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## G = D8×C33order 432 = 24·33

### Direct product of C33 and D8

direct product, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — D8×C33
 Chief series C1 — C2 — C4 — C12 — C3×C12 — C32×C12 — D4×C33 — D8×C33
 Lower central C1 — C2 — C4 — D8×C33
 Upper central C1 — C32×C6 — C32×C12 — D8×C33

Generators and relations for D8×C33
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 532 in 308 conjugacy classes, 196 normal (10 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, D4, C32, C12, C2×C6, D8, C3×C6, C3×C6, C24, C3×D4, C33, C3×C12, C62, C3×D8, C32×C6, C32×C6, C3×C24, D4×C32, C32×C12, C3×C62, C32×D8, C32×C24, D4×C33, D8×C33
Quotients: C1, C2, C3, C22, C6, D4, C32, C2×C6, D8, C3×C6, C3×D4, C33, C62, C3×D8, C32×C6, D4×C32, C3×C62, C32×D8, D4×C33, D8×C33

Smallest permutation representation of D8×C33
On 216 points
Generators in S216
(1 111 130)(2 112 131)(3 105 132)(4 106 133)(5 107 134)(6 108 135)(7 109 136)(8 110 129)(9 214 190)(10 215 191)(11 216 192)(12 209 185)(13 210 186)(14 211 187)(15 212 188)(16 213 189)(17 97 89)(18 98 90)(19 99 91)(20 100 92)(21 101 93)(22 102 94)(23 103 95)(24 104 96)(25 145 33)(26 146 34)(27 147 35)(28 148 36)(29 149 37)(30 150 38)(31 151 39)(32 152 40)(41 113 49)(42 114 50)(43 115 51)(44 116 52)(45 117 53)(46 118 54)(47 119 55)(48 120 56)(57 182 174)(58 183 175)(59 184 176)(60 177 169)(61 178 170)(62 179 171)(63 180 172)(64 181 173)(65 198 137)(66 199 138)(67 200 139)(68 193 140)(69 194 141)(70 195 142)(71 196 143)(72 197 144)(73 206 121)(74 207 122)(75 208 123)(76 201 124)(77 202 125)(78 203 126)(79 204 127)(80 205 128)(81 166 153)(82 167 154)(83 168 155)(84 161 156)(85 162 157)(86 163 158)(87 164 159)(88 165 160)
(1 175 18)(2 176 19)(3 169 20)(4 170 21)(5 171 22)(6 172 23)(7 173 24)(8 174 17)(9 25 65)(10 26 66)(11 27 67)(12 28 68)(13 29 69)(14 30 70)(15 31 71)(16 32 72)(33 137 190)(34 138 191)(35 139 192)(36 140 185)(37 141 186)(38 142 187)(39 143 188)(40 144 189)(41 81 121)(42 82 122)(43 83 123)(44 84 124)(45 85 125)(46 86 126)(47 87 127)(48 88 128)(49 153 206)(50 154 207)(51 155 208)(52 156 201)(53 157 202)(54 158 203)(55 159 204)(56 160 205)(57 97 110)(58 98 111)(59 99 112)(60 100 105)(61 101 106)(62 102 107)(63 103 108)(64 104 109)(73 113 166)(74 114 167)(75 115 168)(76 116 161)(77 117 162)(78 118 163)(79 119 164)(80 120 165)(89 129 182)(90 130 183)(91 131 184)(92 132 177)(93 133 178)(94 134 179)(95 135 180)(96 136 181)(145 198 214)(146 199 215)(147 200 216)(148 193 209)(149 194 210)(150 195 211)(151 196 212)(152 197 213)
(1 167 10)(2 168 11)(3 161 12)(4 162 13)(5 163 14)(6 164 15)(7 165 16)(8 166 9)(17 113 65)(18 114 66)(19 115 67)(20 116 68)(21 117 69)(22 118 70)(23 119 71)(24 120 72)(25 174 73)(26 175 74)(27 176 75)(28 169 76)(29 170 77)(30 171 78)(31 172 79)(32 173 80)(33 182 121)(34 183 122)(35 184 123)(36 177 124)(37 178 125)(38 179 126)(39 180 127)(40 181 128)(41 137 89)(42 138 90)(43 139 91)(44 140 92)(45 141 93)(46 142 94)(47 143 95)(48 144 96)(49 198 97)(50 199 98)(51 200 99)(52 193 100)(53 194 101)(54 195 102)(55 196 103)(56 197 104)(57 206 145)(58 207 146)(59 208 147)(60 201 148)(61 202 149)(62 203 150)(63 204 151)(64 205 152)(81 190 129)(82 191 130)(83 192 131)(84 185 132)(85 186 133)(86 187 134)(87 188 135)(88 189 136)(105 156 209)(106 157 210)(107 158 211)(108 159 212)(109 160 213)(110 153 214)(111 154 215)(112 155 216)
(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 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(17 18)(19 24)(20 23)(21 22)(25 26)(27 32)(28 31)(29 30)(33 34)(35 40)(36 39)(37 38)(41 42)(43 48)(44 47)(45 46)(49 50)(51 56)(52 55)(53 54)(57 58)(59 64)(60 63)(61 62)(65 66)(67 72)(68 71)(69 70)(73 74)(75 80)(76 79)(77 78)(81 82)(83 88)(84 87)(85 86)(89 90)(91 96)(92 95)(93 94)(97 98)(99 104)(100 103)(101 102)(105 108)(106 107)(109 112)(110 111)(113 114)(115 120)(116 119)(117 118)(121 122)(123 128)(124 127)(125 126)(129 130)(131 136)(132 135)(133 134)(137 138)(139 144)(140 143)(141 142)(145 146)(147 152)(148 151)(149 150)(153 154)(155 160)(156 159)(157 158)(161 164)(162 163)(165 168)(166 167)(169 172)(170 171)(173 176)(174 175)(177 180)(178 179)(181 184)(182 183)(185 188)(186 187)(189 192)(190 191)(193 196)(194 195)(197 200)(198 199)(201 204)(202 203)(205 208)(206 207)(209 212)(210 211)(213 216)(214 215)

