direct product, metabelian, nilpotent (class 2), monomial
Aliases: C22⋊C4×C3×C9, C62.14C12, (C2×C6)⋊3C36, (C6×C36)⋊4C2, (C6×C18)⋊2C4, (C2×C36)⋊10C6, (C2×C18)⋊9C12, (C2×C12)⋊2C18, C2.1(C6×C36), C6.18(D4×C9), (C6×C12).23C6, C6.12(C6×C12), C6.13(C2×C36), C22⋊3(C3×C36), C18.34(C3×D4), (C3×C18).52D4, C18.27(C2×C12), C22.2(C6×C18), C62.80(C2×C6), (C22×C6).7C18, (C2×C6).26C62, (C2×C62).20C6, C23.3(C3×C18), C6.12(D4×C32), (C6×C18).52C22, (C22×C18).13C6, C2.1(D4×C3×C9), (C2×C6×C18).2C2, (C2×C4)⋊1(C3×C18), (C2×C12).1(C3×C6), (C2×C6).6(C3×C12), (C3×C6).74(C3×D4), (C3×C6).66(C2×C12), (C2×C6).19(C2×C18), (C3×C18).41(C2×C4), (C2×C18).36(C2×C6), (C22×C6).13(C3×C6), C3.1(C32×C22⋊C4), C32.4(C3×C22⋊C4), (C32×C22⋊C4).2C3, (C3×C22⋊C4).1C32, SmallGroup(432,203)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊C4×C3×C9
G = < a,b,c,d,e | a3=b9=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >
Subgroups: 230 in 170 conjugacy classes, 110 normal (20 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C9, C32, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C18, C18, C3×C6, C3×C6, C3×C6, C2×C12, C22×C6, C22×C6, C3×C9, C36, C2×C18, C2×C18, C3×C12, C62, C62, C62, C3×C22⋊C4, C3×C22⋊C4, C3×C18, C3×C18, C3×C18, C2×C36, C22×C18, C6×C12, C2×C62, C3×C36, C6×C18, C6×C18, C6×C18, C9×C22⋊C4, C32×C22⋊C4, C6×C36, C2×C6×C18, C22⋊C4×C3×C9
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C9, C32, C12, C2×C6, C22⋊C4, C18, C3×C6, C2×C12, C3×D4, C3×C9, C36, C2×C18, C3×C12, C62, C3×C22⋊C4, C3×C18, C2×C36, D4×C9, C6×C12, D4×C32, C3×C36, C6×C18, C9×C22⋊C4, C32×C22⋊C4, C6×C36, D4×C3×C9, C22⋊C4×C3×C9
►On 216 points
Generators in S216
(1 95 115)(2 96 116)(3 97 117)(4 98 109)(5 99 110)(6 91 111)(7 92 112)(8 93 113)(9 94 114)(10 60 100)(11 61 101)(12 62 102)(13 63 103)(14 55 104)(15 56 105)(16 57 106)(17 58 107)(18 59 108)(19 65 87)(20 66 88)(21 67 89)(22 68 90)(23 69 82)(24 70 83)(25 71 84)(26 72 85)(27 64 86)(28 148 52)(29 149 53)(30 150 54)(31 151 46)(32 152 47)(33 153 48)(34 145 49)(35 146 50)(36 147 51)(37 118 203)(38 119 204)(39 120 205)(40 121 206)(41 122 207)(42 123 199)(43 124 200)(44 125 201)(45 126 202)(73 154 139)(74 155 140)(75 156 141)(76 157 142)(77 158 143)(78 159 144)(79 160 136)(80 161 137)(81 162 138)(127 173 195)(128 174 196)(129 175 197)(130 176 198)(131 177 190)(132 178 191)(133 179 192)(134 180 193)(135 172 194)(163 212 181)(164 213 182)(165 214 183)(166 215 184)(167 216 185)(168 208 186)(169 209 187)(170 210 188)(171 211 189)
(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)
(37 135)(38 127)(39 128)(40 129)(41 130)(42 131)(43 132)(44 133)(45 134)(73 181)(74 182)(75 183)(76 184)(77 185)(78 186)(79 187)(80 188)(81 189)(118 172)(119 173)(120 174)(121 175)(122 176)(123 177)(124 178)(125 179)(126 180)(136 209)(137 210)(138 211)(139 212)(140 213)(141 214)(142 215)(143 216)(144 208)(154 163)(155 164)(156 165)(157 166)(158 167)(159 168)(160 169)(161 170)(162 171)(190 199)(191 200)(192 201)(193 202)(194 203)(195 204)(196 205)(197 206)(198 207)
