direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C4⋊2Q16, C12⋊9Q16, C4⋊C8.6C6, C4⋊2(C3×Q16), C4⋊Q8.4C6, C4.34(C6×D4), Q8.6(C3×D4), C2.5(C6×Q16), (C4×Q8).11C6, (C6×Q16).9C2, (C3×Q8).26D4, (C2×Q16).2C6, C6.52(C2×Q16), C12.395(C2×D4), (C2×C12).324D4, C42.17(C2×C6), (Q8×C12).18C2, Q8⋊C4.2C6, C22.86(C6×D4), C12.344(C4○D4), C6.145(C4⋊D4), (C4×C12).259C22, (C2×C24).185C22, (C2×C12).921C23, (C6×Q8).160C22, C6.135(C8.C22), (C2×C8).4(C2×C6), (C3×C4⋊C8).16C2, C4⋊C4.54(C2×C6), C4.43(C3×C4○D4), (C3×C4⋊Q8).19C2, (C2×C4).130(C3×D4), (C2×C6).642(C2×D4), C2.14(C3×C4⋊D4), (C2×Q8).60(C2×C6), (C2×C4).96(C22×C6), (C3×Q8⋊C4).7C2, C2.10(C3×C8.C22), (C3×C4⋊C4).375C22, SmallGroup(192,895)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4⋊2Q16
G = < a,b,c,d | a3=b4=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 170 in 108 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, C12, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×C8, Q16, C2×Q8, C2×Q8, C24, C2×C12, C2×C12, C3×Q8, C3×Q8, Q8⋊C4, C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C3×Q16, C6×Q8, C6×Q8, C4⋊2Q16, C3×Q8⋊C4, C3×C4⋊C8, Q8×C12, C3×C4⋊Q8, C6×Q16, C3×C4⋊2Q16
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, Q16, C2×D4, C4○D4, C3×D4, C22×C6, C4⋊D4, C2×Q16, C8.C22, C3×Q16, C6×D4, C3×C4○D4, C4⋊2Q16, C3×C4⋊D4, C6×Q16, C3×C8.C22, C3×C4⋊2Q16
►Regular action on 192 points
Generators in S192
(1 23 97)(2 24 98)(3 17 99)(4 18 100)(5 19 101)(6 20 102)(7 21 103)(8 22 104)(9 76 141)(10 77 142)(11 78 143)(12 79 144)(13 80 137)(14 73 138)(15 74 139)(16 75 140)(25 45 110)(26 46 111)(27 47 112)(28 48 105)(29 41 106)(30 42 107)(31 43 108)(32 44 109)(33 131 114)(34 132 115)(35 133 116)(36 134 117)(37 135 118)(38 136 119)(39 129 120)(40 130 113)(49 161 96)(50 162 89)(51 163 90)(52 164 91)(53 165 92)(54 166 93)(55 167 94)(56 168 95)(57 169 177)(58 170 178)(59 171 179)(60 172 180)(61 173 181)(62 174 182)(63 175 183)(64 176 184)(65 124 185)(66 125 186)(67 126 187)(68 127 188)(69 128 189)(70 121 190)(71 122 191)(72 123 192)(81 146 154)(82 147 155)(83 148 156)(84 149 157)(85 150 158)(86 151 159)(87 152 160)(88 145 153)
(1 71 90 135)(2 136 91 72)(3 65 92 129)(4 130 93 66)(5 67 94 131)(6 132 95 68)(7 69 96 133)(8 134 89 70)(9 28 177 157)(10 158 178 29)(11 30 179 159)(12 160 180 31)(13 32 181 153)(14 154 182 25)(15 26 183 155)(16 156 184 27)(17 124 53 120)(18 113 54 125)(19 126 55 114)(20 115 56 127)(21 128 49 116)(22 117 50 121)(23 122 51 118)(24 119 52 123)(33 101 187 167)(34 168 188 102)(35 103 189 161)(36 162 190 104)(37 97 191 163)(38 164 192 98)(39 99 185 165)(40 166 186 100)(41 77 85 58)(42 59 86 78)(43 79 87 60)(44 61 88 80)(45 73 81 62)(46 63 82 74)(47 75 83 64)(48 57 84 76)(105 169 149 141)(106 142 150 170)(107 171 151 143)(108 144 152 172)(109 173 145 137)(110 138 146 174)(111 175 147 139)(112 140 148 176)
(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)
(1 107 5 111)(2 106 6 110)(3 105 7 109)(4 112 8 108)(9 116 13 120)(10 115 14 119)(11 114 15 118)(12 113 16 117)(17 28 21 32)(18 27 22 31)(19 26 23 30)(20 25 24 29)(33 74 37 78)(34 73 38 77)(35 80 39 76)(36 79 40 75)(41 102 45 98)(42 101 46 97)(43 100 47 104)(44 99 48 103)(49 153 53 157)(50 160 54 156)(51 159 55 155)(52 158 56 154)(57 189 61 185)(58 188 62 192)(59 187 63 191)(60 186 64 190)(65 169 69 173)(66 176 70 172)(67 175 71 171)(68 174 72 170)(81 164 85 168)(82 163 86 167)(83 162 87 166)(84 161 88 165)(89 152 93 148)(90 151 94 147)(91 150 95 146)(92 149 96 145)(121 180 125 184)(122 179 126 183)(123 178 127 182)(124 177 128 181)(129 141 133 137)(130 140 134 144)(131 139 135 143)(132 138 136 142)
G:=sub<Sym(192)| (1,23,97)(2,24,98)(3,17,99)(4,18,100)(5,19,101)(6,20,102)(7,21,103)(8,22,104)(9,76,141)(10,77,142)(11,78,143)(12,79,144)(13,80,137)(14,73,138)(15,74,139)(16,75,140)(25,45,110)(26,46,111)(27,47,112)(28,48,105)(29,41,106)(30,42,107)(31,43,108)(32,44,109)(33,131,114)(34,132,115)(35,133,116)(36,134,117)(37,135,118)(38,136,119)(39,129,120)(40,130,113)(49,161,96)(50,162,89)(51,163,90)(52,164,91)(53,165,92)(54,166,93)(55,167,94)(56,168,95)(57,169,177)(58,170,178)(59,171,179)(60,172,180)(61,173,181)(62,174,182)(63,175,183)(64,176,184)(65,124,185)(66,125,186)(67,126,187)(68,127,188)(69,128,189)(70,121,190)(71,122,191)(72,123,192)(81,146,154)(82,147,155)(83,148,156)(84,149,157)(85,150,158)(86,151,159)(87,152,160)(88,145,153), (1,71,90,135)(2,136,91,72)(3,65,92,129)(4,130,93,66)(5,67,94,131)(6,132,95,68)(7,69,96,133)(8,134,89,70)(9,28,177,157)(10,158,178,29)(11,30,179,159)(12,160,180,31)(13,32,181,153)(14,154,182,25)(15,26,183,155)(16,156,184,27)(17,124,53,120)(18,113,54,125)(19,126,55,114)(20,115,56,127)(21,128,49,116)(22,117,50,121)(23,122,51,118)(24,119,52,123)(33,101,187,167)(34,168,188,102)(35,103,189,161)(36,162,190,104)(37,97,191,163)(38,164,192,98)(39,99,185,165)(40,166,186,100)(41,77,85,58)(42,59,86,78)(43,79,87,60)(44,61,88,80)(45,73,81,62)(46,63,82,74)(47,75,83,64)(48,57,84,76)(105,169,149,141)(106,142,150,170)(107,171,151,143)(108,144,152,172)(109,173,145,137)(110,138,146,174)(111,175,147,139)(112,140,148,176), (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), (1,107,5,111)(2,106,6,110)(3,105,7,109)(4,112,8,108)(9,116,13,120)(10,115,14,119)(11,114,15,118)(12,113,16,117)(17,28,21,32)(18,27,22,31)(19,26,23,30)(20,25,24,29)(33,74,37,78)(34,73,38,77)(35,80,39,76)(36,79,40,75)(41,102,45,98)(42,101,46,97)(43,100,47,104)(44,99,48,103)(49,153,53,157)(50,160,54,156)(51,159,55,155)(52,158,56,154)(57,189,61,185)(58,188,62,192)(59,187,63,191)(60,186,64,190)(65,169,69,173)(66,176,70,172)(67,175,71,171)(68,174,72,170)(81,164,85,168)(82,163,86,167)(83,162,87,166)(84,161,88,165)(89,152,93,148)(90,151,94,147)(91,150,95,146)(92,149,96,145)(121,180,125,184)(122,179,126,183)(123,178,127,182)(124,177,128,181)(129,141,133,137)(130,140,134,144)(131,139,135,143)(132,138,136,142)>;
G:=Group( (1,23,97)(2,24,98)(3,17,99)(4,18,100)(5,19,101)(6,20,102)(7,21,103)(8,22,104)(9,76,141)(10,77,142)(11,78,143)(12,79,144)(13,80,137)(14,73,138)(15,74,139)(16,75,140)(25,45,110)(26,46,111)(27,47,112)(28,48,105)(29,41,106)(30,42,107)(31,43,108)(32,44,109)(33,131,114)(34,132,115)(35,133,116)(36,134,117)(37,135,118)(38,136,119)(39,129,120)(40,130,113)(49,161,96)(50,162,89)(51,163,90)(52,164,91)(53,165,92)(54,166,93)(55,167,94)(56,168,95)(57,169,177)(58,170,178)(59,171,179)(60,172,180)(61,173,181)(62,174,182)(63,175,183)(64,176,184)(65,124,185)(66,125,186)(67,126,187)(68,127,188)(69,128,189)(70,121,190)(71,122,191)(72,123,192)(81,146,154)(82,147,155)(83,148,156)(84,149,157)(85,150,158)(86,151,159)(87,152,160)(88,145,153), (1,71,90,135)(2,136,91,72)(3,65,92,129)(4,130,93,66)(5,67,94,131)(6,132,95,68)(7,69,96,133)(8,134,89,70)(9,28,177,157)(10,158,178,29)(11,30,179,159)(12,160,180,31)(13,32,181,153)(14,154,182,25)(15,26,183,155)(16,156,184,27)(17,124,53,120)(18,113,54,125)(19,126,55,114)(20,115,56,127)(21,128,49,116)(22,117,50,121)(23,122,51,118)(24,119,52,123)(33,101,187,167)(34,168,188,102)(35,103,189,161)(36,162,190,104)(37,97,191,163)(38,164,192,98)(39,99,185,165)(40,166,186,100)(41,77,85,58)(42,59,86,78)(43,79,87,60)(44,61,88,80)(45,73,81,62)(46,63,82,74)(47,75,83,64)(48,57,84,76)(105,169,149,141)(106,142,150,170)(107,171,151,143)(108,144,152,172)(109,173,145,137)(110,138,146,174)(111,175,147,139)(112,140,148,176), (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), (1,107,5,111)(2,106,6,110)(3,105,7,109)(4,112,8,108)(9,116,13,120)(10,115,14,119)(11,114,15,118)(12,113,16,117)(17,28,21,32)(18,27,22,31)(19,26,23,30)(20,25,24,29)(33,74,37,78)(34,73,38,77)(35,80,39,76)(36,79,40,75)(41,102,45,98)(42,101,46,97)(43,100,47,104)(44,99,48,103)(49,153,53,157)(50,160,54,156)(51,159,55,155)(52,158,56,154)(57,189,61,185)(58,188,62,192)(59,187,63,191)(60,186,64,190)(65,169,69,173)(66,176,70,172)(67,175,71,171)(68,174,72,170)(81,164,85,168)(82,163,86,167)(83,162,87,166)(84,161,88,165)(89,152,93,148)(90,151,94,147)(91,150,95,146)(92,149,96,145)(121,180,125,184)(122,179,126,183)(123,178,127,182)(124,177,128,181)(129,141,133,137)(130,140,134,144)(131,139,135,143)(132,138,136,142) );
G=PermutationGroup([[(1,23,97),(2,24,98),(3,17,99),(4,18,100),(5,19,101),(6,20,102),(7,21,103),(8,22,104),(9,76,141),(10,77,142),(11,78,143),(12,79,144),(13,80,137),(14,73,138),(15,74,139),(16,75,140),(25,45,110),(26,46,111),(27,47,112),(28,48,105),(29,41,106),(30,42,107),(31,43,108),(32,44,109),(33,131,114),(34,132,115),(35,133,116),(36,134,117),(37,135,118),(38,136,119),(39,129,120),(40,130,113),(49,161,96),(50,162,89),(51,163,90),(52,164,91),(53,165,92),(54,166,93),(55,167,94),(56,168,95),(57,169,177),(58,170,178),(59,171,179),(60,172,180),(61,173,181),(62,174,182),(63,175,183),(64,176,184),(65,124,185),(66,125,186),(67,126,187),(68,127,188),(69,128,189),(70,121,190),(71,122,191),(72,123,192),(81,146,154),(82,147,155),(83,148,156),(84,149,157),(85,150,158),(86,151,159),(87,152,160),(88,145,153)], [(1,71,90,135),(2,136,91,72),(3,65,92,129),(4,130,93,66),(5,67,94,131),(6,132,95,68),(7,69,96,133),(8,134,89,70),(9,28,177,157),(10,158,178,29),(11,30,179,159),(12,160,180,31),(13,32,181,153),(14,154,182,25),(15,26,183,155),(16,156,184,27),(17,124,53,120),(18,113,54,125),(19,126,55,114),(20,115,56,127),(21,128,49,116),(22,117,50,121),(23,122,51,118),(24,119,52,123),(33,101,187,167),(34,168,188,102),(35,103,189,161),(36,162,190,104),(37,97,191,163),(38,164,192,98),(39,99,185,165),(40,166,186,100),(41,77,85,58),(42,59,86,78),(43,79,87,60),(44,61,88,80),(45,73,81,62),(46,63,82,74),(47,75,83,64),(48,57,84,76),(105,169,149,141),(106,142,150,170),(107,171,151,143),(108,144,152,172),(109,173,145,137),(110,138,146,174),(111,175,147,139),(112,140,148,176)], [(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)], [(1,107,5,111),(2,106,6,110),(3,105,7,109),(4,112,8,108),(9,116,13,120),(10,115,14,119),(11,114,15,118),(12,113,16,117),(17,28,21,32),(18,27,22,31),(19,26,23,30),(20,25,24,29),(33,74,37,78),(34,73,38,77),(35,80,39,76),(36,79,40,75),(41,102,45,98),(42,101,46,97),(43,100,47,104),(44,99,48,103),(49,153,53,157),(50,160,54,156),(51,159,55,155),(52,158,56,154),(57,189,61,185),(58,188,62,192),(59,187,63,191),(60,186,64,190),(65,169,69,173),(66,176,70,172),(67,175,71,171),(68,174,72,170),(81,164,85,168),(82,163,86,167),(83,162,87,166),(84,161,88,165),(89,152,93,148),(90,151,94,147),(91,150,95,146),(92,149,96,145),(121,180,125,184),(122,179,126,183),(123,178,127,182),(124,177,128,181),(129,141,133,137),(130,140,134,144),(131,139,135,143),(132,138,136,142)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 6A | ··· | 6F | 8A | 8B | 8C | 8D | 12A | ··· | 12H | 12I | ··· | 12R | 12S | 12T | 12U | 12V | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | Q16 | C4○D4 | C3×D4 | C3×D4 | C3×Q16 | C3×C4○D4 | C8.C22 | C3×C8.C22 |
| kernel | C3×C4⋊2Q16 | C3×Q8⋊C4 | C3×C4⋊C8 | Q8×C12 | C3×C4⋊Q8 | C6×Q16 | C4⋊2Q16 | Q8⋊C4 | C4⋊C8 | C4×Q8 | C4⋊Q8 | C2×Q16 | C2×C12 | C3×Q8 | C12 | C12 | C2×C4 | Q8 | C4 | C4 | C6 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 4 | 1 | 2 |
Matrix representation of C3×C4⋊2Q16 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 46 | 54 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 6 | 0 | 0 |
| 6 | 67 | 0 | 0 |
| 0 | 0 | 16 | 57 |
| 0 | 0 | 16 | 16 |
| 1 | 71 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 50 | 45 |
| 0 | 0 | 45 | 23 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,64,0,0,0,0,64],[46,0,0,0,54,27,0,0,0,0,1,0,0,0,0,1],[6,6,0,0,6,67,0,0,0,0,16,16,0,0,57,16],[1,0,0,0,71,72,0,0,0,0,50,45,0,0,45,23] >;
C3×C4⋊2Q16 in GAP, Magma, Sage, TeX
C_3\times C_4\rtimes_2Q_{16} % in TeX
G:=Group("C3xC4:2Q16"); // GroupNames label
G:=SmallGroup(192,895);
// by ID
G=gap.SmallGroup(192,895);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,176,1094,520,6053,1531,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations