Copied to
clipboard

## G = Q8×C3×C9order 216 = 23·33

### Direct product of C3×C9 and Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C3×C9
 Chief series C1 — C3 — C6 — C3×C6 — C3×C18 — C3×C36 — Q8×C3×C9
 Lower central C1 — C2 — Q8×C3×C9
 Upper central C1 — C3×C18 — Q8×C3×C9

Generators and relations for Q8×C3×C9
G = < a,b,c,d | a3=b9=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 60, all normal (12 characteristic)
C1, C2, C3, C3, C4, C6, C6, Q8, C9, C32, C12, C18, C3×C6, C3×Q8, C3×Q8, C3×C9, C36, C3×C12, C3×C18, Q8×C9, Q8×C32, C3×C36, Q8×C3×C9
Quotients: C1, C2, C3, C22, C6, Q8, C9, C32, C2×C6, C18, C3×C6, C3×Q8, C3×C9, C2×C18, C62, C3×C18, Q8×C9, Q8×C32, C6×C18, Q8×C3×C9

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

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

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

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

Q8×C3×C9 is a maximal subgroup of   C36.19D6  C36.20D6  C36.29D6

135 conjugacy classes

 class 1 2 3A ··· 3H 4A 4B 4C 6A ··· 6H 9A ··· 9R 12A ··· 12X 18A ··· 18R 36A ··· 36BB order 1 2 3 ··· 3 4 4 4 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 ··· 1 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

135 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + - image C1 C2 C3 C3 C6 C6 C9 C18 Q8 C3×Q8 C3×Q8 Q8×C9 kernel Q8×C3×C9 C3×C36 Q8×C9 Q8×C32 C36 C3×C12 C3×Q8 C12 C3×C9 C9 C32 C3 # reps 1 3 6 2 18 6 18 54 1 6 2 18

Matrix representation of Q8×C3×C9 in GL4(𝔽37) generated by

 1 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 16 0 0 0 0 26 0 0 0 0 10 0 0 0 0 10
,
 36 0 0 0 0 1 0 0 0 0 36 35 0 0 1 1
,
 36 0 0 0 0 36 0 0 0 0 9 27 0 0 23 28
G:=sub<GL(4,GF(37))| [1,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,26,0,0,0,0,10,0,0,0,0,10],[36,0,0,0,0,1,0,0,0,0,36,1,0,0,35,1],[36,0,0,0,0,36,0,0,0,0,9,23,0,0,27,28] >;

Q8×C3×C9 in GAP, Magma, Sage, TeX

Q_8\times C_3\times C_9
% in TeX

G:=Group("Q8xC3xC9");
// GroupNames label

G:=SmallGroup(216,79);
// by ID

G=gap.SmallGroup(216,79);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-3,216,457,223,338]);
// Polycyclic

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

׿
×
𝔽