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## G = (Q8×C33)⋊C2order 432 = 24·33

### 9th semidirect product of Q8×C33 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — (Q8×C33)⋊C2
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C2×C33⋊C2 — C4×C33⋊C2 — (Q8×C33)⋊C2
 Lower central C33 — C32×C6 — (Q8×C33)⋊C2
 Upper central C1 — C2 — Q8

Generators and relations for (Q8×C33)⋊C2
G = < a,b,c,d,e,f | a3=b3=c3=d4=f2=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, faf=a-1, bc=cb, bd=db, be=eb, fbf=b-1, cd=dc, ce=ec, fcf=c-1, ede-1=fdf=d-1, ef=fe >

Subgroups: 3296 in 560 conjugacy classes, 179 normal (8 characteristic)
C1, C2, C2 [×3], C3 [×13], C4 [×3], C4, C22 [×3], S3 [×39], C6 [×13], C2×C4 [×3], D4 [×3], Q8, C32 [×13], Dic3 [×13], C12 [×39], D6 [×39], C4○D4, C3⋊S3 [×39], C3×C6 [×13], C4×S3 [×39], D12 [×39], C3×Q8 [×13], C33, C3⋊Dic3 [×13], C3×C12 [×39], C2×C3⋊S3 [×39], Q83S3 [×13], C33⋊C2 [×3], C32×C6, C4×C3⋊S3 [×39], C12⋊S3 [×39], Q8×C32 [×13], C335C4, C32×C12 [×3], C2×C33⋊C2 [×3], C12.26D6 [×13], C4×C33⋊C2 [×3], C3312D4 [×3], Q8×C33, (Q8×C33)⋊C2
Quotients: C1, C2 [×7], C22 [×7], S3 [×13], C23, D6 [×39], C4○D4, C3⋊S3 [×13], C22×S3 [×13], C2×C3⋊S3 [×39], Q83S3 [×13], C33⋊C2, C22×C3⋊S3 [×13], C2×C33⋊C2 [×3], C12.26D6 [×13], C22×C33⋊C2, (Q8×C33)⋊C2