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 147)(11 148)(12 149)(13 150)(14 151)(15 152)(16 153)(17 145)(18 146)(19 116)(20 117)(21 109)(22 110)(23 111)(24 112)(25 113)(26 114)(27 115)(28 101)(29 102)(30 103)(31 104)(32 105)(33 106)(34 107)(35 108)(36 100)(37 135)(38 127)(39 128)(40 129)(41 130)(42 131)(43 132)(44 133)(45 134)(46 55)(47 56)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(73 181)(74 182)(75 183)(76 184)(77 185)(78 186)(79 187)(80 188)(81 189)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 97)(89 98)(90 99)(118 172)(119 173)(120 174)(121 175)(122 176)(123 177)(124 178)(125 179)(126 180)(136 209)(137 210)(138 211)(139 212)(140 213)(141 214)(142 215)(143 216)(144 208)(154 163)(155 164)(156 165)(157 166)(158 167)(159 168)(160 169)(161 170)(162 171)(190 199)(191 200)(192 201)(193 202)(194 203)(195 204)(196 205)(197 206)(198 207)
(1 163 55 118)(2 164 56 119)(3 165 57 120)(4 166 58 121)(5 167 59 122)(6 168 60 123)(7 169 61 124)(8 170 62 125)(9 171 63 126)(10 42 111 186)(11 43 112 187)(12 44 113 188)(13 45 114 189)(14 37 115 181)(15 38 116 182)(16 39 117 183)(17 40 109 184)(18 41 110 185)(19 74 152 127)(20 75 153 128)(21 76 145 129)(22 77 146 130)(23 78 147 131)(24 79 148 132)(25 80 149 133)(26 81 150 134)(27 73 151 135)(28 191 83 136)(29 192 84 137)(30 193 85 138)(31 194 86 139)(32 195 87 140)(33 196 88 141)(34 197 89 142)(35 198 90 143)(36 190 82 144)(46 172 64 154)(47 173 65 155)(48 174 66 156)(49 175 67 157)(50 176 68 158)(51 177 69 159)(52 178 70 160)(53 179 71 161)(54 180 72 162)(91 208 100 199)(92 209 101 200)(93 210 102 201)(94 211 103 202)(95 212 104 203)(96 213 105 204)(97 214 106 205)(98 215 107 206)(99 216 108 207)
G:=sub<Sym(216)| (1,95,115)(2,96,116)(3,97,117)(4,98,109)(5,99,110)(6,91,111)(7,92,112)(8,93,113)(9,94,114)(10,60,100)(11,61,101)(12,62,102)(13,63,103)(14,55,104)(15,56,105)(16,57,106)(17,58,107)(18,59,108)(19,65,87)(20,66,88)(21,67,89)(22,68,90)(23,69,82)(24,70,83)(25,71,84)(26,72,85)(27,64,86)(28,148,52)(29,149,53)(30,150,54)(31,151,46)(32,152,47)(33,153,48)(34,145,49)(35,146,50)(36,147,51)(37,118,203)(38,119,204)(39,120,205)(40,121,206)(41,122,207)(42,123,199)(43,124,200)(44,125,201)(45,126,202)(73,154,139)(74,155,140)(75,156,141)(76,157,142)(77,158,143)(78,159,144)(79,160,136)(80,161,137)(81,162,138)(127,173,195)(128,174,196)(129,175,197)(130,176,198)(131,177,190)(132,178,191)(133,179,192)(134,180,193)(135,172,194)(163,212,181)(164,213,182)(165,214,183)(166,215,184)(167,216,185)(168,208,186)(169,209,187)(170,210,188)(171,211,189), (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), (37,135)(38,127)(39,128)(40,129)(41,130)(42,131)(43,132)(44,133)(45,134)(73,181)(74,182)(75,183)(76,184)(77,185)(78,186)(79,187)(80,188)(81,189)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(136,209)(137,210)(138,211)(139,212)(140,213)(141,214)(142,215)(143,216)(144,208)(154,163)(155,164)(156,165)(157,166)(158,167)(159,168)(160,169)(161,170)(162,171)(190,199)(191,200)(192,201)(193,202)(194,203)(195,204)(196,205)(197,206)(198,207), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,147)(11,148)(12,149)(13,150)(14,151)(15,152)(16,153)(17,145)(18,146)(19,116)(20,117)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,101)(29,102)(30,103)(31,104)(32,105)(33,106)(34,107)(35,108)(36,100)(37,135)(38