Copied to
clipboard

## G = C22⋊C4×C3×C9order 432 = 24·33

### Direct product of C3×C9 and C22⋊C4

direct product, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C22⋊C4×C3×C9
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C6×C18 — C6×C36 — C22⋊C4×C3×C9
 Lower central C1 — C2 — C22⋊C4×C3×C9
 Upper central C1 — C6×C18 — C22⋊C4×C3×C9

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 [×2], C2 [×2], C3, C3 [×3], C4 [×2], C22, C22 [×2], C22 [×2], C6, C6 [×11], C6 [×8], C2×C4 [×2], C23, C9 [×3], C32, C12 [×8], C2×C6, C2×C6 [×11], C2×C6 [×8], C22⋊C4, C18 [×9], C18 [×6], C3×C6, C3×C6 [×2], C3×C6 [×2], C2×C12 [×8], C22×C6, C22×C6 [×3], C3×C9, C36 [×6], C2×C18 [×9], C2×C18 [×6], C3×C12 [×2], C62, C62 [×2], C62 [×2], C3×C22⋊C4, C3×C22⋊C4 [×3], C3×C18, C3×C18 [×2], C3×C18 [×2], C2×C36 [×6], C22×C18 [×3], C6×C12 [×2], C2×C62, C3×C36 [×2], C6×C18, C6×C18 [×2], C6×C18 [×2], C9×C22⋊C4 [×3], C32×C22⋊C4, C6×C36 [×2], C2×C6×C18, C22⋊C4×C3×C9
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C2×C4, D4 [×2], C9 [×3], C32, C12 [×8], C2×C6 [×4], C22⋊C4, C18 [×9], C3×C6 [×3], C2×C12 [×4], C3×D4 [×8], C3×C9, C36 [×6], C2×C18 [×3], C3×C12 [×2], C62, C3×C22⋊C4 [×4], C3×C18 [×3], C2×C36 [×3], D4×C9 [×6], C6×C12, D4×C32 [×2], C3×C36 [×2], C6×C18, C9×C22⋊C4 [×3], C32×C22⋊C4, C6×C36, D4×C3×C9 [×2], C22⋊C4×C3×C9

Smallest permutation representation of C22⋊C4×C3×C9
On 216 points
Generators in S216
(1 54 114)(2 46 115)(3 47 116)(4 48 117)(5 49 109)(6 50 110)(7 51 111)(8 52 112)(9 53 113)(10 64 36)(11 65 28)(12 66 29)(13 67 30)(14 68 31)(15 69 32)(16 70 33)(17 71 34)(18 72 35)(19 106 57)(20 107 58)(21 108 59)(22 100 60)(23 101 61)(24 102 62)(25 103 63)(26 104 55)(27 105 56)(37 170 154)(38 171 155)(39 163 156)(40 164 157)(41 165 158)(42 166 159)(43 167 160)(44 168 161)(45 169 162)(73 206 190)(74 207 191)(75 199 192)(76 200 193)(77 201 194)(78 202 195)(79 203 196)(80 204 197)(81 205 198)(82 151 98)(83 152 99)(84 153 91)(85 145 92)(86 146 93)(87 147 94)(88 148 95)(89 149 96)(90 150 97)(118 208 127)(119 209 128)(120 210 129)(121 211 130)(122 212 131)(123 213 132)(124 214 133)(125 215 134)(126 216 135)(136 187 173)(137 188 174)(138 189 175)(139 181 176)(140 182 177)(141 183 178)(142 184 179)(143 185 180)(144 186 172)
(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)
(37 187)(38 188)(39 189)(40 181)(41 182)(42 183)(43 184)(44 185)(45 186)(73 131)(74 132)(75 133)(76 134)(77 135)(78 127)(79 128)(80 129)(81 130)(118 202)(119 203)(120 204)(121 205)(122 206)(123 207)(124 199)(125 200)(126 201)(136 154)(137 155)(138 156)(139 157)(140 158)(141 159)(142 160)(143 161)(144 162)(163 175)(164 176)(165 177)(166 178)(167 179)(168 180)(169 172)(170 173)(171 174)(190 212)(191 213)(192 214)(193 215)(194 216)(195 208)(196 209)(197 210)(198 211)
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 114)(11 115)(12 116)(13 117)(14 109)(15 110)(16 111)(17 112)(18 113)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 187)(38 188)(39 189)(40 181)(41 182)(42 183)(43 184)(44 185)(45 186)(55 151)(56 152)(57 153)(58 145)(59 146)(60 147)(61 148)(62 149)(63 150)(73 131)(74 132)(75 133)(76 134)(77 135)(78 127)(79 128)(80 129)(81 130)(82 104)(83 105)(84 106)(85 107)(86 108)(87 100)(88 101)(89 102)(90 103)(118 202)(119 203)(120 204)(121 205)(122 206)(123 207)(124 199)(125 200)(126 201)(136 154)(137 155)(138 156)(139 157)(140 158)(141 159)(142 160)(143 161)(144 162)(163 175)(164 176)(165 177)(166 178)(167 179)(168 180)(169 172)(170 173)(171 174)(190 212)(191 213)(192 214)(193 215)(194 216)(195 208)(196 209)(197 210)(198 211)
(1 122 26 169)(2 123 27 170)(3 124 19 171)(4 125 20 163)(5 126 21 164)(6 118 22 165)(7 119 23 166)(8 120 24 167)(9 121 25 168)(10 73 151 186)(11 74 152 187)(12 75 153 188)(13 76 145 189)(14 77 146 181)(15 78 147 182)(16 79 148 183)(17 80 149 184)(18 81 150 185)(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)(37 115 132 56)(38 116 133 57)(39 117 134 58)(40 109 135 59)(41 110 127 60)(42 111 128 61)(43 112 129 62)(44 113 130 63)(45 114 131 55)(46 213 105 154)(47 214 106 155)(48 215 107 156)(49 216 108 157)(50 208 100 158)(51 209 101 159)(52 210 102 160)(53 211 103 161)(54 212 104 162)(64 206 98 172)(65 207 99 173)(66 199 91 174)(67 200 92 175)(68 201 93 176)(69 202 94 177)(70 203 95 178)(71 204 96 179)(72 205 97 180)

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

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

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

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

׿
×
𝔽