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G = C2×Q82Dic3order 192 = 26·3

Direct product of C2 and Q82Dic3

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×Q82Dic3, (C6×Q8)⋊5C4, Q84(C2×Dic3), (C2×Q8)⋊5Dic3, (C2×C6).18Q16, C6.46(C2×Q16), C63(Q8⋊C4), (C2×C12).192D4, C12.207(C2×D4), (C2×Q8).189D6, (C2×C6).42SD16, C6.79(C2×SD16), (C22×Q8).7S3, C12.81(C22×C4), (C22×C6).198D4, (C22×C4).370D6, C12.33(C22⋊C4), (C2×C12).474C23, (C6×Q8).200C22, C4.11(C22×Dic3), C22.9(C3⋊Q16), C4.10(C6.D4), C23.108(C3⋊D4), C4⋊Dic3.352C22, (C22×C12).200C22, C22.12(Q82S3), C22.35(C6.D4), (Q8×C2×C6).1C2, C34(C2×Q8⋊C4), (C3×Q8)⋊15(C2×C4), C4.91(C2×C3⋊D4), C2.4(C2×C3⋊Q16), (C2×C6).558(C2×D4), C6.77(C2×C22⋊C4), (C22×C3⋊C8).13C2, C2.4(C2×Q82S3), (C2×C12).119(C2×C4), (C2×C3⋊C8).281C22, (C2×C4⋊Dic3).40C2, (C2×C4).51(C2×Dic3), C22.93(C2×C3⋊D4), (C2×C4).149(C3⋊D4), (C2×C4).560(C22×S3), C2.13(C2×C6.D4), (C2×C6).114(C22⋊C4), SmallGroup(192,783)

Series: Derived Chief Lower central Upper central

C1C12 — C2×Q82Dic3
C1C3C6C2×C6C2×C12C4⋊Dic3C2×C4⋊Dic3 — C2×Q82Dic3
C3C6C12 — C2×Q82Dic3
C1C23C22×C4C22×Q8

Generators and relations for C2×Q82Dic3
 G = < a,b,c,d,e | a2=b4=d6=1, c2=b2, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=d-1 >

Subgroups: 328 in 162 conjugacy classes, 87 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×2], C4 [×6], C22, C22 [×6], C6 [×3], C6 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], Q8 [×4], Q8 [×6], C23, Dic3 [×2], C12 [×2], C12 [×2], C12 [×4], C2×C6, C2×C6 [×6], C4⋊C4 [×3], C2×C8 [×4], C22×C4, C22×C4 [×2], C2×Q8 [×6], C2×Q8 [×3], C3⋊C8 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×6], C3×Q8 [×4], C3×Q8 [×6], C22×C6, Q8⋊C4 [×4], C2×C4⋊C4, C22×C8, C22×Q8, C2×C3⋊C8 [×2], C2×C3⋊C8 [×2], C4⋊Dic3 [×2], C4⋊Dic3, C22×Dic3, C22×C12, C22×C12, C6×Q8 [×6], C6×Q8 [×3], C2×Q8⋊C4, Q82Dic3 [×4], C22×C3⋊C8, C2×C4⋊Dic3, Q8×C2×C6, C2×Q82Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, Dic3 [×4], D6 [×3], C22⋊C4 [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], C2×Dic3 [×6], C3⋊D4 [×4], C22×S3, Q8⋊C4 [×4], C2×C22⋊C4, C2×SD16, C2×Q16, Q82S3 [×2], C3⋊Q16 [×2], C6.D4 [×4], C22×Dic3, C2×C3⋊D4 [×2], C2×Q8⋊C4, Q82Dic3 [×4], C2×Q82S3, C2×C3⋊Q16, C2×C6.D4, C2×Q82Dic3

