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## G = C22⋊C4×C33order 432 = 24·33

### Direct product of C33 and C22⋊C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C22⋊C4×C33
 Chief series C1 — C2 — C22 — C2×C6 — C62 — C3×C62 — C3×C6×C12 — C22⋊C4×C33
 Lower central C1 — C2 — C22⋊C4×C33
 Upper central C1 — C3×C62 — C22⋊C4×C33

Generators and relations for C22⋊C4×C33
G = < a,b,c,d,e,f | a3=b3=c3=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, ef=fe >

Subgroups: 644 in 476 conjugacy classes, 308 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×13], C4 [×2], C22, C22 [×2], C22 [×2], C6 [×39], C6 [×26], C2×C4 [×2], C23, C32 [×13], C12 [×26], C2×C6 [×39], C2×C6 [×26], C22⋊C4, C3×C6 [×39], C3×C6 [×26], C2×C12 [×26], C22×C6 [×13], C33, C3×C12 [×26], C62 [×39], C62 [×26], C3×C22⋊C4 [×13], C32×C6, C32×C6 [×2], C32×C6 [×2], C6×C12 [×26], C2×C62 [×13], C32×C12 [×2], C3×C62, C3×C62 [×2], C3×C62 [×2], C32×C22⋊C4 [×13], C3×C6×C12 [×2], C63, C22⋊C4×C33
Quotients: C1, C2 [×3], C3 [×13], C4 [×2], C22, C6 [×39], C2×C4, D4 [×2], C32 [×13], C12 [×26], C2×C6 [×13], C22⋊C4, C3×C6 [×39], C2×C12 [×13], C3×D4 [×26], C33, C3×C12 [×26], C62 [×13], C3×C22⋊C4 [×13], C32×C6 [×3], C6×C12 [×13], D4×C32 [×26], C32×C12 [×2], C3×C62, C32×C22⋊C4 [×13], C3×C6×C12, D4×C33 [×2], C22⋊C4×C33

