direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C22.7C42, (C2×C24)⋊7C4, (C2×C8)⋊3C12, (C2×C4)⋊2C24, (C2×C12)⋊4C8, C2.2(C4×C24), C6.11(C4×C8), C6.13(C4⋊C8), (C22×C8).3C6, C12.66(C4⋊C4), (C2×C42).4C6, (C2×C12).81Q8, C6.8(C8⋊C4), (C2×C12).532D4, (C2×C6).29C42, C22.7(C2×C24), C22.7(C4×C12), (C22×C24).4C2, C6.24(C22⋊C8), C23.40(C2×C12), (C22×C4).10C12, (C22×C12).14C4, (C2×C6).28M4(2), C12.109(C22⋊C4), C22.7(C3×M4(2)), C6.20(C2.C42), (C22×C12).606C22, C2.1(C3×C4⋊C8), (C2×C4×C12).1C2, C4.17(C3×C4⋊C4), C2.2(C3×C8⋊C4), (C2×C6).38(C2×C8), C2.1(C3×C22⋊C8), (C2×C4).23(C3×Q8), (C2×C6).58(C4⋊C4), (C2×C4).81(C2×C12), (C2×C4).141(C3×D4), C22.15(C3×C4⋊C4), C4.24(C3×C22⋊C4), (C2×C12).343(C2×C4), (C22×C4).140(C2×C6), (C22×C6).144(C2×C4), C22.27(C3×C22⋊C4), C2.1(C3×C2.C42), (C2×C6).129(C22⋊C4), SmallGroup(192,142)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C22.7C42
G = < a,b,c,d,e | a3=b2=c2=e4=1, d4=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, ede-1=bd=db, be=eb, cd=dc, ce=ec >
Subgroups: 154 in 118 conjugacy classes, 82 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C42, C2×C8, C2×C8, C22×C4, C22×C4, C24, C2×C12, C2×C12, C2×C12, C22×C6, C2×C42, C22×C8, C4×C12, C2×C24, C2×C24, C22×C12, C22×C12, C22.7C42, C2×C4×C12, C22×C24, C3×C22.7C42
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, D4, Q8, C12, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C24, C2×C12, C3×D4, C3×Q8, C2.C42, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C3×M4(2), C22.7C42, C3×C2.C42, C4×C24, C3×C8⋊C4, C3×C22⋊C8, C3×C4⋊C8, C3×C22.7C42
►Regular action on 192 points
Generators in S192
(1 39 31)(2 40 32)(3 33 25)(4 34 26)(5 35 27)(6 36 28)(7 37 29)(8 38 30)(9 53 45)(10 54 46)(11 55 47)(12 56 48)(13 49 41)(14 50 42)(15 51 43)(16 52 44)(17 137 112)(18 138 105)(19 139 106)(20 140 107)(21 141 108)(22 142 109)(23 143 110)(24 144 111)(57 73 65)(58 74 66)(59 75 67)(60 76 68)(61 77 69)(62 78 70)(63 79 71)(64 80 72)(81 103 89)(82 104 90)(83 97 91)(84 98 92)(85 99 93)(86 100 94)(87 101 95)(88 102 96)(113 129 121)(114 130 122)(115 131 123)(116 132 124)(117 133 125)(118 134 126)(119 135 127)(120 136 128)(145 161 153)(146 162 154)(147 163 155)(148 164 156)(149 165 157)(150 166 158)(151 167 159)(152 168 160)(169 185 177)(170 186 178)(171 187 179)(172 188 180)(173 189 181)(174 190 182)(175 191 183)(176 192 184)
(1 119)(2 120)(3 113)(4 114)(5 115)(6 116)(7 117)(8 118)(9 108)(10 109)(11 110)(12 111)(13 112)(14 105)(15 106)(16 107)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 121)(26 122)(27 123)(28 124)(29 125)(30 126)(31 127)(32 128)(33 129)(34 130)(35 131)(36 132)(37 133)(38 134)(39 135)(40 136)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(57 145)(58 146)(59 147)(60 148)(61 149)(62 150)(63 151)(64 152)(65 153)(66 154)(67 155)(68 156)(69 157)(70 158)(71 159)(72 160)(73 161)(74 162)(75 163)(76 164)(77 165)(78 166)(79 167)(80 168)(81 169)(82 170)(83 171)(84 172)(85 173)(86 174)(87 175)(88 176)(89 177)(90 178)(91 179)(92 180)(93 181)(94 182)(95 183)(96 184)(97 187)(98 188)(99 189)(100 190)(101 191)(102 192)(103 185)(104 186)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)(161 165)(162 166)(163 167)(164 168)(169 173)(170 174)(171 175)(172 176)(177 181)(178 182)(179 183)(180 184)(185 189)(186 190)(187 191)(188 192)
(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)
