direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C42⋊9C4, C42⋊12C12, C12⋊8(C4⋊C4), (C4×C12)⋊19C4, C6.35(C4⋊Q8), (C2×C12).72Q8, (C2×C12).412D4, (C2×C42).12C6, C22.33(C6×D4), C22.11(C6×Q8), C6.38(C4⋊1D4), C23.61(C22×C6), C22.33(C22×C12), (C22×C6).448C23, (C22×C12).574C22, C4⋊1(C3×C4⋊C4), C2.6(C6×C4⋊C4), (C2×C4⋊C4).5C6, C2.1(C3×C4⋊Q8), C6.61(C2×C4⋊C4), (C2×C4×C12).32C2, (C6×C4⋊C4).34C2, (C2×C4).66(C3×D4), C2.1(C3×C4⋊1D4), (C2×C4).15(C3×Q8), (C2×C4).70(C2×C12), (C2×C6).600(C2×D4), (C2×C6).103(C2×Q8), (C2×C12).331(C2×C4), (C2×C6).220(C22×C4), (C22×C4).114(C2×C6), SmallGroup(192,817)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42⋊9C4
G = < a,b,c,d | a3=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 258 in 186 conjugacy classes, 130 normal (12 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, C2×C12, C2×C12, C22×C6, C2×C42, C2×C4⋊C4, C4×C12, C3×C4⋊C4, C22×C12, C42⋊9C4, C2×C4×C12, C6×C4⋊C4, C3×C42⋊9C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C23, C12, C2×C6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4⋊C4, C4⋊1D4, C4⋊Q8, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C42⋊9C4, C6×C4⋊C4, C3×C4⋊1D4, C3×C4⋊Q8, C3×C42⋊9C4
►Regular action on 192 points
Generators in S192
(1 85 81)(2 86 82)(3 87 83)(4 88 84)(5 192 188)(6 189 185)(7 190 186)(8 191 187)(9 17 13)(10 18 14)(11 19 15)(12 20 16)(21 29 25)(22 30 26)(23 31 27)(24 32 28)(33 41 37)(34 42 38)(35 43 39)(36 44 40)(45 97 101)(46 98 102)(47 99 103)(48 100 104)(49 138 53)(50 139 54)(51 140 55)(52 137 56)(57 65 61)(58 66 62)(59 67 63)(60 68 64)(69 77 73)(70 78 74)(71 79 75)(72 80 76)(89 145 93)(90 146 94)(91 147 95)(92 148 96)(105 109 117)(106 110 118)(107 111 119)(108 112 120)(113 121 125)(114 122 126)(115 123 127)(116 124 128)(129 133 141)(130 134 142)(131 135 143)(132 136 144)(149 157 153)(150 158 154)(151 159 155)(152 160 156)(161 169 165)(162 170 166)(163 171 167)(164 172 168)(173 181 177)(174 182 178)(175 183 179)(176 184 180)
(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 21 34 71)(2 22 35 72)(3 23 36 69)(4 24 33 70)(5 118 56 157)(6 119 53 158)(7 120 54 159)(8 117 55 160)(9 89 59 46)(10 90 60 47)(11 91 57 48)(12 92 58 45)(13 93 63 102)(14 94 64 103)(15 95 61 104)(16 96 62 101)(17 145 67 98)(18 146 68 99)(19 147 65 100)(20 148 66 97)(25 38 75 81)(26 39 76 82)(27 40 73 83)(28 37 74 84)(29 42 79 85)(30 43 80 86)(31 44 77 87)(32 41 78 88)(49 154 189 107)(50 155 190 108)(51 156 191 105)(52 153 192 106)(109 140 152 187)(110 137 149 188)(111 138 150 185)(112 139 151 186)(113 175 164 136)(114 176 161 133)(115 173 162 134)(116 174 163 135)(121 183 172 144)(122 184 169 141)(123 181 170 142)(124 182 171 143)(125 179 168 132)(126 180 165 129)(127 177 166 130)(128 178 167 131)
