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## G = C4○D4×C33order 432 = 24·33

### Direct product of C33 and C4○D4

direct product, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4○D4×C33
 Chief series C1 — C2 — C6 — C3×C6 — C32×C6 — C3×C62 — D4×C33 — C4○D4×C33
 Lower central C1 — C2 — C4○D4×C33
 Upper central C1 — C32×C12 — C4○D4×C33

Generators and relations for C4○D4×C33
G = < a,b,c,d,e,f | a3=b3=c3=d4=f2=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=d2e >

Subgroups: 644 in 560 conjugacy classes, 476 normal (10 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2×C4, D4, Q8, C32, C12, C2×C6, C4○D4, C3×C6, C3×C6, C2×C12, C3×D4, C3×Q8, C33, C3×C12, C62, C3×C4○D4, C32×C6, C32×C6, C6×C12, D4×C32, Q8×C32, C32×C12, C32×C12, C3×C62, C32×C4○D4, C3×C6×C12, D4×C33, Q8×C33, C4○D4×C33
Quotients: C1, C2, C3, C22, C6, C23, C32, C2×C6, C4○D4, C3×C6, C22×C6, C33, C62, C3×C4○D4, C32×C6, C2×C62, C3×C62, C32×C4○D4, C63, C4○D4×C33

Smallest permutation representation of C4○D4×C33
On 216 points
Generators in S216
(1 109 66)(2 110 67)(3 111 68)(4 112 65)(5 105 93)(6 106 94)(7 107 95)(8 108 96)(9 75 70)(10 76 71)(11 73 72)(12 74 69)(13 39 101)(14 40 102)(15 37 103)(16 38 104)(17 59 78)(18 60 79)(19 57 80)(20 58 77)(21 35 97)(22 36 98)(23 33 99)(24 34 100)(25 92 88)(26 89 85)(27 90 86)(28 91 87)(29 43 81)(30 44 82)(31 41 83)(32 42 84)(45 61 49)(46 62 50)(47 63 51)(48 64 52)(53 126 189)(54 127 190)(55 128 191)(56 125 192)(113 216 154)(114 213 155)(115 214 156)(116 215 153)(117 183 179)(118 184 180)(119 181 177)(120 182 178)(121 200 162)(122 197 163)(123 198 164)(124 199 161)(129 196 158)(130 193 159)(131 194 160)(132 195 157)(133 175 171)(134 176 172)(135 173 169)(136 174 170)(137 204 142)(138 201 143)(139 202 144)(140 203 141)(145 186 149)(146 187 150)(147 188 151)(148 185 152)(165 210 206)(166 211 207)(167 212 208)(168 209 205)
(1 87 61)(2 88 62)(3 85 63)(4 86 64)(5 10 35)(6 11 36)(7 12 33)(8 9 34)(13 18 43)(14 19 44)(15 20 41)(16 17 42)(21 93 71)(22 94 72)(23 95 69)(24 96 70)(25 50 110)(26 51 111)(27 52 112)(28 49 109)(29 101 79)(30 102 80)(31 103 77)(32 104 78)(37 58 83)(38 59 84)(39 60 81)(40 57 82)(45 66 91)(46 67 92)(47 68 89)(48 65 90)(53 188 209)(54 185 210)(55 186 211)(56 187 212)(73 98 106)(74 99 107)(75 100 108)(76 97 105)(113 171 196)(114 172 193)(115 169 194)(116 170 195)(117 142 200)(118 143 197)(119 144 198)(120 141 199)(121 179 204)(122 180 201)(123 177 202)(124 178 203)(125 150 208)(126 151 205)(127 152 206)(128 149 207)(129 154 175)(130 155 176)(131 156 173)(132 153 174)(133 158 216)(134 159 213)(135 160 214)(136 157 215)(137 162 183)(138 163 184)(139 164 181)(140 161 182)(145 166 191)(146 167 192)(147 168 189)(148 165 190)
(1 83 7)(2 84 8)(3 81 5)(4 82 6)(9 88 38)(10 85 39)(11 86 40)(12 87 37)(13 71 89)(14 72 90)(15 69 91)(16 70 92)(17 24 46)(18 21 47)(19 22 48)(20 23 45)(25 104 75)(26 101 76)(27 102 73)(28 103 74)(29 105 111)(30 106 112)(31 107 109)(32 108 110)(33 61 58)(34 62 59)(35 63 60)(36 64 57)(41 95 66)(42 96 67)(43 93 68)(44 94 65)(49 77 99)(50 78 100)(51 79 97)(52 80 98)(53 138 213)(54 139 214)(55 140 215)(56 137 216)(113 192 142)(114 189 143)(115 190 144)(116 191 141)(117 196 167)(118 193 168)(119 194 165)(120 195 166)(121 175 150)(122 176 151)(123 173 152)(124 174 149)(125 204 154)(126 201 155)(127 202 156)(128 203 153)(129 208 179)(130 205 180)(131 206 177)(132 207 178)(133 187 162)(134 188 163)(135 185 164)(136 186 161)(145 199 170)(146 200 171)(147 197 172)(148 198 169)(157 211 182)(158 212 183)(159 209 184)(160 210 181)
(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)
(1 189 3 191)(2 190 4 192)(5 116 7 114)(6 113 8 115)(9 169 11 171)(10 170 12 172)(13 124 15 122)(14 121 16 123)(17 177 19 179)(18 178 20 180)(21 132 23 130)(22 129 24 131)(25 185 27 187)(26 186 28 188)(29 140 31 138)(30 137 32 139)(33 193 35 195)(34 194 36 196)(37 197 39 199)(38 198 40 200)(41 201 43 203)(42 202 44 204)(45 205 47 207)(46 206 48 208)(49 209 51 211)(50 210 52 212)(53 111 55 109)(54 112 56 110)(57 117 59 119)(58 118 60 120)(61 168 63 166)(62 165 64 167)(65 125 67 127)(66 126 68 128)(69 176 71 174)(70 173 72 175)(73 133 75 135)(74 134 76 136)(77 184 79 182)(78 181 80 183)(81 141 83 143)(82 142 84 144)(85 145 87 147)(86 146 88 148)(89 149 91 151)(90 150 92 152)(93 153 95 155)(94 154 96 156)(97 157 99 159)(98 158 100 160)(101 161 103 163)(102 162 104 164)(105 215 107 213)(106 216 108 214)
(1 191)(2 192)(3 189)(4 190)(5 114)(6 115)(7 116)(8 113)(9 171)(10 172)(11 169)(12 170)(13 122)(14 123)(15 124)(16 121)(17 179)(18 180)(19 177)(20 178)(21 130)(22 131)(23 132)(24 129)(25 187)(26 188)(27 185)(28 186)(29 138)(30 139)(31 140)(32 137)(33 195)(34 196)(35 193)(36 194)(37 199)(38 200)(39 197)(40 198)(41 203)(42 204)(43 201)(44 202)(45 207)(46 208)(47 205)(48 206)(49 211)(50 212)(51 209)(52 210)(53 111)(54 112)(55 109)(56 110)(57 119)(58 120)(59 117)(60 118)(61 166)(62 167)(63 168)(64 165)(65 127)(66 128)(67 125)(68 126)(69 174)(70 175)(71 176)(72 173)(73 135)(74 136)(75 133)(76 134)(77 182)(78 183)(79 184)(80 181)(81 143)(82 144)(83 141)(84 142)(85 147)(86 148)(87 145)(88 146)(89 151)(90 152)(91 149)(92 150)(93 155)(94 156)(95 153)(96 154)(97 159)(98 160)(99 157)(100 158)(101 163)(102 164)(103 161)(104 162)(105 213)(106 214)(107 215)(108 216)