Smallest permutation representation of C22⋊C4×C33
On 216 points
Generators in S216
(1 6 35)(2 7 36)(3 8 33)(4 5 34)(9 161 141)(10 162 142)(11 163 143)(12 164 144)(13 153 145)(14 154 146)(15 155 147)(16 156 148)(17 45 37)(18 46 38)(19 47 39)(20 48 40)(21 49 29)(22 50 30)(23 51 31)(24 52 32)(25 129 157)(26 130 158)(27 131 159)(28 132 160)(41 121 149)(42 122 150)(43 123 151)(44 124 152)(53 133 125)(54 134 126)(55 135 127)(56 136 128)(57 111 189)(58 112 190)(59 109 191)(60 110 192)(61 213 81)(62 214 82)(63 215 83)(64 216 84)(65 93 197)(66 94 198)(67 95 199)(68 96 200)(69 209 77)(70 210 78)(71 211 79)(72 212 80)(73 101 181)(74 102 182)(75 103 183)(76 104 184)(85 165 105)(86 166 106)(87 167 107)(88 168 108)(89 169 97)(90 170 98)(91 171 99)(92 172 100)(113 137 117)(114 138 118)(115 139 119)(116 140 120)(173 201 193)(174 202 194)(175 203 195)(176 204 196)(177 205 185)(178 206 186)(179 207 187)(180 208 188)
(1 127 15)(2 128 16)(3 125 13)(4 126 14)(5 54 154)(6 55 155)(7 56 156)(8 53 153)(9 21 121)(10 22 122)(11 23 123)(12 24 124)(17 117 129)(18 118 130)(19 119 131)(20 120 132)(25 37 137)(26 38 138)(27 39 139)(28 40 140)(29 41 141)(30 42 142)(31 43 143)(32 44 144)(33 133 145)(34 134 146)(35 135 147)(36 136 148)(45 113 157)(46 114 158)(47 115 159)(48 116 160)(49 149 161)(50 150 162)(51 151 163)(52 152 164)(57 69 169)(58 70 170)(59 71 171)(60 72 172)(61 73 173)(62 74 174)(63 75 175)(64 76 176)(65 165 177)(66 166 178)(67 167 179)(68 168 180)(77 89 189)(78 90 190)(79 91 191)(80 92 192)(81 181 193)(82 182 194)(83 183 195)(84 184 196)(85 185 197)(86 186 198)(87 187 199)(88 188 200)(93 105 205)(94 106 206)(95 107 207)(96 108 208)(97 111 209)(98 112 210)(99 109 211)(100 110 212)(101 201 213)(102 202 214)(103 203 215)(104 204 216)
(1 119 11)(2 120 12)(3 117 9)(4 118 10)(5 114 162)(6 115 163)(7 116 164)(8 113 161)(13 17 121)(14 18 122)(15 19 123)(16 20 124)(21 125 129)(22 126 130)(23 127 131)(24 128 132)(25 29 133)(26 30 134)(27 31 135)(28 32 136)(33 137 141)(34 138 142)(35 139 143)(36 140 144)(37 41 145)(38 42 146)(39 43 147)(40 44 148)(45 149 153)(46 150 154)(47 151 155)(48 152 156)(49 53 157)(50 54 158)(51 55 159)(52 56 160)(57 61 165)(58 62 166)(59 63 167)(60 64 168)(65 169 173)(66 170 174)(67 171 175)(68 172 176)(69 73 177)(70 74 178)(71 75 179)(72 76 180)(77 181 185)(78 182 186)(79 183 187)(80 184 188)(81 85 189)(82 86 190)(83 87 191)(84 88 192)(89 193 197)(90 194 198)(91 195 199)(92 196 200)(93 97 201)(94 98 202)(95 99 203)(96 100 204)(101 205 209)(102 206 210)(103 207 211)(104 208 212)(105 111 213)(106 112 214)(107 109 215)(108 110 216)
(2 60)(4 58)(5 112)(7 110)(10 166)(12 168)(14 170)(16 172)(18 174)(20 176)(22 178)(24 180)(26 182)(28 184)(30 186)(32 188)(34 190)(36 192)(38 194)(40 196)(42 198)(44 200)(46 202)(48 204)(50 206)(52 208)(54 210)(56 212)(62 118)(64 120)(66 122)(68 124)(70 126)(72 128)(74 130)(76 132)(78 134)(80 136)(82 138)(84 140)(86 142)(88 144)(90 146)(92 148)(94 150)(96 152)(98 154)(100 156)(102 158)(104 160)(106 162)(108 164)(114 214)(116 216)
(1 59)(2 60)(3 57)(4 58)(5 112)(6 109)(7 110)(8 111)(9 165)(10 166)(11 167)(12 168)(13 169)(14 170)(15 171)(16 172)(17 173)(18 174)(19 175)(20 176)(21 177)(22 178)(23 179)(24 180)(25 181)(26 182)(27 183)(28 184)(29 185)(30 186)(31 187)(32 188)(33 189)(34 190)(35 191)(36 192)(37 193)(38 194)(39 195)(40 196)(41 197)(42 198)(43 199)(44 200)(45 201)(46 202)(47 203)(48 204)(49 205)(50 206)(51 207)(52 208)(53 209)(54 210)(55 211)(56 212)(61 117)(62 118)(63 119)(64 120)(65 121)(66 122)(67 123)(68 124)(69 125)(70 126)(71 127)(72 128)(73 129)(74 130)(75 131)(76 132)(77 133)(78 134)(79 135)(80 136)(81 137)(82 138)(83 139)(84 140)(85 141)(86 142)(87 143)(88 144)(89 145)(90 146)(91 147)(92 148)(93 149)(94 150)(95 151)(96 152)(97 153)(98 154)(99 155)(100 156)(101 157)(102 158)(103 159)(104 160)(105 161)(106 162)(107 163)(108 164)(113 213)(114 214)(115 215)(116 216)
(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)

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

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

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

270 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A ··· 3Z 4A 4B 4C 4D 6A ··· 6BZ 6CA ··· 6DZ 12A ··· 12CZ order 1 2 2 2 2 2 3 ··· 3 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2

270 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C3 C4 C6 C6 C12 D4 C3×D4 kernel C22⋊C4×C33 C3×C6×C12 C63 C32×C22⋊C4 C3×C62 C6×C12 C2×C62 C62 C32×C6 C3×C6 # reps 1 2 1 26 4 52 26 104 2 52

Matrix representation of C22⋊C4×C33 in GL5(𝔽13)

 3 0 0 0 0 0 9 0 0 0 0 0 1 0 0 0 0 0 3 0 0 0 0 0 3
,
 3 0 0 0 0 0 9 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 3 0 0 0 0 0 3 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 9
,
 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 9 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 12 0 0 0 0 0 8 0 0 0 0 0 9 11 0 0 0 1 4

G:=sub<GL(5,GF(13))| [3,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3],[3,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[3,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,9,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,8,0,0,0,0,0,9,1,0,0,0,11,4] >;

C22⋊C4×C33 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4\times C_3^3
% in TeX

G:=Group("C2^2:C4xC3^3");
// GroupNames label

G:=SmallGroup(432,513);
// by ID

G=gap.SmallGroup(432,513);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,-2,1512,1541]);
// Polycyclic

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

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