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## G = Q8×C9⋊S3order 432 = 24·33

### Direct product of Q8 and C9⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — Q8×C9⋊S3
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C2×C9⋊S3 — C4×C9⋊S3 — Q8×C9⋊S3
 Lower central C3×C9 — C3×C18 — Q8×C9⋊S3
 Upper central C1 — C2 — Q8

Generators and relations for Q8×C9⋊S3
G = < a,b,c,d,e | a4=c9=d3=e2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 1048 in 190 conjugacy classes, 73 normal (14 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, Q8, Q8, C9, C32, Dic3, C12, D6, C2×Q8, D9, C18, C3⋊S3, C3×C6, Dic6, C4×S3, C3×Q8, C3×Q8, C3×C9, Dic9, C36, D18, C3⋊Dic3, C3×C12, C2×C3⋊S3, S3×Q8, C9⋊S3, C3×C18, Dic18, C4×D9, Q8×C9, C324Q8, C4×C3⋊S3, Q8×C32, C9⋊Dic3, C3×C36, C2×C9⋊S3, Q8×D9, Q8×C3⋊S3, C12.D9, C4×C9⋊S3, Q8×C3×C9, Q8×C9⋊S3
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, D9, C3⋊S3, C22×S3, D18, C2×C3⋊S3, S3×Q8, C9⋊S3, C22×D9, C22×C3⋊S3, C2×C9⋊S3, Q8×D9, Q8×C3⋊S3, C22×C9⋊S3, Q8×C9⋊S3

Smallest permutation representation of Q8×C9⋊S3
On 216 points
Generators in S216
(1 103 49 76)(2 104 50 77)(3 105 51 78)(4 106 52 79)(5 107 53 80)(6 108 54 81)(7 100 46 73)(8 101 47 74)(9 102 48 75)(10 140 194 167)(11 141 195 168)(12 142 196 169)(13 143 197 170)(14 144 198 171)(15 136 190 163)(16 137 191 164)(17 138 192 165)(18 139 193 166)(19 152 206 179)(20 153 207 180)(21 145 199 172)(22 146 200 173)(23 147 201 174)(24 148 202 175)(25 149 203 176)(26 150 204 177)(27 151 205 178)(28 109 55 82)(29 110 56 83)(30 111 57 84)(31 112 58 85)(32 113 59 86)(33 114 60 87)(34 115 61 88)(35 116 62 89)(36 117 63 90)(37 118 64 91)(38 119 65 92)(39 120 66 93)(40 121 67 94)(41 122 68 95)(42 123 69 96)(43 124 70 97)(44 125 71 98)(45 126 72 99)(127 181 154 208)(128 182 155 209)(129 183 156 210)(130 184 157 211)(131 185 158 212)(132 186 159 213)(133 187 160 214)(134 188 161 215)(135 189 162 216)
(1 157 49 130)(2 158 50 131)(3 159 51 132)(4 160 52 133)(5 161 53 134)(6 162 54 135)(7 154 46 127)(8 155 47 128)(9 156 48 129)(10 113 194 86)(11 114 195 87)(12 115 196 88)(13 116 197 89)(14 117 198 90)(15 109 190 82)(16 110 191 83)(17 111 192 84)(18 112 193 85)(19 125 206 98)(20 126 207 99)(21 118 199 91)(22 119 200 92)(23 120 201 93)(24 121 202 94)(25 122 203 95)(26 123 204 96)(27 124 205 97)(28 163 55 136)(29 164 56 137)(30 165 57 138)(31 166 58 139)(32 167 59 140)(33 168 60 141)(34 169 61 142)(35 170 62 143)(36 171 63 144)(37 172 64 145)(38 173 65 146)(39 174 66 147)(40 175 67 148)(41 176 68 149)(42 177 69 150)(43 178 70 151)(44 179 71 152)(45 180 72 153)(73 208 100 181)(74 209 101 182)(75 210 102 183)(76 211 103 184)(77 212 104 185)(78 213 105 186)(79 214 106 187)(80 215 107 188)(81 216 108 189)
(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 35 37)(2 36 38)(3 28 39)(4 29 40)(5 30 41)(6 31 42)(7 32 43)(8 33 44)(9 34 45)(10 27 208)(11 19 209)(12 20 210)(13 21 211)(14 22 212)(15 23 213)(16 24 214)(17 25 215)(18 26 216)(46 59 70)(47 60 71)(48 61 72)(49 62 64)(50 63 65)(51 55 66)(52 56 67)(53 57 68)(54 58 69)(73 86 97)(74 87 98)(75 88 99)(76 89 91)(77 90 92)(78 82 93)(79 83 94)(80 84 95)(81 85 96)(100 113 124)(101 114 125)(102 115 126)(103 116 118)(104 117 119)(105 109 120)(106 110 121)(107 111 122)(108 112 123)(127 140 151)(128 141 152)(129 142 153)(130 143 145)(131 144 146)(132 136 147)(133 137 148)(134 138 149)(135 139 150)(154 167 178)(155 168 179)(156 169 180)(157 170 172)(158 171 173)(159 163 174)(160 164 175)(161 165 176)(162 166 177)(181 194 205)(182 195 206)(183 196 207)(184 197 199)(185 198 200)(186 190 201)(187 191 202)(188 192 203)(189 193 204)
(2 9)(3 8)(4 7)(5 6)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 27)(17 26)(18 25)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(36 45)(46 52)(47 51)(48 50)(53 54)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(63 72)(73 79)(74 78)(75 77)(80 81)(82 98)(83 97)(84 96)(85 95)(86 94)(87 93)(88 92)(89 91)(90 99)(100 106)(101 105)(102 104)(107 108)(109 125)(110 124)(111 123)(112 122)(113 121)(114 120)(115 119)(116 118)(117 126)(127 133)(128 132)(129 131)(134 135)(136 152)(137 151)(138 150)(139 149)(140 148)(141 147)(142 146)(143 145)(144 153)(154 160)(155 159)(156 158)(161 162)(163 179)(164 178)(165 177)(166 176)(167 175)(168 174)(169 173)(170 172)(171 180)(181 187)(182 186)(183 185)(188 189)(190 206)(191 205)(192 204)(193 203)(194 202)(195 201)(196 200)(197 199)(198 207)(208 214)(209 213)(210 212)(215 216)

