direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C22.4Q16, C12.28C42, C4⋊C4⋊4C12, (C2×C24)⋊8C4, (C2×C8)⋊4C12, C4.2(C4×C12), (C2×C6).48D8, C12.52(C4⋊C4), (C22×C8).4C6, (C2×C6).19Q16, (C2×C12).70Q8, C6.9(C4.Q8), C22.7(C3×D8), (C2×C12).507D4, (C22×C24).5C2, (C2×C6).43SD16, C6.12(C2.D8), C23.57(C3×D4), C22.4(C3×Q16), (C22×C6).213D4, C6.36(D4⋊C4), C22.9(C3×SD16), C6.18(Q8⋊C4), C12.103(C22⋊C4), C6.24(C2.C42), (C22×C12).571C22, C4.3(C3×C4⋊C4), (C3×C4⋊C4)⋊11C4, (C2×C4⋊C4).2C6, (C6×C4⋊C4).29C2, C2.2(C3×C4.Q8), C2.2(C3×C2.D8), (C2×C4).13(C3×Q8), (C2×C6).59(C4⋊C4), (C2×C4).40(C2×C12), C2.2(C3×D4⋊C4), (C2×C4).112(C3×D4), C22.16(C3×C4⋊C4), C4.19(C3×C22⋊C4), C2.2(C3×Q8⋊C4), (C2×C12).261(C2×C4), (C22×C4).111(C2×C6), C22.29(C3×C22⋊C4), C2.5(C3×C2.C42), (C2×C6).130(C22⋊C4), SmallGroup(192,146)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C22.4Q16
G = < a,b,c,d,e | a3=b2=c2=d8=1, e2=cd4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >
Subgroups: 186 in 114 conjugacy classes, 74 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C12, C2×C6, C2×C6, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C24, C2×C12, C2×C12, C2×C12, C22×C6, C2×C4⋊C4, C22×C8, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C2×C24, C22×C12, C22×C12, C22.4Q16, C6×C4⋊C4, C22×C24, C3×C22.4Q16
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C12, C2×C6, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, C2×C12, C3×D4, C3×Q8, C2.C42, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×D8, C3×SD16, C3×Q16, C22.4Q16, C3×C2.C42, C3×D4⋊C4, C3×Q8⋊C4, C3×C4.Q8, C3×C2.D8, C3×C22.4Q16
►Regular action on 192 points
Generators in S192
(1 63 71)(2 64 72)(3 57 65)(4 58 66)(5 59 67)(6 60 68)(7 61 69)(8 62 70)(9 26 18)(10 27 19)(11 28 20)(12 29 21)(13 30 22)(14 31 23)(15 32 24)(16 25 17)(33 112 41)(34 105 42)(35 106 43)(36 107 44)(37 108 45)(38 109 46)(39 110 47)(40 111 48)(49 133 140)(50 134 141)(51 135 142)(52 136 143)(53 129 144)(54 130 137)(55 131 138)(56 132 139)(73 89 81)(74 90 82)(75 91 83)(76 92 84)(77 93 85)(78 94 86)(79 95 87)(80 96 88)(97 157 145)(98 158 146)(99 159 147)(100 160 148)(101 153 149)(102 154 150)(103 155 151)(104 156 152)(113 128 164)(114 121 165)(115 122 166)(116 123 167)(117 124 168)(118 125 161)(119 126 162)(120 127 163)(169 185 177)(170 186 178)(171 187 179)(172 188 180)(173 189 181)(174 190 182)(175 191 183)(176 192 184)
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 60)(10 61)(11 62)(12 63)(13 64)(14 57)(15 58)(16 59)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 65)(32 66)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 73)(40 74)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 81)(48 82)(49 187)(50 188)(51 189)(52 190)(53 191)(54 192)(55 185)(56 186)(89 110)(90 111)(91 112)(92 105)(93 106)(94 107)(95 108)(96 109)(97 163)(98 164)(99 165)(100 166)(101 167)(102 168)(103 161)(104 162)(113 158)(114 159)(115 160)(116 153)(117 154)(118 155)(119 156)(120 157)(121 147)(122 148)(123 149)(124 150)(125 151)(126 152)(127 145)(128 146)(129 183)(130 184)(131 177)(132 178)(133 179)(134 180)(135 181)(136 182)(137 176)(138 169)(139 170)(140 171)(141 172)(142 173)(143 174)(144 175)