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

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

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

270 conjugacy classes

 class 1 2A 2B 2C 2D 3A ··· 3Z 4A 4B 4C 4D 4E 6A ··· 6Z 6AA ··· 6CZ 12A ··· 12AZ 12BA ··· 12DZ order 1 2 2 2 2 3 ··· 3 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 2 2 2 1 ··· 1 1 1 2 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

270 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C3 C6 C6 C6 C4○D4 C3×C4○D4 kernel C4○D4×C33 C3×C6×C12 D4×C33 Q8×C33 C32×C4○D4 C6×C12 D4×C32 Q8×C32 C33 C32 # reps 1 3 3 1 26 78 78 26 2 52

Matrix representation of C4○D4×C33 in GL4(𝔽13) generated by

 3 0 0 0 0 1 0 0 0 0 3 0 0 0 0 3
,
 3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3
,
 9 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 1 0 0 0 0 5 0 0 0 0 5
,
 1 0 0 0 0 12 0 0 0 0 0 1 0 0 12 0
,
 12 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(13))| [3,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[9,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,1,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,12,0,0,0,0,0,12,0,0,1,0],[12,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;

C4○D4×C33 in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_3^3
% in TeX

G:=Group("C4oD4xC3^3");
// GroupNames label

G:=SmallGroup(432,733);
// by ID

G=gap.SmallGroup(432,733);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-3,-2,3053,1150]);
// Polycyclic

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

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