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G = C3327SD16order 432 = 24·33

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

metabelian, supersoluble, monomial

Aliases: C3327SD16, C337C86C2, (Q8×C33)⋊3C2, (C3×C12).133D6, (Q8×C32)⋊13S3, (C32×C6).82D4, Q82(C33⋊C2), C3312D4.4C2, C33(C3211SD16), C2.6(C3315D4), C6.26(C327D4), C3215(Q82S3), (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

C1C32×C12 — C3327SD16
C1C3C32C33C32×C6C32×C12C3312D4 — C3327SD16
C33C32×C6C32×C12 — C3327SD16
C1C2C4Q8

Generators and relations for C3327SD16
 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 [×13], C4, C4, C22, S3 [×13], C6 [×13], C8, D4, Q8, C32 [×13], C12 [×13], C12 [×13], D6 [×13], SD16, C3⋊S3 [×13], C3×C6 [×13], C3⋊C8 [×13], D12 [×13], C3×Q8 [×13], C33, C3×C12 [×13], C3×C12 [×13], C2×C3⋊S3 [×13], Q82S3 [×13], C33⋊C2, C32×C6, C324C8 [×13], C12⋊S3 [×13], Q8×C32 [×13], C32×C12, C32×C12, C2×C33⋊C2, C3211SD16 [×13], C337C8, C3312D4, Q8×C33, C3327SD16
Quotients: C1, C2 [×3], C22, S3 [×13], D4, D6 [×13], SD16, C3⋊S3 [×13], C3⋊D4 [×13], C2×C3⋊S3 [×13], Q82S3 [×13], C33⋊C2, C327D4 [×13], C2×C33⋊C2, C3211SD16 [×13], C3315D4, C3327SD16

Smallest permutation representation of C3327SD16
On 216 points
Generators in S216
(1 183 81)(2 82 184)(3 177 83)(4 84 178)(5 179 85)(6 86 180)(7 181 87)(8 88 182)(9 127 187)(10 188 128)(11 121 189)(12 190 122)(13 123 191)(14 192 124)(15 125 185)(16 186 126)(17 169 212)(18 213 170)(19 171 214)(20 215 172)(21 173 216)(22 209 174)(23 175 210)(24 211 176)(25 59 79)(26 80 60)(27 61 73)(28 74 62)(29 63 75)(30 76 64)(31 57 77)(32 78 58)(33 147 164)(34 165 148)(35 149 166)(36 167 150)(37 151 168)(38 161 152)(39 145 162)(40 163 146)(41 202 109)(42 110 203)(43 204 111)(44 112 205)(45 206 105)(46 106 207)(47 208 107)(48 108 201)(49 102 91)(50 92 103)(51 104 93)(52 94 97)(53 98 95)(54 96 99)(55 100 89)(56 90 101)(65 156 113)(66 114 157)(67 158 115)(68 116 159)(69 160 117)(70 118 153)(71 154 119)(72 120 155)(129 196 140)(130 141 197)(131 198 142)(132 143 199)(133 200 144)(134 137 193)(135 194 138)(136 139 195)
(1 57 170)(2 171 58)(3 59 172)(4 173 60)(5 61 174)(6 175 62)(7 63 176)(8 169 64)(9 106 164)(10 165 107)(11 108 166)(12 167 109)(13 110 168)(14 161 111)(15 112 162)(16 163 105)(17 76 182)(18 183 77)(19 78 184)(20 177 79)(21 80 178)(22 179 73)(23 74 180)(24 181 75)(25 215 83)(26 84 216)(27 209 85)(28 86 210)(29 211 87)(30 88 212)(31 213 81)(32 82 214)(33 127 207)(34 208 128)(35 121 201)(36 202 122)(37 123 203)(38 204 124)(39 125 205)(40 206 126)(41 190 150)(42 151 191)(43 192 152)(44 145 185)(45 186 146)(46 147 187)(47 188 148)(48 149 189)(49 68 134)(50 135 69)(51 70 136)(52 129 71)(53 72 130)(54 131 65)(55 66 132)(56 133 67)(89 157 199)(90 200 158)(91 159 193)(92 194 160)(93 153 195)(94 196 154)(95 155 197)(96 198 156)(97 140 119)(98 120 141)(99 142 113)(100 114 143)(101 144 115)(102 116 137)(103 138 117)(104 118 139)
(1 99 108)(2 109 100)(3 101 110)(4 111 102)(5 103 112)(6 105 104)(7 97 106)(8 107 98)(9 176 119)(10 120 169)(11 170 113)(12 114 171)(13 172 115)(14 116 173)(15 174 117)(16 118 175)(17 128 72)(18 65 121)(19 122 66)(20 67 123)(21 124 68)(22 69 125)(23 126 70)(24 71 127)(25 200 151)(26 152 193)(27 194 145)(28 146 195)(29 196 147)(30 148 197)(31 198 149)(32 150 199)(33 75 129)(34 130 76)(35 77 131)(36 132 78)(37 79 133)(38 134 80)(39 73 135)(40 136 74)(41 89 82)(42 83 90)(43 91 84)(44 85 92)(45 93 86)(46 87 94)(47 95 88)(48 81 96)(49 178 204)(50 205 179)(51 180 206)(52 207 181)(53 182 208)(54 201 183)(55 184 202)(56 203 177)(57 142 166)(58 167 143)(59 144 168)(60 161 137)(61 138 162)(62 163 139)(63 140 164)(64 165 141)(153 210 186)(154 187 211)(155 212 188)(156 189 213)(157 214 190)(158 191 215)(159 216 192)(160 185 209)
(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)(9 144)(10 139)(11 142)(12 137)(13 140)(14 143)(15 138)(16 141)(17 28)(18 31)(19 26)(20 29)(21 32)(22 27)(23 30)(24 25)(33 158)(34 153)(35 156)(36 159)(37 154)(38 157)(39 160)(40 155)(41 49)(42 52)(43 55)(44 50)(45 53)(46 56)(47 51)(48 54)(57 170)(58 173)(59 176)(60 171)(61 174)(62 169)(63 172)(64 175)(65 149)(66 152)(67 147)(68 150)(69 145)(70 148)(71 151)(72 146)(73 209)(74 212)(75 215)(76 210)(77 213)(78 216)(79 211)(80 214)(81 183)(82 178)(83 181)(84 184)(85 179)(86 182)(87 177)(88 180)(89 204)(90 207)(91 202)(92 205)(93 208)(94 203)(95 206)(96 201)(97 110)(98 105)(99 108)(100 111)(101 106)(102 109)(103 112)(104 107)(113 166)(114 161)(115 164)(116 167)(117 162)(118 165)(119 168)(120 163)(121 198)(122 193)(123 196)(124 199)(125 194)(126 197)(127 200)(128 195)(129 191)(130 186)(131 189)(132 192)(133 187)(134 190)(135 185)(136 188)

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

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

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

72 conjugacy classes

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

72 irreducible representations

dim1111222224
type++++++++
imageC1C2C2C2S3D4D6SD16C3⋊D4Q82S3
kernelC3327SD16C337C8C3312D4Q8×C33Q8×C32C32×C6C3×C12C33C3×C6C32
# reps11111311322613

Matrix representation of C3327SD16 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] >;

C3327SD16 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

׿
×
𝔽