(1 152 59 175)(2 151 60 174)(3 150 57 173)(4 149 58 176)(5 148 169 78)(6 147 170 77)(7 146 171 80)(8 145 172 79)(9 136 34 109)(10 135 35 112)(11 134 36 111)(12 133 33 110)(13 132 38 105)(14 131 39 108)(15 130 40 107)(16 129 37 106)(17 144 42 117)(18 143 43 120)(19 142 44 119)(20 141 41 118)(21 140 46 113)(22 139 47 116)(23 138 48 115)(24 137 45 114)(25 51 102 125)(26 50 103 128)(27 49 104 127)(28 52 101 126)(29 55 98 121)(30 54 99 124)(31 53 100 123)(32 56 97 122)(61 177 83 154)(62 180 84 153)(63 179 81 156)(64 178 82 155)(65 181 87 158)(66 184 88 157)(67 183 85 160)(68 182 86 159)(69 185 91 162)(70 188 92 161)(71 187 89 164)(72 186 90 163)(73 189 95 166)(74 192 96 165)(75 191 93 168)(76 190 94 167)
G:=sub<Sym(192)| (1,85,81)(2,86,82)(3,87,83)(4,88,84)(5,192,188)(6,189,185)(7,190,186)(8,191,187)(9,17,13)(10,18,14)(11,19,15)(12,20,16)(21,29,25)(22,30,26)(23,31,27)(24,32,28)(33,41,37)(34,42,38)(35,43,39)(36,44,40)(45,97,101)(46,98,102)(47,99,103)(48,100,104)(49,138,53)(50,139,54)(51,140,55)(52,137,56)(57,65,61)(58,66,62)(59,67,63)(60,68,64)(69,77,73)(70,78,74)(71,79,75)(72,80,76)(89,145,93)(90,146,94)(91,147,95)(92,148,96)(105,109,117)(106,110,118)(107,111,119)(108,112,120)(113,121,125)(114,122,126)(115,123,127)(116,124,128)(129,133,141)(130,134,142)(131,135,143)(132,136,144)(149,157,153)(150,158,154)(151,159,155)(152,160,156)(161,169,165)(162,170,166)(163,171,167)(164,172,168)(173,181,177)(174,182,178)(175,183,179)(176,184,180), (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), (1,21,34,71)(2,22,35,72)(3,23,36,69)(4,24,33,70)(5,118,56,157)(6,119,53,158)(7,120,54,159)(8,117,55,160)(9,89,59,46)(10,90,60,47)(11,91,57,48)(12,92,58,45)(13,93,63,102)(14,94,64,103)(15,95,61,104)(16,96,62,101)(17,145,67,98)(18,146,68,99)(19,147,65,100)(20,148,66,97)(25,38,75,81)(26,39,76,82)(27,40,73,83)(28,37,74,84)(29,42,79,85)(30,43,80,86)(31,44,77,87)(32,41,78,88)(49,154,189,107)(50,155,190,108)(51,156,191,105)(52,153,192,106)(109,140,152,187)(110,137,149,188)(111,138,150,185)(112,139,151,186)(113,175,164,136)(114,176,161,133)(115,173,162,134)(116,174,163,135)(121,183,172,144)(122,184,169,141)(123,181,170,142)(124,182,171,143)(125,179,168,132)(126,180,165,129)(127,177,166,130)(128,178,167,131), (1,152,59,175)(2,151,60,174)(3,150,57,173)(4,149,58,176)(5,148,169,78)(6,147,170,77)(7,146,171,80)(8,145,172,79)(9,136,34,109)(10,135,35,112)(11,134,36,111)(12,133,33,110)(13,132,38,105)(14,131,39,108)(15,130,40,107)(16,129,37,106)(17,144,42,117)(18,143,43,120)(19,142,44,119)(20,141,41,118)(21,140,46,113)(22,139,47,116)(23,138,48,115)(24,137,45,114)(25,51,102,125)(26,50,103,128)(27,49,104,127)(28,52,101,126)(29,55,98,121)(30,54,99,124)(31,53,100,123)(32,56,97,122)(61,177,83,154)(62,180,84,153)(63,179,81,156)(64,178,82,155)(65,181,87,158)(66,184,88,157)(67,183,85,160)(68,182,86,159)(69,185,91,162)(70,188,92,161)(71,187,89,164)(72,186,90,163)(73,189,95,166)(74,192,96,165)(75,191,93,168)(76,190,94,167)>;
