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G = Q8×C33⋊C2  order 432 = 24·33

Direct product of Q8 and C33⋊C2

direct product, metabelian, supersoluble, monomial, rational

Aliases: Q8×C33⋊C2, C33⋊21(C2×Q8), (Q8×C33)⋊8C2, C32⋊16(S3×Q8), (C3×C12).153D6, (Q8×C32)⋊16S3, C33⋊8Q8⋊10C2, (C32×C12).58C22, (C32×C6).101C23, C33⋊5C4.22C22, C3⋊3(Q8×C3⋊S3), C12.28(C2×C3⋊S3), (C3×Q8)⋊4(C3⋊S3), C6.45(C22×C3⋊S3), C4.6(C2×C33⋊C2), (C4×C33⋊C2).2C2, (C3×C6).190(C22×S3), C2.8(C22×C33⋊C2), (C2×C33⋊C2).22C22, SmallGroup(432,726)

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 | a4=c3=d3=e3=f2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf=c-1, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 2560 in 532 conjugacy classes, 181 normal (8 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C2×C4, Q8, Q8, C32, Dic3, C12, D6, C2×Q8, C3⋊S3, C3×C6, Dic6, C4×S3, C3×Q8, C33, C3⋊Dic3, C3×C12, C2×C3⋊S3, S3×Q8, C33⋊C2, C32×C6, C32⋊4Q8, C4×C3⋊S3, Q8×C32, C33⋊5C4, C32×C12, C2×C33⋊C2, Q8×C3⋊S3, C33⋊8Q8, C4×C33⋊C2, Q8×C33, Q8×C33⋊C2
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C3⋊S3, C22×S3, C2×C3⋊S3, S3×Q8, C33⋊C2, C22×C3⋊S3, C2×C33⋊C2, Q8×C3⋊S3, C22×C33⋊C2, Q8×C33⋊C2

Smallest permutation representation of Q8×C33⋊C2
►On 216 points
Generators in S216
(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)

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

(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)

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

75 conjugacy classes

class 1 2A2B2C3A···3M4A4B4C4D4E4F6A···6M12A···12AM
order12223···34444446···612···12
size1127272···22225454542···24···4

75 irreducible representations

dim11112224
type+++++-+-
imageC1C2C2C2S3Q8D6S3×Q8
kernelQ8×C33⋊C2C33⋊8Q8C4×C33⋊C2Q8×C33Q8×C32C33⋊C2C3×C12C32
# reps13311323913

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

120000000
012000000
00100000
00010000
000012000
000001200
000000123
00000081
,
10000000
01000000
00100000
00010000
00001000
00000100
00000011
0000001112
,
113000000
121000000
00100000
00010000
000012100
000012000
00000010
00000001
,
113000000
121000000
001130000
001210000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000012100
000012000
00000010
00000001
,
110000000
012000000
001230000
00010000
00000100
00001000
00000010
00000001

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,8,0,0,0,0,0,0,3,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,11,0,0,0,0,0,0,1,12],[11,12,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[11,12,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,11,12,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,10,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

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(432,726);
 
// by ID
 
G=gap.SmallGroup(432,726);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,135,58,1124,4037,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^4=c^3=d^3=e^3=f^2=1,b^2=a^2,b*a*b^-1=a^-1,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,f*c*f=c^-1,d*e=e*d,f*d*f=d^-1,f*e*f=e^-1>;
 
// generators/relations
 

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