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G = C32.3GL2(𝔽3)  order 432 = 24·33

2nd non-split extension by C32 of GL2(𝔽3) acting via GL2(𝔽3)/SL2(𝔽3)=C2

non-abelian, soluble

Aliases: C32.3GL2(𝔽3), C3⋊(Q8⋊D9), Q8⋊C9⋊2S3, (C3×Q8)⋊1D9, C6.5(C3⋊S4), (C3×C6).15S4, Q8⋊2(C9⋊S3), C6.6(C3.S4), C3.2(C6.6S4), (Q8×C32).11S3, C2.3(C32.3S4), (C3×Q8⋊C9)⋊4C2, (C3×Q8).4(C3⋊S3), SmallGroup(432,256)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — Q8 — C3×Q8⋊C9 — C32.3GL2(𝔽3)
Chief series
C1 — C2 — Q8 — C3×Q8 — Q8×C32 — C3×Q8⋊C9 — C32.3GL2(𝔽3)
Lower central
C3×Q8⋊C9 — C32.3GL2(𝔽3)
Upper central
C1 — C2

Generators and relations for C32.3GL2(𝔽3)
 G = < a,b,c,d,e,f | a3=b3=c4=f2=1, d2=c2, e3=fbf=b-1, ab=ba, ac=ca, ad=da, ae=ea, faf=a-1, bc=cb, bd=db, be=eb, dcd-1=fdf=c-1, ece-1=cd, fcf=c2d, ede-1=c, fef=be2 >

Subgroups: 796 in 82 conjugacy classes, 23 normal (11 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C32, C12, D6, SD16, D9, C18, C3⋊S3, C3×C6, C3⋊C8, D12, C3×Q8, C3×Q8, C3×C9, D18, C3×C12, C2×C3⋊S3, Q8⋊2S3, C9⋊S3, C3×C18, Q8⋊C9, C32⋊4C8, C12⋊S3, Q8×C32, C2×C9⋊S3, Q8⋊D9, C32⋊11SD16, C3×Q8⋊C9, C32.3GL2(𝔽3)
Quotients: C1, C2, S3, D9, C3⋊S3, S4, GL2(𝔽3), C9⋊S3, C3.S4, C3⋊S4, Q8⋊D9, C6.6S4, C32.3S4, C32.3GL2(𝔽3)

Smallest permutation representation of C32.3GL2(𝔽3)
►On 216 points
Generators in S216
(1 42 194)(2 43 195)(3 44 196)(4 45 197)(5 37 198)(6 38 190)(7 39 191)(8 40 192)(9 41 193)(10 164 66)(11 165 67)(12 166 68)(13 167 69)(14 168 70)(15 169 71)(16 170 72)(17 171 64)(18 163 65)(19 201 208)(20 202 209)(21 203 210)(22 204 211)(23 205 212)(24 206 213)(25 207 214)(26 199 215)(27 200 216)(28 125 153)(29 126 145)(30 118 146)(31 119 147)(32 120 148)(33 121 149)(34 122 150)(35 123 151)(36 124 152)(46 74 174)(47 75 175)(48 76 176)(49 77 177)(50 78 178)(51 79 179)(52 80 180)(53 81 172)(54 73 173)(55 160 98)(56 161 99)(57 162 91)(58 154 92)(59 155 93)(60 156 94)(61 157 95)(62 158 96)(63 159 97)(82 103 182)(83 104 183)(84 105 184)(85 106 185)(86 107 186)(87 108 187)(88 100 188)(89 101 189)(90 102 181)(109 128 136)(110 129 137)(111 130 138)(112 131 139)(113 132 140)(114 133 141)(115 134 142)(116 135 143)(117 127 144)

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

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

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

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

36 conjugacy classes

class 1 2A2B3A3B3C3D 4 6A6B6C6D8A8B9A···9I12A12B12C12D18A···18I
order122333346666889···91212121218···18
size1110822226222254548···8121212128···8

36 irreducible representations

dim112222344466
type+++++++++++
imageC1C2S3S3D9GL2(𝔽3)S4GL2(𝔽3)Q8⋊D9C6.6S4C3.S4C3⋊S4
kernelC32.3GL2(𝔽3)C3×Q8⋊C9Q8⋊C9Q8×C32C3×Q8C32C3×C6C32C3C3C6C6
# reps113192219331

Matrix representation of C32.3GL2(𝔽3) ►in GL6(𝔽73)

020000
36720000
001000
000100
00007169
0000191
,
020000
36720000
001000
000100
000014
00005471
,
100000
010000
0011200
00126200
000010
000001
,
100000
010000
0072200
0072100
000010
000001
,
42170000
14700000
00116300
0066100
00003451
00006814
,
100000
36720000
001000
00687200
0000720
0000191

G:=sub<GL(6,GF(73))| [0,36,0,0,0,0,2,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,71,19,0,0,0,0,69,1],[0,36,0,0,0,0,2,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,54,0,0,0,0,4,71],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,12,0,0,0,0,2,62,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[42,14,0,0,0,0,17,70,0,0,0,0,0,0,11,6,0,0,0,0,63,61,0,0,0,0,0,0,34,68,0,0,0,0,51,14],[1,36,0,0,0,0,0,72,0,0,0,0,0,0,1,68,0,0,0,0,0,72,0,0,0,0,0,0,72,19,0,0,0,0,0,1] >;
 

C32.3GL2(𝔽3) in GAP, Magma, Sage, TeX 

C_3^2._3{\rm GL}_2({\mathbb F}_3)
 
% in TeX
 
G:=Group("C3^2.3GL(2,3)");
 
// GroupNames label
 
G:=SmallGroup(432,256);
 
// by ID
 
G=gap.SmallGroup(432,256);
 
# by ID
 
G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,309,260,254,1011,3784,5681,172,2273,3414,285,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^3=c^4=f^2=1,d^2=c^2,e^3=f*b*f=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f=a^-1,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=f*d*f=c^-1,e*c*e^-1=c*d,f*c*f=c^2*d,e*d*e^-1=c,f*e*f=b*e^2>;
 
// generators/relations
 

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