G:=Group( (1,85,81)(2,86,82)(3,87,83)(4,88,84)(5,192,188)(6,189,185)(7,190,186)(8,191,187)(9,17,13)(10,18,14)(11,19,15)(12,20,16)(21,29,25)(22,30,26)(23,31,27)(24,32,28)(33,41,37)(34,42,38)(35,43,39)(36,44,40)(45,97,101)(46,98,102)(47,99,103)(48,100,104)(49,138,53)(50,139,54)(51,140,55)(52,137,56)(57,65,61)(58,66,62)(59,67,63)(60,68,64)(69,77,73)(70,78,74)(71,79,75)(72,80,76)(89,145,93)(90,146,94)(91,147,95)(92,148,96)(105,109,117)(106,110,118)(107,111,119)(108,112,120)(113,121,125)(114,122,126)(115,123,127)(116,124,128)(129,133,141)(130,134,142)(131,135,143)(132,136,144)(149,157,153)(150,158,154)(151,159,155)(152,160,156)(161,169,165)(162,170,166)(163,171,167)(164,172,168)(173,181,177)(174,182,178)(175,183,179)(176,184,180), (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), (1,21,34,71)(2,22,35,72)(3,23,36,69)(4,24,33,70)(5,118,56,157)(6,119,53,158)(7,120,54,159)(8,117,55,160)(9,89,59,46)(10,90,60,47)(11,91,57,48)(12,92,58,45)(13,93,63,102)(14,94,64,103)(15,95,61,104)(16,96,62,101)(17,145,67,98)(18,146,68,99)(19,147,65,100)(20,148,66,97)(25,38,75,81)(26,39,76,82)(27,40,73,83)(28,37,74,84)(29,42,79,85)(30,43,80,86)(31,44,77,87)(32,41,78,88)(49,154,189,107)(50,155,190,108)(51,156,191,105)(52,153,192,106)(109,140,152,187)(110,137,149,188)(111,138,150,185)(112,139,151,186)(113,175,164,136)(114,176,161,133)(115,173,162,134)(116,174,163,135)(121,183,172,144)(122,184,169,141)(123,181,170,142)(124,182,171,143)(125,179,168,132)(126,180,165,129)(127,177,166,130)(128,178,167,131), (1,152,59,175)(2,151,60,174)(3,150,57,173)(4,149,58,176)(5,148,169,78)(6,147,170,77)(7,146,171,80)(8,145,172,79)(9,136,34,109)(10,135,35,112)(11,134,36,111)(12,133,33,110)(13,132,38,105)(14,131,39,108)(15,130,40,107)(16,129,37,106)(17,144,42,117)(18,143,43,120)(19,142,44,119)(20,141,41,118)(21,140,46,113)(22,139,47,116)(23,138,48,115)(24,137,45,114)(25,51,102,125)(26,50,103,128)(27,49,104,127)(28,52,101,126)(29,55,98,121)(30,54,99,124)(31,53,100,123)(32,56,97,122)(61,177,83,154)(62,180,84,153)(63,179,81,156)(64,178,82,155)(65,181,87,158)(66,184,88,157)(67,183,85,160)(68,182,86,159)(69,185,91,162)(70,188,92,161)(71,187,89,164)(72,186,90,163)(73,189,95,166)(74,192,96,165)(75,191,93,168)(76,190,94,167) );
G=PermutationGroup([[(1,85,81),(2,86,82),(3,87,83),(4,88,84),(5,192,188),(6,189,185),(7,190,186),(8,191,187),(9,17,13),(10,18,14),(11,19,15),(12,20,16),(21,29,25),(22,30,26),(23,31,27),(24,32,28),(33,41,37),(34,42,38),(35,43,39),(36,44,40),(45,97,101),(46,98,102),(47,99,103),(48,100,104),(49,138,53),(50,139,54),(51,140,55),(52,137,56),(57,65,61),(58,66,62),(59,67,63),(60,68,64),(69,77,73),(70,78,74),(71,79,75),(72,80,76),(89,145,93),(90,146,94),(91,147,95),(92,148,96),(105,109,117),(106,110,118),(107,111,119),(108,112,120),(113,121,125),(114,122,126),(115,123,127),(116,124,128),(129,133,141),(130,134,142),(131,135,143),(132,136,144),(149,157,153),(150,158,154),(151,159,155),(152,160,156