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## G = Q8×D27order 432 = 24·33

### Direct product of Q8 and D27

Series: Derived Chief Lower central Upper central

 Derived series C1 — C54 — Q8×D27
 Chief series C1 — C3 — C9 — C27 — C54 — D54 — C4×D27 — Q8×D27
 Lower central C27 — C54 — Q8×D27
 Upper central C1 — C2 — Q8

Generators and relations for Q8×D27
G = < a,b,c,d | a4=c27=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 544 in 76 conjugacy classes, 37 normal (14 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C2×C4, Q8, Q8, C9, Dic3, C12, D6, C2×Q8, D9, C18, Dic6, C4×S3, C3×Q8, C27, Dic9, C36, D18, S3×Q8, D27, C54, Dic18, C4×D9, Q8×C9, Dic27, C108, D54, Q8×D9, Dic54, C4×D27, Q8×C27, Q8×D27
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, D9, C22×S3, D18, S3×Q8, D27, C22×D9, D54, Q8×D9, C22×D27, Q8×D27

Smallest permutation representation of Q8×D27
On 216 points
Generators in S216
(1 104 49 55)(2 105 50 56)(3 106 51 57)(4 107 52 58)(5 108 53 59)(6 82 54 60)(7 83 28 61)(8 84 29 62)(9 85 30 63)(10 86 31 64)(11 87 32 65)(12 88 33 66)(13 89 34 67)(14 90 35 68)(15 91 36 69)(16 92 37 70)(17 93 38 71)(18 94 39 72)(19 95 40 73)(20 96 41 74)(21 97 42 75)(22 98 43 76)(23 99 44 77)(24 100 45 78)(25 101 46 79)(26 102 47 80)(27 103 48 81)(109 188 144 200)(110 189 145 201)(111 163 146 202)(112 164 147 203)(113 165 148 204)(114 166 149 205)(115 167 150 206)(116 168 151 207)(117 169 152 208)(118 170 153 209)(119 171 154 210)(120 172 155 211)(121 173 156 212)(122 174 157 213)(123 175 158 214)(124 176 159 215)(125 177 160 216)(126 178 161 190)(127 179 162 191)(128 180 136 192)(129 181 137 193)(130 182 138 194)(131 183 139 195)(132 184 140 196)(133 185 141 197)(134 186 142 198)(135 187 143 199)
(1 156 49 121)(2 157 50 122)(3 158 51 123)(4 159 52 124)(5 160 53 125)(6 161 54 126)(7 162 28 127)(8 136 29 128)(9 137 30 129)(10 138 31 130)(11 139 32 131)(12 140 33 132)(13 141 34 133)(14 142 35 134)(15 143 36 135)(16 144 37 109)(17 145 38 110)(18 146 39 111)(19 147 40 112)(20 148 41 113)(21 149 42 114)(22 150 43 115)(23 151 44 116)(24 152 45 117)(25 153 46 118)(26 154 47 119)(27 155 48 120)(55 212 104 173)(56 213 105 174)(57 214 106 175)(58 215 107 176)(59 216 108 177)(60 190 82 178)(61 191 83 179)(62 192 84 180)(63 193 85 181)(64 194 86 182)(65 195 87 183)(66 196 88 184)(67 197 89 185)(68 198 90 186)(69 199 91 187)(70 200 92 188)(71 201 93 189)(72 202 94 163)(73 203 95 164)(74 204 96 165)(75 205 97 166)(76 206 98 167)(77 207 99 168)(78 208 100 169)(79 209 101 170)(80 210 102 171)(81 211 103 172)
(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)
(1 49)(2 48)(3 47)(4 46)(5 45)(6 44)(7 43)(8 42)(9 41)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 34)(17 33)(18 32)(19 31)(20 30)(21 29)(22 28)(23 54)(24 53)(25 52)(26 51)(27 50)(55 104)(56 103)(57 102)(58 101)(59 100)(60 99)(61 98)(62 97)(63 96)(64 95)(65 94)(66 93)(67 92)(68 91)(69 90)(70 89)(71 88)(72 87)(73 86)(74 85)(75 84)(76 83)(77 82)(78 108)(79 107)(80 106)(81 105)(109 141)(110 140)(111 139)(112 138)(113 137)(114 136)(115 162)(116 161)(117 160)(118 159)(119 158)(120 157)(121 156)(122 155)(123 154)(124 153)(125 152)(126 151)(127 150)(128 149)(129 148)(130 147)(131 146)(132 145)(133 144)(134 143)(135 142)(163 195)(164 194)(165 193)(166 192)(167 191)(168 190)(169 216)(170 215)(171 214)(172 213)(173 212)(174 211)(175 210)(176 209)(177 208)(178 207)(179 206)(180 205)(181 204)(182 203)(183 202)(184 201)(185 200)(186 199)(187 198)(188 197)(189 196)

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

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

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

75 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6 9A 9B 9C 12A 12B 12C 18A 18B 18C 27A ··· 27I 36A ··· 36I 54A ··· 54I 108A ··· 108AA order 1 2 2 2 3 4 4 4 4 4 4 6 9 9 9 12 12 12 18 18 18 27 ··· 27 36 ··· 36 54 ··· 54 108 ··· 108 size 1 1 27 27 2 2 2 2 54 54 54 2 2 2 2 4 4 4 2 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

75 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + - + + + + + - - - image C1 C2 C2 C2 S3 Q8 D6 D9 D18 D27 D54 S3×Q8 Q8×D9 Q8×D27 kernel Q8×D27 Dic54 C4×D27 Q8×C27 Q8×C9 D27 C36 C3×Q8 C12 Q8 C4 C9 C3 C1 # reps 1 3 3 1 1 2 3 3 9 9 27 1 3 9

Matrix representation of Q8×D27 in GL4(𝔽109) generated by

 1 0 0 0 0 1 0 0 0 0 1 3 0 0 72 108
,
 108 0 0 0 0 108 0 0 0 0 14 84 0 0 82 95
,
 63 79 0 0 30 93 0 0 0 0 1 0 0 0 0 1
,
 102 92 0 0 99 7 0 0 0 0 108 0 0 0 0 108
G:=sub<GL(4,GF(109))| [1,0,0,0,0,1,0,0,0,0,1,72,0,0,3,108],[108,0,0,0,0,108,0,0,0,0,14,82,0,0,84,95],[63,30,0,0,79,93,0,0,0,0,1,0,0,0,0,1],[102,99,0,0,92,7,0,0,0,0,108,0,0,0,0,108] >;

Q8×D27 in GAP, Magma, Sage, TeX

Q_8\times D_{27}
% in TeX

G:=Group("Q8xD27");
// GroupNames label

G:=SmallGroup(432,49);
// by ID

G=gap.SmallGroup(432,49);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,135,58,2804,557,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^27=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