direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C42⋊8C4, C42⋊11C12, (C4×C12)⋊18C4, C12.59(C4⋊C4), (C2×C12).71Q8, (C2×C12).411D4, (C2×C42).11C6, C22.32(C6×D4), C22.10(C6×Q8), C6.63(C4.4D4), C2.C42.4C6, C23.59(C22×C6), C6.26(C42.C2), C6.55(C42⋊C2), C22.31(C22×C12), (C22×C6).446C23, (C22×C12).491C22, C2.5(C6×C4⋊C4), (C2×C4⋊C4).4C6, C6.60(C2×C4⋊C4), C4.10(C3×C4⋊C4), (C2×C4×C12).31C2, (C6×C4⋊C4).33C2, (C2×C4).65(C3×D4), (C2×C4).14(C3×Q8), (C2×C4).69(C2×C12), (C2×C6).599(C2×D4), (C2×C6).102(C2×Q8), C2.1(C3×C4.4D4), (C2×C12).285(C2×C4), C2.1(C3×C42.C2), C2.7(C3×C42⋊C2), C22.17(C3×C4○D4), (C2×C6).207(C4○D4), (C2×C6).218(C22×C4), (C22×C4).113(C2×C6), (C3×C2.C42).3C2, SmallGroup(192,815)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42⋊8C4
G = < a,b,c,d | a3=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >
Subgroups: 226 in 154 conjugacy classes, 98 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C22×C4, C22×C4, C2×C12, C2×C12, C22×C6, C2.C42, C2×C42, C2×C4⋊C4, C4×C12, C3×C4⋊C4, C22×C12, C22×C12, C42⋊8C4, C3×C2.C42, C2×C4×C12, C6×C4⋊C4, C3×C42⋊8C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C23, C12, C2×C6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4⋊C4, C42⋊C2, C4.4D4, C42.C2, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C42⋊8C4, C6×C4⋊C4, C3×C42⋊C2, C3×C4.4D4, C3×C42.C2, C3×C42⋊8C4
►Regular action on 192 points
Generators in S192
(1 83 79)(2 84 80)(3 81 77)(4 82 78)(5 13 9)(6 14 10)(7 15 11)(8 16 12)(17 25 21)(18 26 22)(19 27 23)(20 28 24)(29 37 33)(30 38 34)(31 39 35)(32 40 36)(41 101 45)(42 102 46)(43 103 47)(44 104 48)(49 140 144)(50 137 141)(51 138 142)(52 139 143)(53 61 57)(54 62 58)(55 63 59)(56 64 60)(65 75 69)(66 76 70)(67 73 71)(68 74 72)(85 93 89)(86 94 90)(87 95 91)(88 96 92)(97 105 109)(98 106 110)(99 107 111)(100 108 112)(113 117 125)(114 118 126)(115 119 127)(116 120 128)(121 129 133)(122 130 134)(123 131 135)(124 132 136)(145 153 149)(146 154 150)(147 155 151)(148 156 152)(157 165 161)(158 166 162)(159 167 163)(160 168 164)(169 177 173)(170 178 174)(171 179 175)(172 180 176)(181 189 185)(182 190 186)(183 191 187)(184 192 188)
(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 67 30 18)(2 68 31 19)(3 65 32 20)(4 66 29 17)(5 41 54 86)(6 42 55 87)(7 43 56 88)(8 44 53 85)(9 45 58 90)(10 46 59 91)(11 47 60 92)(12 48 57 89)(13 101 62 94)(14 102 63 95)(15 103 64 96)(16 104 61 93)(21 78 70 33)(22 79 71 34)(23 80 72 35)(24 77 69 36)(25 82 76 37)(26 83 73 38)(27 84 74 39)(28 81 75 40)(49 154 189 105)(50 155 190 106)(51 156 191 107)(52 153 192 108)(97 144 146 181)(98 141 147 182)(99 142 148 183)(100 143 145 184)(109 140 150 185)(110 137 151 186)(111 138 152 187)(112 139 149 188)(113 173 162 136)(114 174 163 133)(115 175 164 134)(116 176 161 135)(117 169 158 124)(118 170 159 121)(119 171 160 122)(120 172 157 123)(125 177 166 132)(126 178 167 129)(127 179 168 130)(128 180 165 131)