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 73)(8 74)(9 109)(10 110)(11 111)(12 112)(13 105)(14 106)(15 107)(16 108)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)(49 168)(50 161)(51 162)(52 163)(53 164)(54 165)(55 166)(56 167)(57 93)(58 94)(59 95)(60 96)(61 89)(62 90)(63 91)(64 92)(65 85)(66 86)(67 87)(68 88)(69 81)(70 82)(71 83)(72 84)(97 190)(98 191)(99 192)(100 185)(101 186)(102 187)(103 188)(104 189)(113 129)(114 130)(115 131)(116 132)(117 133)(118 134)(119 135)(120 136)(121 137)(122 138)(123 139)(124 140)(125 141)(126 142)(127 143)(128 144)(145 174)(146 175)(147 176)(148 169)(149 170)(150 171)(151 172)(152 173)(153 178)(154 179)(155 180)(156 181)(157 182)(158 183)(159 184)(160 177)
(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)
(1 138 79 126)(2 176 80 151)(3 144 73 124)(4 174 74 149)(5 142 75 122)(6 172 76 147)(7 140 77 128)(8 170 78 145)(9 50 105 165)(10 187 106 98)(11 56 107 163)(12 185 108 104)(13 54 109 161)(14 191 110 102)(15 52 111 167)(16 189 112 100)(17 173 33 148)(18 141 34 121)(19 171 35 146)(20 139 36 127)(21 169 37 152)(22 137 38 125)(23 175 39 150)(24 143 40 123)(25 181 41 160)(26 134 42 114)(27 179 43 158)(28 132 44 120)(29 177 45 156)(30 130 46 118)(31 183 47 154)(32 136 48 116)(49 93 164 61)(51 91 166 59)(53 89 168 57)(55 95 162 63)(58 190 90 101)(60 188 92 99)(62 186 94 97)(64 192 96 103)(65 129 81 117)(66 182 82 153)(67 135 83 115)(68 180 84 159)(69 133 85 113)(70 178 86 157)(71 131 87 119)(72 184 88 155)
G:=sub<Sym(192)| (1,63,71)(2,64,72)(3,57,65)(4,58,66)(5,59,67)(6,60,68)(7,61,69)(8,62,70)(9,26,18)(10,27,19)(11,28,20)(12,29,21)(13,30,22)(14,31,23)(15,32,24)(16,25,17)(33,112,41)(34,105,42)(35,106,43)(36,107,44)(37,108,45)(38,109,46)(39,110,47)(40,111,48)(49,133,140)(50,134,141)(51,135,142)(52,136,143)(53,129,144)(54,130,137)(55,131,138)(56,132,139)(73,89,81)(74,90,82)(75,91,83)(76,92,84)(77,93,85)(78,94,86)(79,95,87)(80,96,88)(97,157,145)(98,158,146)(99,159,147)(100,160,148)(101,153,149)(102,154,150)(103,155,151)(104,156,152)(113,128,164)(114,121,165)(115,122,166)(116,123,167)(117,124,168)(118,125,161)(119,126,162)(120,127,163)(169,185,177)(170,186,178)(171,187,179)(172,188,180)(173,189,181)(174,190,182)(175,191,183)(176,192,184), (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,65)(32,66)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,73)(40,74)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,81)(48,82)(49,187)(50,188)(51,189)(52,190)(53,191)(54,192)(55,185)(56,186)(89,110)(90,111)(91,112)(92,105)(93,106)(94,107)(95,108)(96,109)(97,163)(98,164)(99,165)(100,166)(101,167)(102,168)(103,161)(104,162)(113,158)(114,159)(115,160)(116,153)(117,154)(118,155)(119,156)(120,157)(121,147)(122,148)(123,149)(124,150)(125,151)(126,152)(127,145)(128,146)(129,183)(130,184)(131,177)(132,178)(133,179)(134,180)(135,181)(136,182)(137,176)(138,169)(139,170)(140,171)(141,172)(142,173)(143,174)(144,175), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,73)(8,74)(9,109)(10,110)(11,111)(12,112)(13,105)(14,106)(15,107)(16,108)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44)(49,168)(50,161)(51,162)(52,163)(53,164)(54,165)(55,166)(56,167)(57,93)(58,94)(59,95)(60,96)(61,89)(62,90)(63,91)(64,92)(65,85)(66,86)(67,87)(68,88)(69,81)(70,82)(71,83)(72,84)(97,190)(98,191)(99,192)(100,185)(101,186)(102,187)(103,188)(104,189)(113,129)(114,130)(115,131)(116,132)(117,133)(118,134)(119,135)(120,136)(121,137)(122,138)(123,139)(124,140)(125,141)(126,142)(127,143)(128,144)(145,174)(146,175)(147,176)(148,169)(149,170)(150,171)(151,172)(152,173)(153,178)(154,179)(155,180)(156,181)(157,182)(158,183)(159,184)(160,177), (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), (1,138,79,126)(2,176,80,151)(3,144,73,124)(4,174,74,149)(5,142,75,122)(6,172,76,147)(7,140,77,128)(8,170,78,145)(9,50,105,165)(10,187,106,98)(11,56,107,163)(12,185,108,104)(13,54,109,161)(14,191,110,102)(15,52,111,167)(16,189,112,100)(17,173,33,148)(18,141,34,121)(19,171,35,146)(20,139,36,127)(21,169,37,152)(22,137,38,125)(23,175,39,150)(24,143,40,123)(25,181,41,160)(26,134,42,114)(27,179,43,158)(28,132,44,120)(29,177,45,156)(30,130,46,118)(31,183,47,154)(32,136,48,116)(49,93,164,61)(51,91,166,59)(53,89,168,57)(55,95,162,63)(58,190,90,101)(60,188,92,99)(62,186,94,97)(64,192,96,103)(65,129,81,117)(66,182,82,153)(67,135,83,115)(68,180,84,159)(69,133,85,113)(70,178,86,157)(71,131,87,119)(72,184,88,155)>;
G:=Group( (1,63,71)(2,64,72)(3,57,65)(4,58,66)(5,59,67)(6,60,68)(7,61,69)(8,62,70)(9,26,18)(10,27,19)(11,28,20)(12,29,21)(13,30,22)(14,31,23)(15,32,24)(16,25,17)(33,112,41)(34,105,42)(35,106,43)(36,107,44)(37,108,45)(38,109,46)(39,110,47)(40,111,48)(49,133,140)(50,134,141)(51,135,142)(52,136,143)(53,129,144)(54,130,137)(55,131,138)(56,132,139)(73,89,81)(74,90,82)(75,91,83)(76,92,84)(77,93,85)(78,94,86)(79,95,87)(80,96,88)(97,157,145)(98,158,146)(99,159,147)(100,160,148)(101,153,149)(102,154,150)(103,155,151)(104,156,152)(113,128,164)(114,121,165)(115,122,166)(116,123,167)(117,124,168)(118,125,161)(119,126,162)(120,127,163)(169,185,177)(170,186,178)(171,187,179)(172,188,180)(173,189,181)(174,190,182)(175,191,183)(176,192,184), (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,65)(32,66)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,73)(40,74)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,81)(48,82)(49,187)(50,188)(51,189)(52,190)(53,191)(54,192)(55,185)(56,186)(89,110)(90,111)(91,112)(92,105)(93,106)(94,107)(95,108)(96,109)(97,163)(98,164)(99,165)(100,166)(101,167)(102,168)(103,161)(104,162)(113,158)(114,159)(115,160)(116,153)(117,154)(118,155)(119,156)(120,157)(121,147)(122,148)(123,149)(124,150)(125,151)(126,152)(127,145)(128,146)(129,183)(130,184)(131,177)(132,178)(133,179)(134,180)(135,181)(136,182)(137,176)(138,169)(139,170)(140,171)(141,172)(142,173)(143,174)(144,175), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,73)(8,74)(9,109)(10,110)(11,111)(12,112)(13,105)(14,106)(15,107)(16,108)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44)(49,168)(50,161