),(161,169,165),(162,170,166),(163,171,167),(164,172,168),(173,181,177),(174,182,178),(175,183,179),(176,184,180)], [(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)], [(1,21,34,71),(2,22,35,72),(3,23,36,69),(4,24,33,70),(5,118,56,157),(6,119,53,158),(7,120,54,159),(8,117,55,160),(9,89,59,46),(10,90,60,47),(11,91,57,48),(12,92,58,45),(13,93,63,102),(14,94,64,103),(15,95,61,104),(16,96,62,101),(17,145,67,98),(18,146,68,99),(19,147,65,100),(20,148,66,97),(25,38,75,81),(26,39,76,82),(27,40,73,83),(28,37,74,84),(29,42,79,85),(30,43,80,86),(31,44,77,87),(32,41,78,88),(49,154,189,107),(50,155,190,108),(51,156,191,105),(52,153,192,106),(109,140,152,187),(110,137,149,188),(111,138,150,185),(112,139,151,186),(113,175,164,136),(114,176,161,133),(115,173,162,134),(116,174,163,135),(121,183,172,144),(122,184,169,141),(123,181,170,142),(124,182,171,143),(125,179,168,132),(126,180,165,129),(127,177,166,130),(128,178,167,131)], [(1,152,59,175),(2,151,60,174),(3,150,57,173),(4,149,58,176),(5,148,169,78),(6,147,170,77),(7,146,171,80),(8,145,172,79),(9,136,34,109),(10,135,35,112),(11,134,36,111),(12,133,33,110),(13,132,38,105),(14,131,39,108),(15,130,40,107),(16,129,37,106),(17,144,42,117),(18,143,43,120),(19,142,44,119),(20,141,41,118),(21,140,46,113),(22,139,47,116),(23,138,48,115),(24,137,45,114),(25,51,102,125),(26,50,103,128),(27,49,104,127),(28,52,101,126),(29,55,98,121),(30,54,99,124),(31,53,100,123),(32,56,97,122),(61,177,83,154),(62,180,84,153),(63,179,81,156),(64,178,82,155),(65,181,87,158),(66,184,88,157),(67,183,85,160),(68,182,86,159),(69,185,91,162),(70,188,92,161),(71,187,89,164),(72,186,90,163),(73,189,95,166),(74,192,96,165),(75,191,93,168),(76,190,94,167)]])
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 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D4 | Q8 | C3×D4 | C3×Q8 |
| kernel | C3×C42⋊9C4 | C2×C4×C12 | C6×C4⋊C4 | C42⋊9C4 | C4×C12 | C2×C42 | C2×C4⋊C4 | C42 | C2×C12 | C2×C12 | C2×C4 | C2×C4 |
| # reps | 1 | 1 | 6 | 2 | 8 | 2 | 12 | 16 | 6 | 6 | 12 | 12 |
Matrix representation of C3×C42⋊9C4 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 12 | 0 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 11 | 6 | 0 | 0 |
| 0 | 6 | 2 | 0 | 0 |
| 0 | 0 | 0 | 7 | 2 |
| 0 | 0 | 0 | 2 | 6 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,3,0,0,0,0,0,3],[1,0,0,0,0,0,0,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,12,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,1,0],[8,0,0,0,0,0,11,6,0,0,0,6,2,0,0,0,0,0,7,2,0,0,0,2,6] >;
C3×C42⋊9C4 in GAP, Magma, Sage, TeX
C_3\times C_4^2\rtimes_9C_4
% in TeX
G:=Group("C3xC4^2:9C4"); // GroupNames label
G:=SmallGroup(192,817);
// by ID
G=gap.SmallGroup(192,817);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,176,1094,268]);
// 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^-1,d*c*d^-1=c^-1>;
// generators/relations