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

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

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

189 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3Z 4 6A ··· 6Z 6AA ··· 6BZ 8A 8B 12A ··· 12Z 24A ··· 24AZ order 1 2 2 2 3 ··· 3 4 6 ··· 6 6 ··· 6 8 8 12 ··· 12 24 ··· 24 size 1 1 4 4 1 ··· 1 2 1 ··· 1 4 ··· 4 2 2 2 ··· 2 2 ··· 2

189 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 D4 D8 C3×D4 C3×D8 kernel D8×C33 C32×C24 D4×C33 C32×D8 C3×C24 D4×C32 C32×C6 C33 C3×C6 C32 # reps 1 1 2 26 26 52 1 2 26 52

Matrix representation of D8×C33 in GL5(𝔽73)

 64 0 0 0 0 0 8 0 0 0 0 0 8 0 0 0 0 0 1 0 0 0 0 0 1
,
 8 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 8 0 0 0 0 0 8
,
 1 0 0 0 0 0 64 0 0 0 0 0 64 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 72 0 0 0 0 0 0 16 57 0 0 0 16 16
,
 72 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 16 57 0 0 0 57 57

G:=sub<GL(5,GF(73))| [64,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,0,1],[8,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,8],[1,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,57,16],[72,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,16,57,0,0,0,57,57] >;

D8×C33 in GAP, Magma, Sage, TeX

D_8\times C_3^3
% in TeX

G:=Group("D8xC3^3");
// GroupNames label

G:=SmallGroup(432,517);
// by ID

G=gap.SmallGroup(432,517);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,-2,1541,13613,6816,124]);
// 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,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