,127)(39,128)(40,129)(41,130)(42,131)(43,132)(44,133)(45,134)(46,55)(47,56)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(73,181)(74,182)(75,183)(76,184)(77,185)(78,186)(79,187)(80,188)(81,189)(82,91)(83,92)(84,93)(85,94)(86,95)(87,96)(88,97)(89,98)(90,99)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(136,209)(137,210)(138,211)(139,212)(140,213)(141,214)(142,215)(143,216)(144,208)(154,163)(155,164)(156,165)(157,166)(158,167)(159,168)(160,169)(161,170)(162,171)(190,199)(191,200)(192,201)(193,202)(194,203)(195,204)(196,205)(197,206)(198,207), (1,163,55,118)(2,164,56,119)(3,165,57,120)(4,166,58,121)(5,167,59,122)(6,168,60,123)(7,169,61,124)(8,170,62,125)(9,171,63,126)(10,42,111,186)(11,43,112,187)(12,44,113,188)(13,45,114,189)(14,37,115,181)(15,38,116,182)(16,39,117,183)(17,40,109,184)(18,41,110,185)(19,74,152,127)(20,75,153,128)(21,76,145,129)(22,77,146,130)(23,78,147,131)(24,79,148,132)(25,80,149,133)(26,81,150,134)(27,73,151,135)(28,191,83,136)(29,192,84,137)(30,193,85,138)(31,194,86,139)(32,195,87,140)(33,196,88,141)(34,197,89,142)(35,198,90,143)(36,190,82,144)(46,172,64,154)(47,173,65,155)(48,174,66,156)(49,175,67,157)(50,176,68,158)(51,177,69,159)(52,178,70,160)(53,179,71,161)(54,180,72,162)(91,208,100,199)(92,209,101,200)(93,210,102,201)(94,211,103,202)(95,212,104,203)(96,213,105,204)(97,214,106,205)(98,215,107,206)(99,216,108,207)>;
G:=Group( (1,95,115)(2,96,116)(3,97,117)(4,98,109)(5,99,110)(6,91,111)(7,92,112)(8,93,113)(9,94,114)(10,60,100)(11,61,101)(12,62,102)(13,63,103)(14,55,104)(15,56,105)(16,57,106)(17,58,107)(18,59,108)(19,65,87)(20,66,88)(21,67,89)(22,68,90)(23,69,82)(24,70,83)(25,71,84)(26,72,85)(27,64,86)(28,148,52)(29,149,53)(30,150,54)(31,151,46)(32,152,47)(33,153,48)(34,145,49)(35,146,50)(36,147,51)(37,118,203)(38,119,204)(39,120,205)(40,121,206)(41,122,207)(42,123,199)(43,124,200)(44,125,201)(45,126,202)(73,154,139)(74,155,140)(75,156,141)(76,157,142)(77,158,143)(78,159,144)(79,160,136)(80,161,137)(81,162,138)(127,173,195)(128,174,196)(129,175,197)(130,176,198)(131,177,190)(132,178,191)(133,179,192)(134,180,193)(135,172,194)(163,212,181)(164,213,182)(165,214,183)(166,215,184)(167,216,185)(168,208,186)(169,209,187)(170,210,188)(171,211,189), (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), (37,135)(38,127)(39,128)(40,129)(41,130)(42,131)(43,132)(44,133)(45,134)(73,181)(74,182)(75,183)(76,184)(77,185)(78,186)(79,187)(80,188)(81,189)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(136,209)(137,210)(138,211)(139,212)(140,213)(141,214)(142,215)(143,216)(144,208)(154,163)(155,164)(156,165)(157,166)(158,167)(159,168)(160,169)(161,170)(162,171)(190,199)(191,200)(192,201)(193,202)(194,203)(195,204)(196,205)(197,206)(198,207), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,147)(11,148)(12,149)(13,150)(14,151)(15,152)(16,153)(17,145)(18,146)(19,116)(20,117)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,101)(29,102)(30,103)(31,104)(32,105)(33,106)(34,107)(35,108)(36,100)(37,135)(38,127)(39,128)(40,129)(41,130)(42,131)(43,132)(44,133)(45,134)(46,55)(47,56)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(73,181)(74,182)(75,183)(76,184)(77,185)(78,186)(79,187)(80,188)(81,189)(82,91)(83,92)(84,93)(85,94)(86,95)(87,96)(88,97)(89,98)(90,99)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(136,209)(137,210)(138,211)(139,212)(140,213)(141,214)(142,215)(143,216)(144,208)(154,163)(155,164)(156,165)(157,166)(158,167)(159,168)(160,169)(161,170)(162,171)(190,199)(191,200)(192,201)(193,202)(194,203)(195,204)(196,205)(197,206)(198,207), (1,163,55,118)(2,164,56,119)(3,165,57,120)(4,166,58,121)(5,167,59,122)(6,168,60,123)(7,169,61,124)(8,170,62,125)(9,171,63,126)(10,42,111,186)(11,43,112,187)(12,44,113,188)(13,45,114,189)(14,37,115,181)(15,38,116,182)(16,39,117,183)(17,40,109,184)(18,41,110,185)(19,74,152,127)(20,75,153,128)(21,76,145,129)(22,77,146,130)(23,78,147,131)(24,79,148,132)(25,80,149,133)(26,81,150,134)(27,73,151,135)(28,191,83,136)(29,192,84,137)(30,193,85,138)(31,194,86,139)(32,195,87,140)(33,196,88,141)(34,197,89,142)(35,198,90,143)(36,190,82,144)(46,172,64,154)(47,173,65,155)(48,174,66,156)(49,175,67,157)(50,176,68,158)(51,177,69,159)(52,178,70,160)(53,179,71,161)(54,180,72,162)(91,208,100,199)(92,209,101,200)(93,210,102,201)(94,211,103,202)(95,212,104,203)(96,213,105,204)(97,214,106,205)(98,215,107,206)(99,216,108,207) );
G=PermutationGroup([[(1,95,115),(2,96,116),(3,97,117),(4,98,109),(5,99,110),(6,91,111),(7,92,112),(8,93,113),(9,94,114),(10,60,100),(11,61,101),(12,62,102),(13,63,103),(14,55,104),(15,56,105),(16,57,106),(17,58,107),(18,59,108),(19,65,87),(20,66,88),(21,67,89),(22,68,90),(23,69,82),(24,70,83),(25,71,84),(26,72,85),(27,64,86),(28,148,52),(29,149,53),(30,150,54),(31,151,46),(32,152,47),(33,153,48),(34,145,49),(35,146,50),(36,147,51),(37,118,203),(38,119,204),(39,120,205),(40,121,206),(41,122,207),(42,123,199),(43,124,200),(44,125,201),(45,126,202),(73,154,139),(74,155,140),(75,156,141),(76,157,142),(77,158,143),(78,159,144),(79,160,136),(80,161,137),(81,162,138),(127,173,195),(128,174,196),(129,175,197),(130,176,198),(131,177,190),(132,178,191),(133,179,192),(134,180,193),(135,172,194),(163,212,181),(164,213,182),(165,214,183),(166,215,184),(167,216,185),(168,208,186),(169,209,187),(170,210,188),(171,211,189)], [(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)], [(37,135),(38,127),(39,128),(40,129),(41,130),(42,131),(43,132),(44,133),(45,134),(73,181),(74,182),(75,183),(76,184),(77,185),(78,186),(79,187),(80,188),(81,189),(118,172),(119,173),(120,174),(121,175),(122,176),(123,177),(124,178),(125,179),(126,180),(136,209),(137,210),(138,211),(139,212),(140,213),(141,214),(142,215),(143,216),(144,208),(154,163),(155,164),(156,165),(157,166),(158,167),(159,168),(160,169),(161,170),(162,171),(190,199),(191,200),(192,201),(193,202),(194,203),(195,204),(196,205),(197,206),(198,207)], [(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,72),(10,147),(11,148),(12,149),(13,150),(14,151),(15,152),(16,153),(17,145),(18,146),(19,116),(20,117),(21,109),(22,110),(23,111),(24,112),(25,113),(26,114),(27,115),(28,101),(29,102),(30,103),(31,104),(32,105),(33,106),(34,107),(35,108),(36,100),(37,135),(38,127),(39,128),(40,129),(41,130),(42,131),(43,132),(44,133),(45,134),(46,55),(47,56),(48,57),