(1 146 6 122)(2 98 7 172)(3 148 8 124)(4 100 5 170)(9 174 78 112)(10 134 79 150)(11 176 80 110)(12 136 77 152)(13 178 82 108)(14 130 83 154)(15 180 84 106)(16 132 81 156)(17 182 86 120)(18 142 87 158)(19 184 88 118)(20 144 85 160)(21 186 90 116)(22 138 91 162)(23 188 92 114)(24 140 89 164)(25 190 94 128)(26 51 95 166)(27 192 96 126)(28 49 93 168)(29 145 54 121)(30 97 55 171)(31 147 56 123)(32 99 53 169)(33 149 58 133)(34 109 59 175)(35 151 60 135)(36 111 57 173)(37 153 62 129)(38 105 63 179)(39 155 64 131)(40 107 61 177)(41 157 66 141)(42 117 67 183)(43 159 68 143)(44 119 65 181)(45 161 70 137)(46 113 71 187)(47 163 72 139)(48 115 69 185)(50 101 165 76)(52 103 167 74)(73 191 102 125)(75 189 104 127)
G:=sub<Sym(192)| (1,83,79)(2,84,80)(3,81,77)(4,82,78)(5,13,9)(6,14,10)(7,15,11)(8,16,12)(17,25,21)(18,26,22)(19,27,23)(20,28,24)(29,37,33)(30,38,34)(31,39,35)(32,40,36)(41,101,45)(42,102,46)(43,103,47)(44,104,48)(49,140,144)(50,137,141)(51,138,142)(52,139,143)(53,61,57)(54,62,58)(55,63,59)(56,64,60)(65,75,69)(66,76,70)(67,73,71)(68,74,72)(85,93,89)(86,94,90)(87,95,91)(88,96,92)(97,105,109)(98,106,110)(99,107,111)(100,108,112)(113,117,125)(114,118,126)(115,119,127)(116,120,128)(121,129,133)(122,130,134)(123,131,135)(124,132,136)(145,153,149)(146,154,150)(147,155,151)(148,156,152)(157,165,161)(158,166,162)(159,167,163)(160,168,164)(169,177,173)(170,178,174)(171,179,175)(172,180,176)(181,189,185)(182,190,186)(183,191,187)(184,192,188), (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), (1,67,30,18)(2,68,31,19)(3,65,32,20)(4,66,29,17)(5,41,54,86)(6,42,55,87)(7,43,56,88)(8,44,53,85)(9,45,58,90)(10,46,59,91)(11,47,60,92)(12,48,57,89)(13,101,62,94)(14,102,63,95)(15,103,64,96)(16,104,61,93)(21,78,70,33)(22,79,71,34)(23,80,72,35)(24,77,69,36)(25,82,76,37)(26,83,73,38)(27,84,74,39)(28,81,75,40)(49,154,189,105)(50,155,190,106)(51,156,191,107)(52,153,192,108)(97,144,146,181)(98,141,147,182)(99,142,148,183)(100,143,145,184)(109,140,150,185)(110,137,151,186)(111,138,152,187)(112,139,149,188)(113,173,162,136)(114,174,163,133)(115,175,164,134)(116,176,161,135)(117,169,158,124)(118,170,159,121)(119,171,160,122)(120,172,157,123)(125,177,166,132)(126,178,167,129)(127,179,168,130)(128,180,165,131), (1,146,6,122)(2,98,7,172)(3,148,8,124)(4,100,5,170)(9,174,78,112)(10,134,79,150)(11,176,80,110)(12,136,77,152)(13,178,82,108)(14,130,83,154)(15,180,84,106)(16,132,81,156)(17,182,86,120)(18,142,87,158)(19,184,88,118)(20,144,85,160)(21,186,90,116)(22,138,91,162)(23,188,92,114)(24,140,89,164)(25,190,94,128)(26,51,95,166)(27,192,96,126)(28,49,93,168)(29,145,54,121)(30,97,55,171)(31,147,56,123)(32,99,53,169)(33,149,58,133)(34,109,59,175)(35,151,60,135)(36,111,57,173)(37,153,62,129)(38,105,63,179)(39,155,64,131)(40,107,61,177)(41,157,66,141)(42,117,67,183)(43,159,68,143)(44,119,65,181)(45,161,70,137)(46,113,71,187)(47,163,72,139)(48,115,69,185)(50,101,165,76)(52,103,167,74)(73,191,102,125)(75,189,104,127)>;