)(51,162)(52,163)(53,164)(54,165)(55,166)(56,167)(57,93)(58,94)(59,95)(60,96)(61,89)(62,90)(63,91)(64,92)(65,85)(66,86)(67,87)(68,88)(69,81)(70,82)(71,83)(72,84)(97,190)(98,191)(99,192)(100,185)(101,186)(102,187)(103,188)(104,189)(113,129)(114,130)(115,131)(116,132)(117,133)(118,134)(119,135)(120,136)(121,137)(122,138)(123,139)(124,140)(125,141)(126,142)(127,143)(128,144)(145,174)(146,175)(147,176)(148,169)(149,170)(150,171)(151,172)(152,173)(153,178)(154,179)(155,180)(156,181)(157,182)(158,183)(159,184)(160,177), (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), (1,138,79,126)(2,176,80,151)(3,144,73,124)(4,174,74,149)(5,142,75,122)(6,172,76,147)(7,140,77,128)(8,170,78,145)(9,50,105,165)(10,187,106,98)(11,56,107,163)(12,185,108,104)(13,54,109,161)(14,191,110,102)(15,52,111,167)(16,189,112,100)(17,173,33,148)(18,141,34,121)(19,171,35,146)(20,139,36,127)(21,169,37,152)(22,137,38,125)(23,175,39,150)(24,143,40,123)(25,181,41,160)(26,134,42,114)(27,179,43,158)(28,132,44,120)(29,177,45,156)(30,130,46,118)(31,183,47,154)(32,136,48,116)(49,93,164,61)(51,91,166,59)(53,89,168,57)(55,95,162,63)(58,190,90,101)(60,188,92,99)(62,186,94,97)(64,192,96,103)(65,129,81,117)(66,182,82,153)(67,135,83,115)(68,180,84,159)(69,133,85,113)(70,178,86,157)(71,131,87,119)(72,184,88,155) );
G=PermutationGroup([[(1,63,71),(2,64,72),(3,57,65),(4,58,66),(5,59,67),(6,60,68),(7,61,69),(8,62,70),(9,26,18),(10,27,19),(11,28,20),(12,29,21),(13,30,22),(14,31,23),(15,32,24),(16,25,17),(33,112,41),(34,105,42),(35,106,43),(36,107,44),(37,108,45),(38,109,46),(39,110,47),(40,111,48),(49,133,140),(50,134,141),(51,135,142),(52,136,143),(53,129,144),(54,130,137),(55,131,138),(56,132,139),(73,89,81),(74,90,82),(75,91,83),(76,92,84),(77,93,85),(78,94,86),(79,95,87),(80,96,88),(97,157,145),(98,158,146),(99,159,147),(100,160,148),(101,153,149),(102,154,150),(103,155,151),(104,156,152),(113,128,164),(114,121,165),(115,122,166),(116,123,167),(117,124,168),(118,125,161),(119,126,162),(120,127,163),(169,185,177),(170,186,178),(171,187,179),(172,188,180),(173,189,181),(174,190,182),(175,191,183),(176,192,184)], [(1,21),(2,22),(3,23),(4,24),(5,17),(6,18),(7,19),(8,20),(9,60),(10,61),(11,62),(12,63),(13,64),(14,57),(15,58),(16,59),(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,65),(32,66),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,73),(40,74),(41,83),(42,84),(43,85),(44,86),(45,87),(46,88),(47,81),(48,82),(49,187),(50,188),(51,189),(52,190),(53,191),(54,192),(55,185),(56,186),(89,110),(90,111),(91,112),(92,105),(93,106),(94,107),(95,108),(96,109),(97,163),(98,164),(99,165),(100,166),(101,167),(102,168),(103,161),(104,162),(113,158),(114,159),(115,160),(116,153),(117,154),(118,155),(119,156),(120,157),(121,147),(122,148),(123,149),(124,150),(125,151),(126,152),(127,145),(128,146),(129,183),(130,184),(131,177),(132,178),(133,179),(134,180),(135,181),(136,182),(137,176),(138,169),(139,170),(140,171),(141,172),(142,173),(143,174),(144,175)], [(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,73),(8,74),(9,109),(10,110),(