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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 [×3], C4 [×2], C22, C6, C6 [×3], C6 [×4], C8 [×2], C2×C4, C9 [×3], C32, C12 [×2], C12 [×6], C2×C6, C2×C6 [×3], M4(2), C18 [×3], C18 [×3], C3×C6, C3×C6, C24 [×8], C2×C12, C2×C12 [×3], C3×C9, C36 [×6], C2×C18 [×3], C3×C12 [×2], C62, C3×M4(2), C3×M4(2) [×3], C3×C18, C3×C18, C72 [×6], C2×C36 [×3], C3×C24 [×2], C6×C12, C3×C36 [×2], C6×C18, C9×M4(2) [×3], C32×M4(2), C3×C72 [×2], C6×C36, M4(2)×C3×C9
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C2×C4, C9 [×3], C32, C12 [×8], C2×C6 [×4], M4(2), C18 [×9], C3×C6 [×3], C2×C12 [×4], C3×C9, C36 [×6], C2×C18 [×3], C3×C12 [×2], C62, C3×M4(2) [×4], C3×C18 [×3], C2×C36 [×3], C6×C12, C3×C36 [×2], C6×C18, C9×M4(2) [×3], C32×M4(2), C6×C36, M4(2)×C3×C9

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

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

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

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

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

׿
×
𝔽