G:=Group( (1,83,79)(2,84,80)(3,81,77)(4,82,78)(5,13,9)(6,14,10)(7,15,11)(8,16,12)(17,25,21)(18,26,22)(19,27,23)(20,28,24)(29,37,33)(30,38,34)(31,39,35)(32,40,36)(41,101,45)(42,102,46)(43,103,47)(44,104,48)(49,140,144)(50,137,141)(51,138,142)(52,139,143)(53,61,57)(54,62,58)(55,63,59)(56,64,60)(65,75,69)(66,76,70)(67,73,71)(68,74,72)(85,93,89)(86,94,90)(87,95,91)(88,96,92)(97,105,109)(98,106,110)(99,107,111)(100,108,112)(113,117,125)(114,118,126)(115,119,127)(116,120,128)(121,129,133)(122,130,134)(123,131,135)(124,132,136)(145,153,149)(146,154,150)(147,155,151)(148,156,152)(157,165,161)(158,166,162)(159,167,163)(160,168,164)(169,177,173)(170,178,174)(171,179,175)(172,180,176)(181,189,185)(182,190,186)(183,191,187)(184,192,188), (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), (1,67,30,18)(2,68,31,19)(3,65,32,20)(4,66,29,17)(5,41,54,86)(6,42,55,87)(7,43,56,88)(8,44,53,85)(9,45,58,90)(10,46,59,91)(11,47,60,92)(12,48,57,89)(13,101,62,94)(14,102,63,95)(15,103,64,96)(16,104,61,93)(21,78,70,33)(22,79,71,34)(23,80,72,35)(24,77,69,36)(25,82,76,37)(26,83,73,38)(27,84,74,39)(28,81,75,40)(49,154,189,105)(50,155,190,106)(51,156,191,107)(52,153,192,108)(97,144,146,181)(98,141,147,182)(99,142,148,183)(100,143,145,184)(109,140,150,185)(110,137,151,186)(111,138,152,187)(112,139,149,188)(113,173,162,136)(114,174,163,133)(115,175,164,134)(116,176,161,135)(117,169,158,124)(118,170,159,121)(119,171,160,122)(120,172,157,123)(125,177,166,132)(126,178,167,129)(127,179,168,130)(128,180,165,131), (1,146,6,122)(2,98,7,172)(3,148,8,124)(4,100,5,170)(9,174,78,112)(10,134,79,150)(11,176,80,110)(12,136,77,152)(13,178,82,108)(14,130,83,154)(15,180,84,106)(16,132,81,156)(17,182,86,120)(18,142,87,158)(19,184,88,118)(20,144,85,160)(21,186,90,116)(22,138,91,162)(23,188,92,114)(24,140,89,164)(25,190,94,128)(26,51,95,166)(27,192,96,126)(28,49,93,168)(29,145,54,121)(30,97,55,171)(31,147,56,123)(32,99,53,169)(33,149,58,133)(34,109,59,175)(35,151,60,135)(36,111,57,173)(37,153,62,129)(38,105,63,179)(39,155,64,131)(40,107,61,177)(41,157,66,141)(42,117,67,183)(43,159,68,143)(44,119,65,181)(45,161,70,137)(46,113,71,187)(47,163,72,139)(48,115,69,185)(50,101,165,76)(52,103,167,74)(73,191,102,125)(75,189,104,127) );
G=PermutationGroup([[(1,83,79),(2,84,80),(3,81,77),(4,82,78),(5,13,9),(6,14,10),(7,15,11),(8,16,12),(17,25,21),(18,26,22),(19,27,23),(20,28,24),(29,37,33),(30,38,34),(31,39,35),(32,40,36),(41,101,45),(42,102,46),(43,103,47),(44,104,48),(49,140,144),(50,137,141),(51,138,142),(52,139,143),(53,61,57),(54,62,58),(55,63,59),(56,64,60),(65,75,69),(66,76,70),(67,73,71),(68,74,72),(85,93,89),(86,94,90),(87,95,91),(88,96,92),(97,105,109),(98,106,110),(99,107,111),(100,108,112),(113,117,125),(114,118,126),(115,119,127),(116,120,128),(121,129,133),(122,130,134),(123,131,135),(124,132,136),(145,153,149),(146,154,150),(147,155,151),(148,156,152),(157,165,161),(158,166,162),(159,167,163),(160,168,164),(169,177,173),(170,178,174),(171,179,175),(172,180,176),(181,189,185),(182,190,186),(183,191,