11,111),(12,112),(13,105),(14,106),(15,107),(16,108),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,36),(25,45),(26,46),(27,47),(28,48),(29,41),(30,42),(31,43),(32,44),(49,168),(50,161),(51,162),(52,163),(53,164),(54,165),(55,166),(56,167),(57,93),(58,94),(59,95),(60,96),(61,89),(62,90),(63,91),(64,92),(65,85),(66,86),(67,87),(68,88),(69,81),(70,82),(71,83),(72,84),(97,190),(98,191),(99,192),(100,185),(101,186),(102,187),(103,188),(104,189),(113,129),(114,130),(115,131),(116,132),(117,133),(118,134),(119,135),(120,136),(121,137),(122,138),(123,139),(124,140),(125,141),(126,142),(127,143),(128,144),(145,174),(146,175),(147,176),(148,169),(149,170),(150,171),(151,172),(152,173),(153,178),(154,179),(155,180),(156,181),(157,182),(158,183),(159,184),(160,177)], [(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)], [(1,138,79,126),(2,176,80,151),(3,144,73,124),(4,174,74,149),(5,142,75,122),(6,172,76,147),(7,140,77,128),(8,170,78,145),(9,50,105,165),(10,187,106,98),(11,56,107,163),(12,185,108,104),(13,54,109,161),(14,191,110,102),(15,52,111,167),(16,189,112,100),(17,173,33,148),(18,141,34,121),(19,171,35,146),(20,139,36,127),(21,169,37,152),(22,137,38,125),(23,175,39,150),(24,143,40,123),(25,181,41,160),(26,134,42,114),(27,179,43,158),(28,132,44,120),(29,177,45,156),(30,130,46,118),(31,183,47,154),(32,136,48,116),(49,93,164,61),(51,91,166,59),(53,89,168,57),(55,95,162,63),(58,190,90,101),(60,188,92,99),(62,186,94,97),(64,192,96,103),(65,129,81,117),(66,182,82,153),(67,135,83,115),(68,180,84,159),(69,133,85,113),(70,178,86,157),(71,131,87,119),(72,184,88,155)]])
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6N | 8A | ··· | 8H | 12A | ··· | 12H | 12I | ··· | 12X | 24A | ··· | 24P |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | D4 | Q8 | D4 | D8 | SD16 | Q16 | C3×D4 | C3×Q8 | C3×D4 | C3×D8 | C3×SD16 | C3×Q16 |
| kernel | C3×C22.4Q16 | C6×C4⋊C4 | C22×C24 | C22.4Q16 | C3×C4⋊C4 | C2×C24 | C2×C4⋊C4 | C22×C8 | C4⋊C4 | C2×C8 | C2×C12 | C2×C12 | C22×C6 | C2×C6 | C2×C6 | C2×C6 | C2×C4 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 8 | 4 | 4 | 2 | 16 | 8 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 2 | 4 | 8 | 4 |
Matrix representation of C3×C22.4Q16 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 27 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 6 | 67 |
| 0 | 0 | 6 | 6 |
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 13 | 66 |
| 0 | 0 | 66 | 60 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,8,0,0,0,0,64,0,0,0,0,64],[72,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[27,0,0,0,0,72,0,0,0,0,6,6,0,0,67,6],[27,0,0,0,0,27,0,0,0,0,13,66,0,0,66,60] >;
C3×C22.4Q16 in GAP, Magma, Sage, TeX
C_3\times C_2^2._4Q_{16} % in TeX
G:=Group("C3xC2^2.4Q16"); // GroupNames label
G:=SmallGroup(192,146);
// by ID
G=gap.SmallGroup(192,146);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,344,3027,248,6053,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^2=d^8=1,e^2=c*d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^-1>;
// generators/relations