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## G = C3×Q8⋊C9order 216 = 23·33

### Direct product of C3 and Q8⋊C9

Aliases: C3×Q8⋊C9, C32.2SL2(𝔽3), (C3×Q8)⋊C9, Q82(C3×C9), C6.2(C3×A4), (C3×C6).4A4, C6.3(C3.A4), (Q8×C32).1C3, (C3×Q8).3C32, C3.2(C3×SL2(𝔽3)), C2.(C3×C3.A4), SmallGroup(216,40)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C3×Q8⋊C9
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8⋊C9 — C3×Q8⋊C9
 Lower central Q8 — C3×Q8⋊C9
 Upper central C1 — C3×C6

Generators and relations for C3×Q8⋊C9
G = < a,b,c,d | a3=b4=d9=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >

Smallest permutation representation of C3×Q8⋊C9
Regular action on 216 points
Generators in S216
(1 215 190)(2 216 191)(3 208 192)(4 209 193)(5 210 194)(6 211 195)(7 212 196)(8 213 197)(9 214 198)(10 83 111)(11 84 112)(12 85 113)(13 86 114)(14 87 115)(15 88 116)(16 89 117)(17 90 109)(18 82 110)(19 188 92)(20 189 93)(21 181 94)(22 182 95)(23 183 96)(24 184 97)(25 185 98)(26 186 99)(27 187 91)(28 153 124)(29 145 125)(30 146 126)(31 147 118)(32 148 119)(33 149 120)(34 150 121)(35 151 122)(36 152 123)(37 65 172)(38 66 173)(39 67 174)(40 68 175)(41 69 176)(42 70 177)(43 71 178)(44 72 179)(45 64 180)(46 162 142)(47 154 143)(48 155 144)(49 156 136)(50 157 137)(51 158 138)(52 159 139)(53 160 140)(54 161 141)(55 171 75)(56 163 76)(57 164 77)(58 165 78)(59 166 79)(60 167 80)(61 168 81)(62 169 73)(63 170 74)(100 128 201)(101 129 202)(102 130 203)(103 131 204)(104 132 205)(105 133 206)(106 134 207)(107 135 199)(108 127 200)
(1 52 152 94)(2 163 153 135)(3 178 145 82)(4 46 146 97)(5 166 147 129)(6 172 148 85)(7 49 149 91)(8 169 150 132)(9 175 151 88)(10 100 72 58)(11 143 64 185)(12 195 65 32)(13 103 66 61)(14 137 67 188)(15 198 68 35)(16 106 69 55)(17 140 70 182)(18 192 71 29)(19 115 157 39)(20 74 158 206)(21 215 159 123)(22 109 160 42)(23 77 161 200)(24 209 162 126)(25 112 154 45)(26 80 155 203)(27 212 156 120)(28 107 191 56)(30 184 193 142)(31 101 194 59)(33 187 196 136)(34 104 197 62)(36 181 190 139)(37 119 113 211)(38 81 114 204)(40 122 116 214)(41 75 117 207)(43 125 110 208)(44 78 111 201)(47 180 98 84)(48 130 99 167)(50 174 92 87)(51 133 93 170)(53 177 95 90)(54 127 96 164)(57 141 108 183)(60 144 102 186)(63 138 105 189)(73 121 205 213)(76 124 199 216)(79 118 202 210)(83 128 179 165)(86 131 173 168)(89 134 176 171)
(1 171 152 134)(2 177 153 90)(3 54 145 96)(4 165 146 128)(5 180 147 84)(6 48 148 99)(7 168 149 131)(8 174 150 87)(9 51 151 93)(10 142 72 184)(11 194 64 31)(12 102 65 60)(13 136 66 187)(14 197 67 34)(15 105 68 63)(16 139 69 181)(17 191 70 28)(18 108 71 57)(19 73 157 205)(20 214 158 122)(21 117 159 41)(22 76 160 199)(23 208 161 125)(24 111 162 44)(25 79 154 202)(26 211 155 119)(27 114 156 38)(29 183 192 141)(30 100 193 58)(32 186 195 144)(33 103 196 61)(35 189 198 138)(36 106 190 55)(37 80 113 203)(39 121 115 213)(40 74 116 206)(42 124 109 216)(43 77 110 200)(45 118 112 210)(46 179 97 83)(47 129 98 166)(49 173 91 86)(50 132 92 169)(52 176 94 89)(53 135 95 163)(56 140 107 182)(59 143 101 185)(62 137 104 188)(75 123 207 215)(78 126 201 209)(81 120 204 212)(82 127 178 164)(85 130 172 167)(88 133 175 170)
(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)

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

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

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

C3×Q8⋊C9 is a maximal subgroup of   C32.3CSU2(𝔽3)  C32.3GL2(𝔽3)  Q8⋊C93S3

63 conjugacy classes

 class 1 2 3A ··· 3H 4 6A ··· 6H 9A ··· 9R 12A ··· 12H 18A ··· 18R order 1 2 3 ··· 3 4 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 size 1 1 1 ··· 1 6 1 ··· 1 4 ··· 4 6 ··· 6 4 ··· 4

63 irreducible representations

 dim 1 1 1 1 2 2 2 2 3 3 3 type + - + image C1 C3 C3 C9 SL2(𝔽3) SL2(𝔽3) Q8⋊C9 C3×SL2(𝔽3) A4 C3.A4 C3×A4 kernel C3×Q8⋊C9 Q8⋊C9 Q8×C32 C3×Q8 C32 C32 C3 C3 C3×C6 C6 C6 # reps 1 6 2 18 1 2 18 6 1 6 2

Matrix representation of C3×Q8⋊C9 in GL3(𝔽37) generated by

 26 0 0 0 10 0 0 0 10
,
 1 0 0 0 5 14 0 14 32
,
 1 0 0 0 0 36 0 1 0
,
 10 0 0 0 28 11 0 4 2
G:=sub<GL(3,GF(37))| [26,0,0,0,10,0,0,0,10],[1,0,0,0,5,14,0,14,32],[1,0,0,0,0,1,0,36,0],[10,0,0,0,28,4,0,11,2] >;

C3×Q8⋊C9 in GAP, Magma, Sage, TeX

C_3\times Q_8\rtimes C_9
% in TeX

G:=Group("C3xQ8:C9");
// GroupNames label

G:=SmallGroup(216,40);
// by ID

G=gap.SmallGroup(216,40);
# by ID

G:=PCGroup([6,-3,-3,-3,-2,2,-2,54,1299,117,2434,202,88]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=d^9=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=c,d*c*d^-1=b*c>;
// generators/relations

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