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63),(73,181),(74,182),(75,183),(76,184),(77,185),(78,186),(79,187),(80,188),(81,189),(82,91),(83,92),(84,93),(85,94),(86,95),(87,96),(88,97),(89,98),(90,99),(118,172),(119,173),(120,174),(121,175),(122,176),(123,177),(124,178),(125,179),(126,180),(136,209),(137,210),(138,211),(139,212),(140,213),(141,214),(142,215),(143,216),(144,208),(154,163),(155,164),(156,165),(157,166),(158,167),(159,168),(160,169),(161,170),(162,171),(190,199),(191,200),(192,201),(193,202),(194,203),(195,204),(196,205),(197,206),(198,207)], [(1,163,55,118),(2,164,56,119),(3,165,57,120),(4,166,58,121),(5,167,59,122),(6,168,60,123),(7,169,61,124),(8,170,62,125),(9,171,63,126),(10,42,111,186),(11,43,112,187),(12,44,113,188),(13,45,114,189),(14,37,115,181),(15,38,116,182),(16,39,117,183),(17,40,109,184),(18,41,110,185),(19,74,152,127),(20,75,153,128),(21,76,145,129),(22,77,146,130),(23,78,147,131),(24,79,148,132),(25,80,149,133),(26,81,150,134),(27,73,151,135),(28,191,83,136),(29,192,84,137),(30,193,85,138),(31,194,86,139),(32,195,87,140),(33,196,88,141),(34,197,89,142),(35,198,90,143),(36,190,82,144),(46,172,64,154),(47,173,65,155),(48,174,66,156),(49,175,67,157),(50,176,68,158),(51,177,69,159),(52,178,70,160),(53,179,71,161),(54,180,72,162),(91,208,100,199),(92,209,101,200),(93,210,102,201),(94,211,103,202),(95,212,104,203),(96,213,105,204),(97,214,106,205),(98,215,107,206),(99,216,108,207)]])
270 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | ··· | 3H | 4A | 4B | 4C | 4D | 6A | ··· | 6X | 6Y | ··· | 6AN | 9A | ··· | 9R | 12A | ··· | 12AF | 18A | ··· | 18BB | 18BC | ··· | 18CL | 36A | ··· | 36BT |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 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 | C3 | C3 | C4 | C6 | C6 | C6 | C6 | C9 | C12 | C12 | C18 | C18 | C36 | D4 | C3×D4 | C3×D4 | D4×C9 |
| kernel | C22⋊C4×C3×C9 | C6×C36 | C2×C6×C18 | C9×C22⋊C4 | C32×C22⋊C4 | C6×C18 | C2×C36 | C22×C18 | C6×C12 | C2×C62 | C3×C22⋊C4 | C2×C18 | C62 | C2×C12 | C22×C6 | C2×C6 | C3×C18 | C18 | C3×C6 | C6 |
| # reps | 1 | 2 | 1 | 6 | 2 | 4 | 12 | 6 | 4 | 2 | 18 | 24 | 8 | 36 | 18 | 72 | 2 | 12 | 4 | 36 |
Matrix representation of C22⋊C4×C3×C9 ►in GL4(𝔽37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 7 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 36 | 36 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 36 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 |
| 0 | 0 | 36 | 35 |
| 0 | 0 | 1 | 1 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[7,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,36,0,0,0,0,1,36,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[36,0,0,0,0,31,0,0,0,0,36,1,0,0,35,1] >;
C22⋊C4×C3×C9 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4\times C_3\times C_9
% in TeX
G:=Group("C2^2:C4xC3xC9"); // GroupNames label
G:=SmallGroup(432,203);
// by ID
G=gap.SmallGroup(432,203);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,504,533,772]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^9=c^2=d^2=e^4=1,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,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations