direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C3×C2.Q32, C6.6Q32, Q16⋊1C12, C24.89D4, C6.10SD32, C12.31SD16, C8.8(C2×C12), (C2×C48).3C2, (C2×C16).1C6, (C3×Q16)⋊7C4, (C2×C6).50D8, C8.15(C3×D4), C2.1(C3×Q32), C2.D8.1C6, C24.54(C2×C4), (C2×Q16).1C6, (C6×Q16).7C2, C2.2(C3×SD32), C4.2(C3×SD16), C22.9(C3×D8), (C2×C12).406D4, C6.38(D4⋊C4), C12.70(C22⋊C4), (C2×C24).395C22, (C2×C8).72(C2×C6), (C2×C4).60(C3×D4), C4.2(C3×C22⋊C4), (C3×C2.D8).8C2, C2.7(C3×D4⋊C4), SmallGroup(192,164)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C2.Q32
G = < a,b,c,d | a3=b2=c16=1, d2=c8, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=bc-1 >
Subgroups: 114 in 58 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, C12, C12, C2×C6, C16, C4⋊C4, C2×C8, Q16, Q16, C2×Q8, C24, C2×C12, C2×C12, C3×Q8, C2.D8, C2×C16, C2×Q16, C48, C3×C4⋊C4, C2×C24, C3×Q16, C3×Q16, C6×Q8, C2.Q32, C3×C2.D8, C2×C48, C6×Q16, C3×C2.Q32
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, D8, SD16, C2×C12, C3×D4, D4⋊C4, SD32, Q32, C3×C22⋊C4, C3×D8, C3×SD16, C2.Q32, C3×D4⋊C4, C3×SD32, C3×Q32, C3×C2.Q32
►Regular action on 192 points
Generators in S192
(1 96 113)(2 81 114)(3 82 115)(4 83 116)(5 84 117)(6 85 118)(7 86 119)(8 87 120)(9 88 121)(10 89 122)(11 90 123)(12 91 124)(13 92 125)(14 93 126)(15 94 127)(16 95 128)(17 147 177)(18 148 178)(19 149 179)(20 150 180)(21 151 181)(22 152 182)(23 153 183)(24 154 184)(25 155 185)(26 156 186)(27 157 187)(28 158 188)(29 159 189)(30 160 190)(31 145 191)(32 146 192)(33 59 170)(34 60 171)(35 61 172)(36 62 173)(37 63 174)(38 64 175)(39 49 176)(40 50 161)(41 51 162)(42 52 163)(43 53 164)(44 54 165)(45 55 166)(46 56 167)(47 57 168)(48 58 169)(65 102 129)(66 103 130)(67 104 131)(68 105 132)(69 106 133)(70 107 134)(71 108 135)(72 109 136)(73 110 137)(74 111 138)(75 112 139)(76 97 140)(77 98 141)(78 99 142)(79 100 143)(80 101 144)
(1 159)(2 160)(3 145)(4 146)(5 147)(6 148)(7 149)(8 150)(9 151)(10 152)(11 153)(12 154)(13 155)(14 156)(15 157)(16 158)(17 117)(18 118)(19 119)(20 120)(21 121)(22 122)(23 123)(24 124)(25 125)(26 126)(27 127)(28 128)(29 113)(30 114)(31 115)(32 116)(33 78)(34 79)(35 80)(36 65)(37 66)(38 67)(39 68)(40 69)(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(57 97)(58 98)(59 99)(60 100)(61 101)(62 102)(63 103)(64 104)(81 190)(82 191)(83 192)(84 177)(85 178)(86 179)(87 180)(88 181)(89 182)(90 183)(91 184)(92 185)(93 186)(94 187)(95 188)(96 189)(129 173)(130 174)(131 175)(132 176)(133 161)(134 162)(135 163)(136 164)(137 165)(138 166)(139 167)(140 168)(141 169)(142 170)(143 171)(144 172)
(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 59 9 51)(2 98 10 106)(3 57 11 49)(4 112 12 104)(5 55 13 63)(6 110 14 102)(7 53 15 61)(8 108 16 100)(17 74 25 66)(18 44 26 36)(19 72 27 80)(20 42 28 34)(21 70 29 78)(22 40 30 48)(23 68 31 76)(24 38 32 46)(33 121 41 113)(35 119 43 127)(37 117 45 125)(39 115 47 123)(50 160 58 152)(52 158 60 150)(54 156 62 148)(56 154 64 146)(65 118 73 126)(67 116 75 124)(69 114 77 122)(71 128 79 120)(81 141 89 133)(82 168 90 176)(83 139 91 131)(84 166 92 174)(85 137 93 129)(86 164 94 172)(87 135 95 143)(88 162 96 170)(97 153 105 145)(99 151 107 159)(101 149 109 157)(103 147 111 155)(130 177 138 185)(132 191 140 183)(134 189 142 181)(136 187 144 179)(161 190 169 182)(163 188 171 180)(165 186 173 178)(167 184 175 192)
G:=sub<Sym(192)| (1,96,113)(2,81,114)(3,82,115)(4,83,116)(5,84,117)(6,85,118)(7,86,119)(8,87,120)(9,88,121)(10,89,122)(11,90,123)(12,91,124)(13,92,125)(14,93,126)(15,94,127)(16,95,128)(17,147,177)(18,148,178)(19,149,179)(20,150,180)(21,151,181)(22,152,182)(23,153,183)(24,154,184)(25,155,185)(26,156,186)(27,157,187)(28,158,188)(29,159,189)(30,160,190)(31,145,191)(32,146,192)(33,59,170)(34,60,171)(35,61,172)(36,62,173)(37,63,174)(38,64,175)(39,49,176)(40,50,161)(41,51,162)(42,52,163)(43,53,164)(44,54,165)(45,55,166)(46,56,167)(47,57,168)(48,58,169)(65,102,129)(66,103,130)(67,104,131)(68,105,132)(69,106,133)(70,107,134)(71,108,135)(72,109,136)(73,110,137)(74,111,138)(75,112,139)(76,97,140)(77,98,141)(78,99,142)(79,100,143)(80,101,144), (1,159)(2,160)(3,145)(4,146)(5,147)(6,148)(7,149)(8,150)(9,151)(10,152)(11,153)(12,154)(13,155)(14,156)(15,157)(16,158)(17,117)(18,118)(19,119)(20,120)(21,121)(22,122)(23,123)(24,124)(25,125)(26,126)(27,127)(28,128)(29,113)(30,114)(31,115)(32,116)(33,78)(34,79)(35,80)(36,65)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,97)(58,98)(59,99)(60,100)(61,101)(62,102)(63,103)(64,104)(81,190)(82,191)(83,192)(84,177)(85,178)(86,179)(87,180)(88,181)(89,182)(90,183)(91,184)(92,185)(93,186)(94,187)(95,188)(96,189)(129,173)(130,174)(131,175)(132,176)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(141,169)(142,170)(143,171)(144,172), (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), (1,59,9,51)(2,98,10,106)(3,57,11,49)(4,112,12,104)(5,55,13,63)(6,110,14,102)(7,53,15,61)(8,108,16,100)(17,74,25,66)(18,44,26,36)(19,72,27,80)(20,42,28,34)(21,70,29,78)(22,40,30,48)(23,68,31,76)(24,38,32,46)(33,121,41,113)(35,119,43,127)(37,117,45,125)(39,115,47,123)(50,160,58,152)(52,158,60,150)(54,156,62,148)(56,154,64,146)(65,118,73,126)(67,116,75,124)(69,114,77,122)(71,128,79,120)(81,141,89,133)(82,168,90,176)(83,139,91,131)(84,166,92,174)(85,137,93,129)(86,164,94,172)(87,135,95,143)(88,162,96,170)(97,153,105,145)(99,151,107,159)(101,149,109,157)(103,147,111,155)(130,177,138,185)(132,191,140,183)(134,189,142,181)(136,187,144,179)(161,190,169,182)(163,188,171,180)(165,186,173,178)(167,184,175,192)>;
G:=Group( (1,96,113)(2,81,114)(3,82,115)(4,83,116)(5,84,117)(6,85,118)(7,86,119)(8,87,120)(9,88,121)(10,89,122)(11,90,123)(12,91,124)(13,92,125)(14,93,126)(15,94,127)(16,95,128)(17,147,177)(18,148,178)(19,149,179)(20,150,180)(21,151,181)(22,152,182)(23,153,183)(24,154,184)(25,155,185)(26,156,186)(27,157,187)(28,158,188)(29,159,189)(30,160,190)(31,145,191)(32,146,192)(33,59,170)(34,60,171)(35,61,172)(36,62,173)(37,63,174)(38,64,175)(39,49,176)(40,50,161)(41,51,162)(42,52,163)(43,53,164)(44,54,165)(45,55,166)(46,56,167)(47,57,168)(48,58,169)(65,102,129)(66,103,130)(67,104,131)(68,105,132)(69,106,133)(70,107,134)(71,108,135)(72,109,136)(73,110,137)(74,111,138)(75,112,139)(76,97,140)(77,98,141)(78,99,142)(79,100,143)(80,101,144), (1,159)(2,160)(3,145)(4,146)(5,147)(6,148)(7,149)(8,150)(9,151)(10,152)(11,153)(12,154)(13,155)(14,156)(15,157)(16,158)(17,117)(18,118)(19,119)(20,120)(21,121)(22,122)(23,123)(24,124)(25,125)(26,126)(27,127)(28,128)(29,113)(30,114)(31,115)(32,116)(33,78)(34,79)(35,80)(36,65)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,97)(58,98)(59,99)(60,100)(61,101)(62,102)(63,103)(64,104)(81,190)(82,191)(83,192)(84,177)(85,178)(86,179)(87,180)(88,181)(89,182)(90,183)(91,184)(92,185)(93,186)(94,187)(95,188)(96,189)(129,173)(130,174)(131,175)(132,176)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(141,169)(142,170)(143,171)(144,172), (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), (1,59,9,51)(2,98,10,106)(3,57,11,49)(4,112,12,104)(5,55,13,63)(6,110,14,102)(7,53,15,61)(8,108,16,100)(17,74,25,66)(18,44,26,36)(19,72,27,80)(20,42,28,34)(21,70,29,78)(22,40,30,48)(23,68,31,76)(24,38,32,46)(33,121,41,113)(35,119,43,127)(37,117,45,125)(39,115,47,123)(50,160,58,152)(52,158,60,150)(54,156,62,148)(56,154,64,146)(65,118,73,126)(67,116,75,124)(69,114,77,122)(71,128,79,120)(81,141,89,133)(82,168,90,176)(83,139,91,131)(84,166,92,174)(85,137,93,129)(86,164,94,172)(87,135,95,143)(88,162,96,170)(97,153,105,145)(99,151,107,159)(101,149,109,157)(103,147,111,155)(130,177,138,185)(132,191,140,183)(134,189,142,181)(136,187,144,179)(161,190,169,182)(163,188,171,180)(165,186,173,178)(167,184,175,192) );
G=PermutationGroup([[(1,96,113),(2,81,114),(3,82,115),(4,83,116),(5,84,117),(6,85,118),(7,86,119),(8,87,120),(9,88,121),(10,89,122),(11,90,123),(12,91,124),(13,92,125),(14,93,126),(15,94,127),(16,95,128),(17,147,177),(18,148,178),(19,149,179),(20,150,180),(21,151,181),(22,152,182),(23,153,183),(24,154,184),(25,155,185),(26,156,186),(27,157,187),(28,158,188),(29,159,189),(30,160,190),(31,145,191),(32,146,192),(33,59,170),(34,60,171),(35,61,172),(36,62,173),(37,63,174),(38,64,175),(39,49,176),(40,50,161),(41,51,162),(42,52,163),(43,53,164),(44,54,165),(45,55,166),(46,56,167),(47,57,168),(48,58,169),(65,102,129),(66,103,130),(67,104,131),(68,105,132),(69,106,133),(70,107,134),(71,108,135),(72,109,136),(73,110,137),(74,111,138),(75,112,139),(76,97,140),(77,98,141),(78,99,142),(79,100,143),(80,101,144)], [(1,159),(2,160),(3,145),(4,146),(5,147),(6,148),(7,149),(8,150),(9,151),(10,152),(11,153),(12,154),(13,155),(14,156),(15,157),(16,158),(17,117),(18,118),(19,119),(20,120),(21,121),(22,122),(23,123),(24,124),(25,125),(26,126),(27,127),(28,128),(29,113),(30,114),(31,115),(32,116),(33,78),(34,79),(35,80),(36,65),(37,66),(38,67),(39,68),(40,69),(41,70),(42,71),(43,72),(44,73),(45,74),(46,75),(47,76),(48,77),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112),(57,97),(58,98),(59,99),(60,100),(61,101),(62,102),(63,103),(64,104),(81,190),(82,191),(83,192),(84,177),(85,178),(86,179),(87,180),(88,181),(89,182),(90,183),(91,184),(92,185),(93,186),(94,187),(95,188),(96,189),(129,173),(130,174),(131,175),(132,176),(133,161),(134,162),(135,163),(136,164),(137,165),(138,166),(139,167),(140,168),(141,169),(142,170),(143,171),(144,172)], [(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)], [(1,59,9,51),(2,98,10,106),(3,57,11,49),(4,112,12,104),(5,55,13,63),(6,110,14,102),(7,53,15,61),(8,108,16,100),(17,74,25,66),(18,44,26,36),(19,72,27,80),(20,42,28,34),(21,70,29,78),(22,40,30,48),(23,68,31,76),(24,38,32,46),(33,121,41,113),(35,119,43,127),(37,117,45,125),(39,115,47,123),(50,160,58,152),(52,158,60,150),(54,156,62,148),(56,154,64,146),(65,118,73,126),(67,116,75,124),(69,114,77,122),(71,128,79,120),(81,141,89,133),(82,168,90,176),(83,139,91,131),(84,166,92,174),(85,137,93,129),(86,164,94,172),(87,135,95,143),(88,162,96,170),(97,153,105,145),(99,151,107,159),(101,149,109,157),(103,147,111,155),(130,177,138,185),(132,191,140,183),(134,189,142,181),(136,187,144,179),(161,190,169,182),(163,188,171,180),(165,186,173,178),(167,184,175,192)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 16A | ··· | 16H | 24A | ··· | 24H | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
66 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 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | D4 | SD16 | D8 | C3×D4 | C3×D4 | SD32 | Q32 | C3×SD16 | C3×D8 | C3×SD32 | C3×Q32 |
| kernel | C3×C2.Q32 | C3×C2.D8 | C2×C48 | C6×Q16 | C2.Q32 | C3×Q16 | C2.D8 | C2×C16 | C2×Q16 | Q16 | C24 | C2×C12 | C12 | C2×C6 | C8 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 |
Matrix representation of C3×C2.Q32 ►in GL4(𝔽97) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 61 | 0 |
| 0 | 0 | 0 | 61 |
| 96 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 22 | 53 | 0 | 0 |
| 22 | 75 | 0 | 0 |
| 0 | 0 | 71 | 2 |
| 0 | 0 | 95 | 71 |
| 1 | 95 | 0 | 0 |
| 0 | 96 | 0 | 0 |
| 0 | 0 | 1 | 17 |
| 0 | 0 | 17 | 96 |
G:=sub<GL(4,GF(97))| [1,0,0,0,0,1,0,0,0,0,61,0,0,0,0,61],[96,0,0,0,0,96,0,0,0,0,1,0,0,0,0,1],[22,22,0,0,53,75,0,0,0,0,71,95,0,0,2,71],[1,0,0,0,95,96,0,0,0,0,1,17,0,0,17,96] >;
C3×C2.Q32 in GAP, Magma, Sage, TeX
C_3\times C_2.Q_{32} % in TeX
G:=Group("C3xC2.Q32"); // GroupNames label
G:=SmallGroup(192,164);
// by ID
G=gap.SmallGroup(192,164);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,680,1683,850,360,6053,3036,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^16=1,d^2=c^8,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations