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G = C33⋊27SD16  order 432 = 24·33

3rd semidirect product of C33 and SD16 acting via SD16/Q8=C2

metabelian, supersoluble, monomial

Aliases: C33⋊27SD16, C33⋊7C8⋊6C2, (Q8×C33)⋊3C2, (C3×C12).133D6, (Q8×C32)⋊13S3, (C32×C6).82D4, Q8⋊2(C33⋊C2), C33⋊12D4.4C2, C3⋊3(C32⋊11SD16), C2.6(C33⋊15D4), C6.26(C32⋊7D4), C32⋊15(Q8⋊2S3), (C32×C12).33C22, C12.19(C2×C3⋊S3), (C3×Q8)⋊3(C3⋊S3), C4.3(C2×C33⋊C2), (C3×C6).117(C3⋊D4), SmallGroup(432,509)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C12 — C33⋊27SD16
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C32×C12 — C33⋊12D4 — C33⋊27SD16
Lower central
C33 — C32×C6 — C32×C12 — C33⋊27SD16
Upper central
C1 — C2 — C4 — Q8

Generators and relations for C33⋊27SD16
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, bc=cb, dbd-1=ebe=b-1, dcd-1=ece=c-1, ede=d3 >

Subgroups: 1824 in 280 conjugacy classes, 115 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, D4, Q8, C32, C12, C12, D6, SD16, C3⋊S3, C3×C6, C3⋊C8, D12, C3×Q8, C33, C3×C12, C3×C12, C2×C3⋊S3, Q8⋊2S3, C33⋊C2, C32×C6, C32⋊4C8, C12⋊S3, Q8×C32, C32×C12, C32×C12, C2×C33⋊C2, C32⋊11SD16, C33⋊7C8, C33⋊12D4, Q8×C33, C33⋊27SD16
Quotients: C1, C2, C22, S3, D4, D6, SD16, C3⋊S3, C3⋊D4, C2×C3⋊S3, Q8⋊2S3, C33⋊C2, C32⋊7D4, C2×C33⋊C2, C32⋊11SD16, C33⋊15D4, C33⋊27SD16

Smallest permutation representation of C33⋊27SD16
►On 216 points
Generators in S216
(1 74 162)(2 163 75)(3 76 164)(4 165 77)(5 78 166)(6 167 79)(7 80 168)(8 161 73)(9 151 56)(10 49 152)(11 145 50)(12 51 146)(13 147 52)(14 53 148)(15 149 54)(16 55 150)(17 112 198)(18 199 105)(19 106 200)(20 193 107)(21 108 194)(22 195 109)(23 110 196)(24 197 111)(25 181 33)(26 34 182)(27 183 35)(28 36 184)(29 177 37)(30 38 178)(31 179 39)(32 40 180)(41 210 113)(42 114 211)(43 212 115)(44 116 213)(45 214 117)(46 118 215)(47 216 119)(48 120 209)(57 173 131)(58 132 174)(59 175 133)(60 134 176)(61 169 135)(62 136 170)(63 171 129)(64 130 172)(65 121 144)(66 137 122)(67 123 138)(68 139 124)(69 125 140)(70 141 126)(71 127 142)(72 143 128)(81 201 97)(82 98 202)(83 203 99)(84 100 204)(85 205 101)(86 102 206)(87 207 103)(88 104 208)(89 186 153)(90 154 187)(91 188 155)(92 156 189)(93 190 157)(94 158 191)(95 192 159)(96 160 185)

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

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

(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)

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

72 conjugacy classes

class 1 2A2B3A···3M4A4B6A···6M8A8B12A···12AM
order1223···3446···68812···12
size111082···2242···254544···4

72 irreducible representations

dim1111222224
type++++++++
imageC1C2C2C2S3D4D6SD16C3⋊D4Q8⋊2S3
kernelC33⋊27SD16C33⋊7C8C33⋊12D4Q8×C33Q8×C32C32×C6C3×C12C33C3×C6C32
# reps11111311322613

Matrix representation of C33⋊27SD16 ►in GL8(𝔽73)

072000000
172000000
00010000
0072720000
0000727200
00001000
00000010
00000001
,
721000000
720000000
00100000
00010000
00000100
0000727200
00000010
00000001
,
721000000
720000000
00010000
0072720000
00001000
00000100
00000010
00000001
,
721000000
01000000
0030600000
0030430000
00001000
0000727200
000000069
0000001812
,
721000000
01000000
007200000
00110000
00001000
0000727200
00000010
0000007072

G:=sub<GL(8,GF(73))| [0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,30,30,0,0,0,0,0,0,60,43,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,69,12],[72,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,70,0,0,0,0,0,0,0,72] >;
 

C33⋊27SD16 in GAP, Magma, Sage, TeX 

C_3^3\rtimes_{27}{\rm SD}_{16}
 
% in TeX
 
G:=Group("C3^3:27SD16");
 
// GroupNames label
 
G:=SmallGroup(432,509);
 
// by ID
 
G=gap.SmallGroup(432,509);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,64,254,135,58,1124,4037,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a^-1,b*c=c*b,d*b*d^-1=e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^3>;
 
// generators/relations
 

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