116),(118,136,185),(119,137,186),(120,138,187),(121,139,188),(122,140,189),(123,141,181),(124,142,182),(125,143,183),(126,144,184),(145,158,171),(146,159,163),(147,160,164),(148,161,165),(149,162,166),(150,154,167),(151,155,168),(152,156,169),(153,157,170)], [(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)], [(1,66,30,107),(2,67,31,108),(3,68,32,100),(4,69,33,101),(5,70,34,102),(6,71,35,103),(7,72,36,104),(8,64,28,105),(9,65,29,106),(10,128,74,98),(11,129,75,99),(12,130,76,91),(13,131,77,92),(14,132,78,93),(15,133,79,94),(16,134,80,95),(17,135,81,96),(18,127,73,97),(19,169,196,142),(20,170,197,143),(21,171,198,144),(22,163,190,136),(23,164,191,137),(24,165,192,138),(25,166,193,139),(26,167,194,140),(27,168,195,141),(37,187,202,148),(38,188,203,149),(39,189,204,150),(40,181,205,151),(41,182,206,152),(42,183,207,153),(43,184,199,145),(44,185,200,146),(45,186,201,147),(46,110,56,83),(47,111,57,84),(48,112,58,85),(49,113,59,86),(50,114,60,87),(51,115,61,88),(52,116,62,89),(53,117,63,90),(54,109,55,82),(118,213,159,172),(119,214,160,173),(120,215,161,174),(121,216,162,175),(122,208,154,176),(123,209,155,177),(124,210,156,178),(125,211,157,179),(126,212,158,180)], [(1,107,30,66),(2,108,31,67),(3,100,32,68),(4,101,33,69),(5,102,34,70),(6,103,35,71),(7,104,36,72),(8,105,28,64),(9,106,29,65),(10,98,74,128),(11,99,75,129),(12,91,76,130),(13,92,77,131),(14,93,78,132),(15,94,79,133),(16,95,80,134),(17,96,81,135),(18,97,73,127),(19,169,196,142),(20,170,197,143),(21,171,198,144),(22,163,190,136),(23,164,191,137),(24,165,192,138),(25,166,193,139),(26,167,194,140),(27,168,195,141),(37,187,202,148),(38,188,203,149),(39,189,204,150),(40,181,205,151),(41,182,206,152),(42,183,207,153),(43,184,199,145),(44,185,200,146),(45,186,201,147),(46,83,56,110),(47,84,57,111),(48,85,58,112),(49,86,59,113),(50,87,60,114),(51,88,61,115),(52,89,62,116),(53,90,63,117),(54,82,55,109),(118,213,159,172),(119,214,160,173),(120,215,161,174),(121,216,162,175),(122,208,154,176),(123,209,155,177),(124,210,156,178),(125,211,157,179),(126,212,158,180)], [(1,215),(2,216),(3,208),(4,209),(5,210),(6,211),(7,212),(8,213),(9,214),(10,206),(11,207),(12,199),(13,200),(14,201),(15,202),(16,203),(17,204),(18,205),(19,61),(20,62),(21,63),(22,55),(23,56),(24,57),(25,58),(26,59),(27,60),(28,172),(29,173),(30,174),(31,175),(32,176),(33,177),(34,178),(35,179),(36,180),(37,79),(38,80),(39,81),(40,73),(41,74),(42,75),(43,76),(44,77),(45,78),(46,191),(47,192),(48,193),(49,194),(50,195),(51,196),(52,197),(53,198),(54,190),(64,159),(65,160),(66,161),(67,162),(68,154),(69,155),(70,156),(71,157),(72,158),(82,163),(83,164),(84,165),(85,166),(86,167),(87,168),(88,169),(89,170),(90,171),(91,184),(92,185),(93,186),(94,187),(95,188),(96,189),(97,181),(98,182),(99,183),(100,122),(101,123),(102,124),(103,125),(104,126),(105,118),(106,119),(107,120),(108,121),(109,136),(110,137),(111,138),(112,139),(113,140),(114,141),(115,142),(116,143),(117,144),(127,151),(128,152),(129,153),(130,145),(131,146),(132,147),(133,148),(134,149),(135,150)]])
270 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | ··· | 3H | 4A | 4B | 4C | 4D | 4E | 6A | ··· | 6H | 6I | ··· | 6AF | 9A | ··· | 9R | 12A | ··· | 12P | 12Q | ··· | 12AN | 18A | ··· | 18R | 18S | ··· | 18BT | 36A | ··· | 36AJ | 36AK | ··· | 36CL |
| order | 1 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
270 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | C4○D4 | C3×C4○D4 | C3×C4○D4 | C9×C4○D4 |
| kernel | C4○D4×C3×C9 | C6×C36 | D4×C3×C9 | Q8×C3×C9 | C9×C4○D4 | C32×C4○D4 | C2×C36 | D4×C9 | Q8×C9 | C6×C12 | D4×C32 | Q8×C32 | C3×C4○D4 | C2×C12 | C3×D4 | C3×Q8 | C3×C9 | C9 | C32 | C3 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 18 | 18 | 6 | 6 | 6 | 2 | 18 | 54 | 54 | 18 | 2 | 12 | 4 | 36 |
Matrix representation of C4○D4×C3×C9 ►in GL4(𝔽37) generated by
| 26 | 0 | 0 | 0 |
| 0 | 26 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 26 | 0 |
| 0 | 0 | 0 | 26 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 6 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 6 |
| 0 | 0 | 31 | 0 |
G:=sub<GL(4,GF(37))| [26,0,0,0,0,26,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,12,0,0,0,0,26,0,0,0,0,26],[1,0,0,0,0,1,0,0,0,0,6,0,0,0,0,6],[36,0,0,0,0,36,0,0,0,0,31,0,0,0,0,6],[36,0,0,0,0,36,0,0,0,0,0,31,0,0,6,0] >;
C4○D4×C3×C9 in GAP, Magma, Sage, TeX
C_4\circ D_4\times C_3\times C_9
% in TeX
G:=Group("C4oD4xC3xC9"); // GroupNames label
G:=SmallGroup(432,409);
// by ID
G=gap.SmallGroup(432,409);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,1037,394,528]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^9=c^4=e^2=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d>;
// generators/relations