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## G = M4(2)×C3×C9order 432 = 24·33

### Direct product of C3×C9 and M4(2)

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — M4(2)×C3×C9
 Chief series C1 — C2 — C6 — C12 — C3×C12 — C3×C36 — C3×C72 — M4(2)×C3×C9
 Lower central C1 — C2 — M4(2)×C3×C9
 Upper central C1 — C3×C36 — M4(2)×C3×C9

Generators and relations for M4(2)×C3×C9
G = < a,b,c,d | a3=b9=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >

Subgroups: 110 in 100 conjugacy classes, 90 normal (28 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, C9, C32, C12, C12, C2×C6, C2×C6, M4(2), C18, C18, C3×C6, C3×C6, C24, C2×C12, C2×C12, C3×C9, C36, C2×C18, C3×C12, C62, C3×M4(2), C3×M4(2), C3×C18, C3×C18, C72, C2×C36, C3×C24, C6×C12, C3×C36, C6×C18, C9×M4(2), C32×M4(2), C3×C72, C6×C36, M4(2)×C3×C9
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C9, C32, C12, C2×C6, M4(2), C18, C3×C6, C2×C12, C3×C9, C36, C2×C18, C3×C12, C62, C3×M4(2), C3×C18, C2×C36, C6×C12, C3×C36, C6×C18, C9×M4(2), C32×M4(2), C6×C36, M4(2)×C3×C9

Smallest permutation representation of M4(2)×C3×C9
On 216 points
Generators in S216
(1 51 111)(2 52 112)(3 53 113)(4 54 114)(5 46 115)(6 47 116)(7 48 117)(8 49 109)(9 50 110)(10 101 59)(11 102 60)(12 103 61)(13 104 62)(14 105 63)(15 106 55)(16 107 56)(17 108 57)(18 100 58)(19 93 70)(20 94 71)(21 95 72)(22 96 64)(23 97 65)(24 98 66)(25 99 67)(26 91 68)(27 92 69)(28 86 153)(29 87 145)(30 88 146)(31 89 147)(32 90 148)(33 82 149)(34 83 150)(35 84 151)(36 85 152)(37 201 154)(38 202 155)(39 203 156)(40 204 157)(41 205 158)(42 206 159)(43 207 160)(44 199 161)(45 200 162)(73 141 190)(74 142 191)(75 143 192)(76 144 193)(77 136 194)(78 137 195)(79 138 196)(80 139 197)(81 140 198)(118 189 173)(119 181 174)(120 182 175)(121 183 176)(122 184 177)(123 185 178)(124 186 179)(125 187 180)(126 188 172)(127 208 166)(128 209 167)(129 210 168)(130 211 169)(131 212 170)(132 213 171)(133 214 163)(134 215 164)(135 216 165)
(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)
(1 177 69 131 13 142 34 206)(2 178 70 132 14 143 35 207)(3 179 71 133 15 144 36 199)(4 180 72 134 16 136 28 200)(5 172 64 135 17 137 29 201)(6 173 65 127 18 138 30 202)(7 174 66 128 10 139 31 203)(8 175 67 129 11 140 32 204)(9 176 68 130 12 141 33 205)(19 213 105 192 84 160 52 123)(20 214 106 193 85 161 53 124)(21 215 107 194 86 162 54 125)(22 216 108 195 87 154 46 126)(23 208 100 196 88 155 47 118)(24 209 101 197 89 156 48 119)(25 210 102 198 90 157 49 120)(26 211 103 190 82 158 50 121)(27 212 104 191 83 159 51 122)(37 115 188 96 165 57 78 145)(38 116 189 97 166 58 79 146)(39 117 181 98 167 59 80 147)(40 109 182 99 168 60 81 148)(41 110 183 91 169 61 73 149)(42 111 184 92 170 62 74 150)(43 112 185 93 171 63 75 151)(44 113 186 94 163 55 76 152)(45 114 187 95 164 56 77 153)
(37 165)(38 166)(39 167)(40 168)(41 169)(42 170)(43 171)(44 163)(45 164)(73 183)(74 184)(75 185)(76 186)(77 187)(78 188)(79 189)(80 181)(81 182)(118 196)(119 197)(120 198)(121 190)(122 191)(123 192)(124 193)(125 194)(126 195)(127 202)(128 203)(129 204)(130 205)(131 206)(132 207)(133 199)(134 200)(135 201)(136 180)(137 172)(138 173)(139 174)(140 175)(141 176)(142 177)(143 178)(144 179)(154 216)(155 208)(156 209)(157 210)(158 211)(159 212)(160 213)(161 214)(162 215)

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

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

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

270 conjugacy classes

 class 1 2A 2B 3A ··· 3H 4A 4B 4C 6A ··· 6H 6I ··· 6P 8A 8B 8C 8D 9A ··· 9R 12A ··· 12P 12Q ··· 12X 18A ··· 18R 18S ··· 18AJ 24A ··· 24AF 36A ··· 36AJ 36AK ··· 36BB 72A ··· 72BT order 1 2 2 3 ··· 3 4 4 4 6 ··· 6 6 ··· 6 8 8 8 8 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 18 ··· 18 24 ··· 24 36 ··· 36 36 ··· 36 72 ··· 72 size 1 1 2 1 ··· 1 1 1 2 1 ··· 1 2 ··· 2 2 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 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 1 1 1 1 2 2 2 2 type + + + image C1 C2 C2 C3 C3 C4 C4 C6 C6 C6 C6 C9 C12 C12 C12 C12 C18 C18 C36 C36 M4(2) C3×M4(2) C3×M4(2) C9×M4(2) kernel M4(2)×C3×C9 C3×C72 C6×C36 C9×M4(2) C32×M4(2) C3×C36 C6×C18 C72 C2×C36 C3×C24 C6×C12 C3×M4(2) C36 C2×C18 C3×C12 C62 C24 C2×C12 C12 C2×C6 C3×C9 C9 C32 C3 # reps 1 2 1 6 2 2 2 12 6 4 2 18 12 12 4 4 36 18 36 36 2 12 4 36

Matrix representation of M4(2)×C3×C9 in GL3(𝔽73) generated by

 64 0 0 0 8 0 0 0 8
,
 55 0 0 0 8 0 0 0 8
,
 46 0 0 0 72 71 0 60 1
,
 72 0 0 0 1 0 0 72 72
G:=sub<GL(3,GF(73))| [64,0,0,0,8,0,0,0,8],[55,0,0,0,8,0,0,0,8],[46,0,0,0,72,60,0,71,1],[72,0,0,0,1,72,0,0,72] >;

M4(2)×C3×C9 in GAP, Magma, Sage, TeX

M_4(2)\times C_3\times C_9
% in TeX

G:=Group("M4(2)xC3xC9");
// GroupNames label

G:=SmallGroup(432,212);
// by ID

G=gap.SmallGroup(432,212);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-3,-2,252,3053,394,242]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^9=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^5>;
// generators/relations

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