(1 175 151 47)(2 88 152 144)(3 169 145 41)(4 82 146 138)(5 171 147 43)(6 84 148 140)(7 173 149 45)(8 86 150 142)(9 37 189 165)(10 134 190 78)(11 39 191 167)(12 136 192 80)(13 33 185 161)(14 130 186 74)(15 35 187 163)(16 132 188 76)(17 121 89 65)(18 26 90 154)(19 123 91 67)(20 28 92 156)(21 125 93 69)(22 30 94 158)(23 127 95 71)(24 32 96 160)(25 177 153 49)(27 179 155 51)(29 181 157 53)(31 183 159 55)(34 104 162 105)(36 98 164 107)(38 100 166 109)(40 102 168 111)(42 114 170 58)(44 116 172 60)(46 118 174 62)(48 120 176 64)(50 122 178 66)(52 124 180 68)(54 126 182 70)(56 128 184 72)(57 137 113 81)(59 139 115 83)(61 141 117 85)(63 143 119 87)(73 112 129 103)(75 106 131 97)(77 108 133 99)(79 110 135 101)
G:=sub<Sym(192)| (1,39,31)(2,40,32)(3,33,25)(4,34,26)(5,35,27)(6,36,28)(7,37,29)(8,38,30)(9,53,45)(10,54,46)(11,55,47)(12,56,48)(13,49,41)(14,50,42)(15,51,43)(16,52,44)(17,137,112)(18,138,105)(19,139,106)(20,140,107)(21,141,108)(22,142,109)(23,143,110)(24,144,111)(57,73,65)(58,74,66)(59,75,67)(60,76,68)(61,77,69)(62,78,70)(63,79,71)(64,80,72)(81,103,89)(82,104,90)(83,97,91)(84,98,92)(85,99,93)(86,100,94)(87,101,95)(88,102,96)(113,129,121)(114,130,122)(115,131,123)(116,132,124)(117,133,125)(118,134,126)(119,135,127)(120,136,128)(145,161,153)(146,162,154)(147,163,155)(148,164,156)(149,165,157)(150,166,158)(151,167,159)(152,168,160)(169,185,177)(170,186,178)(171,187,179)(172,188,180)(173,189,181)(174,190,182)(175,191,183)(176,192,184), (1,119)(2,120)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,108)(10,109)(11,110)(12,111)(13,112)(14,105)(15,106)(16,107)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,127)(32,128)(33,129)(34,130)(35,131)(36,132)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(57,145)(58,146)(59,147)(60,148)(61,149)(62,150)(63,151)(64,152)(65,153)(66,154)(67,155)(68,156)(69,157)(70,158)(71,159)(72,160)(73,161)(74,162)(75,163)(76,164)(77,165)(78,166)(79,167)(80,168)(81,169)(82,170)(83,171)(84,172)(85,173)(86,174)(87,175)(88,176)(89,177)(90,178)(91,179)(92,180)(93,181)(94,182)(95,183)(96,184)(97,187)(98,188)(99,189)(100,190)(101,191)(102,192)(103,185)(104,186), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160)(161,165)(162,166)(163,167)(164,168)(169,173)(170,174)(171,175)(172,176)(177,181)(178,182)(179,183)(180,184)(185,189)(186,190)(187,191)(188,192), (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), (1,175,151,47)(2,88,152,144)(3,169,145,41)(4,82,146,138)(5,171,147,43)(6,84,148,140)(7,173,149,45)(8,86,150,142)(9,37,189,165)(10,134,190,78)(11,39,191,167)(12,136,192,80)(13,33,185,161)(14,130,186,74)(15,35,187,163)(16,132,188,76)(17,121,89,65)(18,26,90,154)(19,123,91,67)(20,28,92,156)(21,125,93,69)(22,30,94,158)(23,127,95,71)(24,32,96,160)(25,177,153,49)(27,179,155,51)(29,181,157,53)(31,183,159,55)(34,104,162,105)(36,98,164,107)(38,100,166,109)(40,102,168,111)(42,114,170,58)(44,116,172,60)(46,118,174,62)(48,120,176,64)(50,122,178,66)(52,124,180,68)(54,126,182,70)(56,128,184,72)(57,137,113,81)(59,139,115,83)(61,141,117,85)(63,143,119,87)(73,112,129,103)(75,106,131,97)(77,108,133,99)(79,110,135,101)>;