Smallest permutation representation of (Q8×C33)⋊C2
On 216 points
Generators in S216
(1 110 74)(2 111 75)(3 112 76)(4 109 73)(5 95 63)(6 96 64)(7 93 61)(8 94 62)(9 137 47)(10 138 48)(11 139 45)(12 140 46)(13 90 60)(14 91 57)(15 92 58)(16 89 59)(17 68 165)(18 65 166)(19 66 167)(20 67 168)(21 37 77)(22 38 78)(23 39 79)(24 40 80)(25 207 152)(26 208 149)(27 205 150)(28 206 151)(29 88 100)(30 85 97)(31 86 98)(32 87 99)(33 102 52)(34 103 49)(35 104 50)(36 101 51)(41 69 105)(42 70 106)(43 71 107)(44 72 108)(53 185 199)(54 186 200)(55 187 197)(56 188 198)(81 159 119)(82 160 120)(83 157 117)(84 158 118)(113 171 132)(114 172 129)(115 169 130)(116 170 131)(121 142 202)(122 143 203)(123 144 204)(124 141 201)(125 145 195)(126 146 196)(127 147 193)(128 148 194)(133 191 174)(134 192 175)(135 189 176)(136 190 173)(153 214 181)(154 215 182)(155 216 183)(156 213 184)(161 209 177)(162 210 178)(163 211 179)(164 212 180)
(1 49 60)(2 50 57)(3 51 58)(4 52 59)(5 38 18)(6 39 19)(7 40 20)(8 37 17)(9 43 85)(10 44 86)(11 41 87)(12 42 88)(13 110 34)(14 111 35)(15 112 36)(16 109 33)(21 165 62)(22 166 63)(23 167 64)(24 168 61)(25 146 211)(26 147 212)(27 148 209)(28 145 210)(29 46 106)(30 47 107)(31 48 108)(32 45 105)(53 153 131)(54 154 132)(55 155 129)(56 156 130)(65 95 78)(66 96 79)(67 93 80)(68 94 77)(69 99 139)(70 100 140)(71 97 137)(72 98 138)(73 102 89)(74 103 90)(75 104 91)(76 101 92)(81 175 204)(82 176 201)(83 173 202)(84 174 203)(113 186 215)(114 187 216)(115 188 213)(116 185 214)(117 190 142)(118 191 143)(119 192 144)(120 189 141)(121 157 136)(122 158 133)(123 159 134)(124 160 135)(125 162 151)(126 163 152)(127 164 149)(128 161 150)(169 198 184)(170 199 181)(171 200 182)(172 197 183)(177 205 194)(178 206 195)(179 207 196)(180 208 193)
(1 80 43)(2 77 44)(3 78 41)(4 79 42)(5 45 92)(6 46 89)(7 47 90)(8 48 91)(9 60 93)(10 57 94)(11 58 95)(12 59 96)(13 61 137)(14 62 138)(15 63 139)(16 64 140)(17 31 104)(18 32 101)(19 29 102)(20 30 103)(21 72 111)(22 69 112)(23 70 109)(24 71 110)(25 130 173)(26 131 174)(27 132 175)(28 129 176)(33 167 100)(34 168 97)(35 165 98)(36 166 99)(37 108 75)(38 105 76)(39 106 73)(40 107 74)(49 67 85)(50 68 86)(51 65 87)(52 66 88)(53 203 147)(54 204 148)(55 201 145)(56 202 146)(81 209 154)(82 210 155)(83 211 156)(84 212 153)(113 134 205)(114 135 206)(115 136 207)(116 133 208)(117 163 184)(118 164 181)(119 161 182)(120 162 183)(121 196 188)(122 193 185)(123 194 186)(124 195 187)(125 197 141)(126 198 142)(127 199 143)(128 200 144)(149 170 191)(150 171 192)(151 172 189)(152 169 190)(157 179 213)(158 180 214)(159 177 215)(160 178 216)
(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)
(1 119 3 117)(2 118 4 120)(5 196 7 194)(6 195 8 193)(9 200 11 198)(10 199 12 197)(13 204 15 202)(14 203 16 201)(17 208 19 206)(18 207 20 205)(21 212 23 210)(22 211 24 209)(25 168 27 166)(26 167 28 165)(29 114 31 116)(30 113 32 115)(33 176 35 174)(34 175 36 173)(37 180 39 178)(38 179 40 177)(41 184 43 182)(42 183 44 181)(45 188 47 186)(46 187 48 185)(49 192 51 190)(50 191 52 189)(53 140 55 138)(54 139 56 137)(57 143 59 141)(58 142 60 144)(61 148 63 146)(62 147 64 145)(65 152 67 150)(66 151 68 149)(69 156 71 154)(70 155 72 153)(73 160 75 158)(74 159 76 157)(77 164 79 162)(78 163 80 161)(81 112 83 110)(82 111 84 109)(85 171 87 169)(86 170 88 172)(89 124 91 122)(90 123 92 121)(93 128 95 126)(94 127 96 125)(97 132 99 130)(98 131 100 129)(101 136 103 134)(102 135 104 133)(105 213 107 215)(106 216 108 214)
(1 4)(2 3)(5 98)(6 97)(7 100)(8 99)(9 66)(10 65)(11 68)(12 67)(13 102)(14 101)(15 104)(16 103)(17 139)(18 138)(19 137)(20 140)(21 105)(22 108)(23 107)(24 106)(25 185)(26 188)(27 187)(28 186)(29 61)(30 64)(31 63)(32 62)(33 90)(34 89)(35 92)(36 91)(37 69)(38 72)(39 71)(40 70)(41 77)(42 80)(43 79)(44 78)(45 165)(46 168)(47 167)(48 166)(49 59)(50 58)(51 57)(52 60)(53 207)(54 206)(55 205)(56 208)(73 110)(74 109)(75 112)(76 111)(81 160)(82 159)(83 158)(84 157)(85 96)(86 95)(87 94)(88 93)(113 145)(114 148)(115 147)(116 146)(117 118)(119 120)(121 174)(122 173)(123 176)(124 175)(125 171)(126 170)(127 169)(128 172)(129 194)(130 193)(131 196)(132 195)(133 202)(134 201)(135 204)(136 203)(141 192)(142 191)(143 190)(144 189)(149 198)(150 197)(151 200)(152 199)(153 179)(154 178)(155 177)(156 180)(161 183)(162 182)(163 181)(164 184)(209 216)(210 215)(211 214)(212 213)

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

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

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

75 conjugacy classes

 class 1 2A 2B 2C 2D 3A ··· 3M 4A 4B 4C 4D 4E 6A ··· 6M 12A ··· 12AM order 1 2 2 2 2 3 ··· 3 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 54 54 54 2 ··· 2 2 2 2 27 27 2 ··· 2 4 ··· 4

75 irreducible representations

 dim 1 1 1 1 2 2 2 4 type + + + + + + + image C1 C2 C2 C2 S3 D6 C4○D4 Q8⋊3S3 kernel (Q8×C33)⋊C2 C4×C33⋊C2 C33⋊12D4 Q8×C33 Q8×C32 C3×C12 C33 C32 # reps 1 3 3 1 13 39 2 13

Matrix representation of (Q8×C33)⋊C2 in GL8(𝔽13)

 0 1 0 0 0 0 0 0 12 12 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 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 1 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 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12
,
 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 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 8 0 0 0 0 0 0 3 12 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 12 0 0 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 8 12 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 1 8 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 1

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

(Q8×C33)⋊C2 in GAP, Magma, Sage, TeX

(Q_8\times C_3^3)\rtimes C_2
% in TeX

G:=Group("(Q8xC3^3):C2");
// GroupNames label

G:=SmallGroup(432,727);
// by ID

G=gap.SmallGroup(432,727);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,58,1124,4037,14118]);
// 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,f*a*f=a^-1,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=b^-1,c*d=d*c,c*e=e*c,f*c*f=c^-1,e*d*e^-1=f*d*f=d^-1,e*f=f*e>;
// generators/relations

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