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

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

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

75 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 9A ··· 9I 12A ··· 12L 18A ··· 18I 36A ··· 36AA order 1 2 2 2 3 3 3 3 4 4 4 4 4 4 6 6 6 6 9 ··· 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 27 27 2 2 2 2 2 2 2 54 54 54 2 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 S3 Q8 D6 D6 D9 D18 S3×Q8 S3×Q8 Q8×D9 kernel Q8×C9⋊S3 C12.D9 C4×C9⋊S3 Q8×C3×C9 Q8×C9 Q8×C32 C9⋊S3 C36 C3×C12 C3×Q8 C12 C9 C32 C3 # reps 1 3 3 1 3 1 2 9 3 9 27 3 1 9

Matrix representation of Q8×C9⋊S3 in GL6(𝔽37)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 24 0 0 0 0 3 36
,
 36 0 0 0 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 5 2 0 0 0 0 24 32
,
 26 6 0 0 0 0 31 20 0 0 0 0 0 0 17 11 0 0 0 0 26 6 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 36 36 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 36 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 36 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,24,36],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,24,0,0,0,0,2,32],[26,31,0,0,0,0,6,20,0,0,0,0,0,0,17,26,0,0,0,0,11,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,1,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,1,0,0,0,0,0,1,0,0,0,0,0,0,1,36,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Q8×C9⋊S3 in GAP, Magma, Sage, TeX

Q_8\times C_9\rtimes S_3
% in TeX

G:=Group("Q8xC9:S3");
// GroupNames label

G:=SmallGroup(432,392);
// by ID

G=gap.SmallGroup(432,392);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,135,58,6164,662,4037,14118]);
// Polycyclic

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

׿
×
𝔽