G:=Group( (1,39,31)(2,40,32)(3,33,25)(4,34,26)(5,35,27)(6,36,28)(7,37,29)(8,38,30)(9,53,45)(10,54,46)(11,55,47)(12,56,48)(13,49,41)(14,50,42)(15,51,43)(16,52,44)(17,137,112)(18,138,105)(19,139,106)(20,140,107)(21,141,108)(22,142,109)(23,143,110)(24,144,111)(57,73,65)(58,74,66)(59,75,67)(60,76,68)(61,77,69)(62,78,70)(63,79,71)(64,80,72)(81,103,89)(82,104,90)(83,97,91)(84,98,92)(85,99,93)(86,100,94)(87,101,95)(88,102,96)(113,129,121)(114,130,122)(115,131,123)(116,132,124)(117,133,125)(118,134,126)(119,135,127)(120,136,128)(145,161,153)(146,162,154)(147,163,155)(148,164,156)(149,165,157)(150,166,158)(151,167,159)(152,168,160)(169,185,177)(170,186,178)(171,187,179)(172,188,180)(173,189,181)(174,190,182)(175,191,183)(176,192,184), (1,119)(2,120)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,108)(10,109)(11,110)(12,111)(13,112)(14,105)(15,106)(16,107)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,127)(32,128)(33,129)(34,130)(35,131)(36,132)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(57,145)(58,146)(59,147)(60,148)(61,149)(62,150)(63,151)(64,152)(65,153)(66,154)(67,155)(68,156)(69,157)(70,158)(71,159)(72,160)(73,161)(74,162)(75,163)(76,164)(77,165)(78,166)(79,167)(80,168)(81,169)(82,170)(83,171)(84,172)(85,173)(86,174)(87,175)(88,176)(89,177)(90,178)(91,179)(92,180)(93,181)(94,182)(95,183)(96,184)(97,187)(98,188)(99,189)(100,190)(101,191)(102,192)(103,185)(104,186), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160)(161,165)(162,166)(163,167)(164,168)(169,173)(170,174)(171,175)(172,176)(177,181)(178,182)(179,183)(180,184)(185,189)(186,190)(187,191)(188,192), (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), (1,175,151,47)(2,88,152,144)(3,169,145,41)(4,82,146,138)(5,171,147,43)(6,84,148,140)(7,173,149,45)(8,86,150,142)(9,37,189,165)(10,134,190,78)(11,39,191,167)(12,136,192,80)(13,33,185,161)(14,130,186,74)(15,35,187,163)(16,132,188,76)(17,121,89,65)(18,26,90,154)(19,123,91,67)(20,28,92,156)(21,125,93,69)(22,30,94,158)(23,127,95,71)(24,32,96,160)(25,177,153,49)(27,179,155,51)(29,181,157,53)(31,183,159,55)(34,104,162,105)(36,98,164,107)(38,100,166,109)(40,102,168,111)(42,114,170,58)(44,116,172,60)(46,118,174,62)(48,120,176,64)(50,122,178,66)(52,124,180,68)(54,126,182,70)(56,128,184,72)(57,137,113,81)(59,139,115,83)(61,141,117,85)(63,143,119,87)(73,112,129,103)(75,106,131,97)(77,108,133,99)(79,110,135,101) );
G=PermutationGroup([[(1,39,31),(2,40,32),(3,33,25),(4,34,26),(5,35,27),(6,36,28),(7,37,29),(8,38,30),(9,53,45),(10,54,46),(11,55,47),(12,56,48),(13,49,41),(14,50,42),(15,51,43),(16,52,44),(17,137,112),(18,138,105),(19,139,106),(20,140,107),(21,141,108),(22,142,109),(23,143,110),(24,144,111),(57,73,65),(58,74,66),(59,75,67),(60,76,68),(61,77,69),(62,78,70),(63,79,71),(64,80,72),(81,103,89),(82,104,90),(83,97,91),(84,98,92),(85,99,93),(86,100,94),(87,101,95),(88,102,96),(113,129,121),(114,130,122),(115,131,123),(116,132,124),(117,133,125),(118,134,126),(119,135,127),(120,136,128),(145,161,153),(146,162,154),(147,163,155),(148,164,156),(149,165,157),(150,166,158),(151,167,159),(152,168,160),(169,185,177),(170,186,178),(171,187,179),(172,188,180),(173,189,181),(174,190,182),(175,191,183),(176,192,184)], [(1,119),(2,120),(3,113),(4,114),(5,115),(6,116),(7,117),(8,118),(9,108),(10,109),(11,110),(12,111),(13,112),(14,105),(15,106),(16,107),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,121),(26,122),(27,123),(28,124),(29,125),(30,126),(31,127),(32,128),(33,129),(34,130),(35,131),(36,132),(37,133),(38,134),(39,135),(40,136