187),(184,192,188)], [(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)], [(1,67,30,18),(2,68,31,19),(3,65,32,20),(4,66,29,17),(5,41,54,86),(6,42,55,87),(7,43,56,88),(8,44,53,85),(9,45,58,90),(10,46,59,91),(11,47,60,92),(12,48,57,89),(13,101,62,94),(14,102,63,95),(15,103,64,96),(16,104,61,93),(21,78,70,33),(22,79,71,34),(23,80,72,35),(24,77,69,36),(25,82,76,37),(26,83,73,38),(27,84,74,39),(28,81,75,40),(49,154,189,105),(50,155,190,106),(51,156,191,107),(52,153,192,108),(97,144,146,181),(98,141,147,182),(99,142,148,183),(100,143,145,184),(109,140,150,185),(110,137,151,186),(111,138,152,187),(112,139,149,188),(113,173,162,136),(114,174,163,133),(115,175,164,134),(116,176,161,135),(117,169,158,124),(118,170,159,121),(119,171,160,122),(120,172,157,123),(125,177,166,132),(126,178,167,129),(127,179,168,130),(128,180,165,131)], [(1,146,6,122),(2,98,7,172),(3,148,8,124),(4,100,5,170),(9,174,78,112),(10,134,79,150),(11,176,80,110),(12,136,77,152),(13,178,82,108),(14,130,83,154),(15,180,84,106),(16,132,81,156),(17,182,86,120),(18,142,87,158),(19,184,88,118),(20,144,85,160),(21,186,90,116),(22,138,91,162),(23,188,92,114),(24,140,89,164),(25,190,94,128),(26,51,95,166),(27,192,96,126),(28,49,93,168),(29,145,54,121),(30,97,55,171),(31,147,56,123),(32,99,53,169),(33,149,58,133),(34,109,59,175),(35,151,60,135),(36,111,57,173),(37,153,62,129),(38,105,63,179),(39,155,64,131),(40,107,61,177),(41,157,66,141),(42,117,67,183),(43,159,68,143),(44,119,65,181),(45,161,70,137),(46,113,71,187),(47,163,72,139),(48,115,69,185),(50,101,165,76),(52,103,167,74),(73,191,102,125),(75,189,104,127)]])
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 4A | ··· | 4L | 4M | ··· | 4T | 6A | ··· | 6N | 12A | ··· | 12X | 12Y | ··· | 12AN |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | Q8 | C4○D4 | C3×D4 | C3×Q8 | C3×C4○D4 |
| kernel | C3×C42⋊8C4 | C3×C2.C42 | C2×C4×C12 | C6×C4⋊C4 | C42⋊8C4 | C4×C12 | C2.C42 | C2×C42 | C2×C4⋊C4 | C42 | C2×C12 | C2×C12 | C2×C6 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 4 | 1 | 2 | 2 | 8 | 8 | 2 | 4 | 16 | 2 | 2 | 8 | 4 | 4 | 16 |
Matrix representation of C3×C42⋊8C4 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 12 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 7 | 11 | 0 | 0 |
| 0 | 11 | 6 | 0 | 0 |
| 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 12 | 12 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[1,0,0,0,0,0,0,1,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,8],[12,0,0,0,0,0,0,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,2,12],[5,0,0,0,0,0,7,11,0,0,0,11,6,0,0,0,0,0,1,12,0,0,0,2,12] >;
C3×C42⋊8C4 in GAP, Magma, Sage, TeX
C_3\times C_4^2\rtimes_8C_4
% in TeX
G:=Group("C3xC4^2:8C4"); // GroupNames label
G:=SmallGroup(192,815);
// by ID
G=gap.SmallGroup(192,815);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,344,1094,142]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^2,d*c*d^-1=b^2*c>;
// generators/relations