Smallest permutation representation of C2×Q82Dic3
Regular action on 192 points
Generators in S192
(1 56)(2 57)(3 58)(4 59)(5 60)(6 55)(7 65)(8 66)(9 61)(10 62)(11 63)(12 64)(13 145)(14 146)(15 147)(16 148)(17 149)(18 150)(19 70)(20 71)(21 72)(22 67)(23 68)(24 69)(25 76)(26 77)(27 78)(28 73)(29 74)(30 75)(31 82)(32 83)(33 84)(34 79)(35 80)(36 81)(37 88)(38 89)(39 90)(40 85)(41 86)(42 87)(43 94)(44 95)(45 96)(46 91)(47 92)(48 93)(49 101)(50 102)(51 97)(52 98)(53 99)(54 100)(103 154)(104 155)(105 156)(106 151)(107 152)(108 153)(109 160)(110 161)(111 162)(112 157)(113 158)(114 159)(115 166)(116 167)(117 168)(118 163)(119 164)(120 165)(121 172)(122 173)(123 174)(124 169)(125 170)(126 171)(127 178)(128 179)(129 180)(130 175)(131 176)(132 177)(133 184)(134 185)(135 186)(136 181)(137 182)(138 183)(139 190)(140 191)(141 192)(142 187)(143 188)(144 189)
(1 29 19 9)(2 30 20 10)(3 25 21 11)(4 26 22 12)(5 27 23 7)(6 28 24 8)(13 191 182 178)(14 192 183 179)(15 187 184 180)(16 188 185 175)(17 189 186 176)(18 190 181 177)(31 42 45 49)(32 37 46 50)(33 38 47 51)(34 39 48 52)(35 40 43 53)(36 41 44 54)(55 73 69 66)(56 74 70 61)(57 75 71 62)(58 76 72 63)(59 77 67 64)(60 78 68 65)(79 90 93 98)(80 85 94 99)(81 86 95 100)(82 87 96 101)(83 88 91 102)(84 89 92 97)(103 113 116 121)(104 114 117 122)(105 109 118 123)(106 110 119 124)(107 111 120 125)(108 112 115 126)(127 145 140 137)(128 146 141 138)(129 147 142 133)(130 148 143 134)(131 149 144 135)(132 150 139 136)(151 161 164 169)(152 162 165 170)(153 157 166 171)(154 158 167 172)(155 159 168 173)(156 160 163 174)
(1 46 19 32)(2 47 20 33)(3 48 21 34)(4 43 22 35)(5 44 23 36)(6 45 24 31)(7 54 27 41)(8 49 28 42)(9 50 29 37)(10 51 30 38)(11 52 25 39)(12 53 26 40)(13 164 182 151)(14 165 183 152)(15 166 184 153)(16 167 185 154)(17 168 186 155)(18 163 181 156)(55 96 69 82)(56 91 70 83)(57 92 71 84)(58 93 72 79)(59 94 67 80)(60 95 68 81)(61 102 74 88)(62 97 75 89)(63 98 76 90)(64 99 77 85)(65 100 78 86)(66 101 73 87)(103 148 116 134)(104 149 117 135)(105 150 118 136)(106 145 119 137)(107 146 120 138)(108 147 115 133)(109 132 123 139)(110 127 124 140)(111 128 125 141)(112 129 126 142)(113 130 121 143)(114 131 122 144)(157 180 171 187)(158 175 172 188)(159 176 173 189)(160 177 174 190)(161 178 169 191)(162 179 170 192)
(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)
(1 151 4 154)(2 156 5 153)(3 155 6 152)(7 157 10 160)(8 162 11 159)(9 161 12 158)(13 53 16 50)(14 52 17 49)(15 51 18 54)(19 164 22 167)(20 163 23 166)(21 168 24 165)(25 173 28 170)(26 172 29 169)(27 171 30 174)(31 179 34 176)(32 178 35 175)(33 177 36 180)(37 182 40 185)(38 181 41 184)(39 186 42 183)(43 188 46 191)(44 187 47 190)(45 192 48 189)(55 107 58 104)(56 106 59 103)(57 105 60 108)(61 110 64 113)(62 109 65 112)(63 114 66 111)(67 116 70 119)(68 115 71 118)(69 120 72 117)(73 125 76 122)(74 124 77 121)(75 123 78 126)(79 131 82 128)(80 130 83 127)(81 129 84 132)(85 134 88 137)(86 133 89 136)(87 138 90 135)(91 140 94 143)(92 139 95 142)(93 144 96 141)(97 150 100 147)(98 149 101 146)(99 148 102 145)

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

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

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

48 conjugacy classes

class 1 2A···2G 3 4A4B4C4D4E4F4G4H4I4J4K4L6A···6G8A···8H12A···12L
order12···234444444444446···68···812···12
size11···1222224444121212122···26···64···4

48 irreducible representations

dim111111222222222244
type+++++++++-+-+-
imageC1C2C2C2C2C4S3D4D4D6Dic3D6SD16Q16C3⋊D4C3⋊D4Q82S3C3⋊Q16
kernelC2×Q82Dic3Q82Dic3C22×C3⋊C8C2×C4⋊Dic3Q8×C2×C6C6×Q8C22×Q8C2×C12C22×C6C22×C4C2×Q8C2×Q8C2×C6C2×C6C2×C4C23C22C22
# reps141118131142446222

Matrix representation of C2×Q82Dic3 in GL6(𝔽73)

7200000
0720000
001000
000100
000010
000001
,
100000
010000
001000
000100
0000171
0000172
,
100000
0720000
0072000
0007200
00004035
00002133
,
7200000
010000
0007200
0017200
0000720
0000072
,
2700000
0720000
0032800
00317000
00002464
00005649

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,71,72],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,40,21,0,0,0,0,35,33],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[27,0,0,0,0,0,0,72,0,0,0,0,0,0,3,31,0,0,0,0,28,70,0,0,0,0,0,0,24,56,0,0,0,0,64,49] >;

C2×Q82Dic3 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes_2{\rm Dic}_3
% in TeX

G:=Group("C2xQ8:2Dic3");
// GroupNames label

G:=SmallGroup(192,783);
// by ID

G=gap.SmallGroup(192,783);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,184,1684,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^6=1,c^2=b^2,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