),(41,137),(42,138),(43,139),(44,140),(45,141),(46,142),(47,143),(48,144),(57,145),(58,146),(59,147),(60,148),(61,149),(62,150),(63,151),(64,152),(65,153),(66,154),(67,155),(68,156),(69,157),(70,158),(71,159),(72,160),(73,161),(74,162),(75,163),(76,164),(77,165),(78,166),(79,167),(80,168),(81,169),(82,170),(83,171),(84,172),(85,173),(86,174),(87,175),(88,176),(89,177),(90,178),(91,179),(92,180),(93,181),(94,182),(95,183),(96,184),(97,187),(98,188),(99,189),(100,190),(101,191),(102,192),(103,185),(104,186)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112),(113,117),(114,118),(115,119),(116,120),(121,125),(122,126),(123,127),(124,128),(129,133),(130,134),(131,135),(132,136),(137,141),(138,142),(139,143),(140,144),(145,149),(146,150),(147,151),(148,152),(153,157),(154,158),(155,159),(156,160),(161,165),(162,166),(163,167),(164,168),(169,173),(170,174),(171,175),(172,176),(177,181),(178,182),(179,183),(180,184),(185,189),(186,190),(187,191),(188,192)], [(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)], [(1,175,151,47),(2,88,152,144),(3,169,145,41),(4,82,146,138),(5,171,147,43),(6,84,148,140),(7,173,149,45),(8,86,150,142),(9,37,189,165),(10,134,190,78),(11,39,191,167),(12,136,192,80),(13,33,185,161),(14,130,186,74),(15,35,187,163),(16,132,188,76),(17,121,89,65),(18,26,90,154),(19,123,91,67),(20,28,92,156),(21,125,93,69),(22,30,94,158),(23,127,95,71),(24,32,96,160),(25,177,153,49),(27,179,155,51),(29,181,157,53),(31,183,159,55),(34,104,162,105),(36,98,164,107),(38,100,166,109),(40,102,168,111),(42,114,170,58),(44,116,172,60),(46,118,174,62),(48,120,176,64),(50,122,178,66),(52,124,180,68),(54,126,182,70),(56,128,184,72),(57,137,113,81),(59,139,115,83),(61,141,117,85),(63,143,119,87),(73,112,129,103),(75,106,131,97),(77,108,133,99),(79,110,135,101)]])
120 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 4A | ··· | 4H | 4I | ··· | 4P | 6A | ··· | 6N | 8A | ··· | 8P | 12A | ··· | 12P | 12Q | ··· | 12AF | 24A | ··· | 24AF |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C12 | C12 | C24 | D4 | Q8 | M4(2) | C3×D4 | C3×Q8 | C3×M4(2) |
| kernel | C3×C22.7C42 | C2×C4×C12 | C22×C24 | C22.7C42 | C2×C24 | C22×C12 | C2×C42 | C22×C8 | C2×C12 | C2×C8 | C22×C4 | C2×C4 | C2×C12 | C2×C12 | C2×C6 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 1 | 2 | 2 | 8 | 4 | 2 | 4 | 16 | 16 | 8 | 32 | 3 | 1 | 4 | 6 | 2 | 8 |
Matrix representation of C3×C22.7C42 ►in GL5(𝔽73)
| 64 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 46 | 0 | 0 | 0 | 0 |
| 0 | 63 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 47 | 12 |
| 0 | 0 | 0 | 47 | 26 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 52 | 71 |
| 0 | 0 | 0 | 2 | 21 |
G:=sub<GL(5,GF(73))| [64,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[46,0,0,0,0,0,63,0,0,0,0,0,27,0,0,0,0,0,47,47,0,0,0,12,26],[1,0,0,0,0,0,72,0,0,0,0,0,46,0,0,0,0,0,52,2,0,0,0,71,21] >;
C3×C22.7C42 in GAP, Magma, Sage, TeX
C_3\times C_2^2._7C_4^2
% in TeX
G:=Group("C3xC2^2.7C4^2"); // GroupNames label
G:=SmallGroup(192,142);
// by ID
G=gap.SmallGroup(192,142);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,344,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^2=e^4=1,d^4=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,e*d*e^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c>;
// generators/relations