direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C4.SD16, C12.20Q16, C12.33SD16, (C4×C8).4C6, C4⋊Q8.7C6, (C4×C24).7C2, C2.9(C6×Q16), C4.3(C3×Q16), C6.56(C2×Q16), C4.4(C3×SD16), (C2×C12).420D4, C42.77(C2×C6), Q8⋊C4.1C6, C6.95(C2×SD16), C2.15(C6×SD16), C22.108(C6×D4), C12.269(C4○D4), (C4×C12).361C22, (C2×C24).367C22, (C2×C12).943C23, C6.72(C4.4D4), (C6×Q8).172C22, C4⋊C4.18(C2×C6), (C2×C8).69(C2×C6), C4.14(C3×C4○D4), (C2×C4).76(C3×D4), (C3×C4⋊Q8).22C2, (C2×C6).664(C2×D4), (C2×Q8).17(C2×C6), (C3×Q8⋊C4).1C2, C2.10(C3×C4.4D4), (C3×C4⋊C4).238C22, (C2×C4).118(C22×C6), SmallGroup(192,920)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4.SD16
G = < a,b,c,d | a3=b4=c8=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c3 >
Subgroups: 162 in 98 conjugacy classes, 58 normal (26 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C2×Q8, C24, C2×C12, C2×C12, C3×Q8, C4×C8, Q8⋊C4, C4⋊Q8, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C6×Q8, C6×Q8, C4.SD16, C4×C24, C3×Q8⋊C4, C3×C4⋊Q8, C3×C4.SD16
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, SD16, Q16, C2×D4, C4○D4, C3×D4, C22×C6, C4.4D4, C2×SD16, C2×Q16, C3×SD16, C3×Q16, C6×D4, C3×C4○D4, C4.SD16, C3×C4.4D4, C6×SD16, C6×Q16, C3×C4.SD16
►Regular action on 192 points
Generators in S192
(1 83 39)(2 84 40)(3 85 33)(4 86 34)(5 87 35)(6 88 36)(7 81 37)(8 82 38)(9 25 63)(10 26 64)(11 27 57)(12 28 58)(13 29 59)(14 30 60)(15 31 61)(16 32 62)(17 74 69)(18 75 70)(19 76 71)(20 77 72)(21 78 65)(22 79 66)(23 80 67)(24 73 68)(41 101 93)(42 102 94)(43 103 95)(44 104 96)(45 97 89)(46 98 90)(47 99 91)(48 100 92)(49 186 138)(50 187 139)(51 188 140)(52 189 141)(53 190 142)(54 191 143)(55 192 144)(56 185 137)(105 118 155)(106 119 156)(107 120 157)(108 113 158)(109 114 159)(110 115 160)(111 116 153)(112 117 154)(121 161 151)(122 162 152)(123 163 145)(124 164 146)(125 165 147)(126 166 148)(127 167 149)(128 168 150)(129 169 177)(130 170 178)(131 171 179)(132 172 180)(133 173 181)(134 174 182)(135 175 183)(136 176 184)
(1 42 11 23)(2 43 12 24)(3 44 13 17)(4 45 14 18)(5 46 15 19)(6 47 16 20)(7 48 9 21)(8 41 10 22)(25 78 81 100)(26 79 82 101)(27 80 83 102)(28 73 84 103)(29 74 85 104)(30 75 86 97)(31 76 87 98)(32 77 88 99)(33 96 59 69)(34 89 60 70)(35 90 61 71)(36 91 62 72)(37 92 63 65)(38 93 64 66)(39 94 57 67)(40 95 58 68)(49 147 117 177)(50 148 118 178)(51 149 119 179)(52 150 120 180)(53 151 113 181)(54 152 114 182)(55 145 115 183)(56 146 116 184)(105 170 139 166)(106 171 140 167)(107 172 141 168)(108 173 142 161)(109 174 143 162)(110 175 144 163)(111 176 137 164)(112 169 138 165)(121 158 133 190)(122 159 134 191)(123 160 135 192)(124 153 136 185)(125 154 129 186)(126 155 130 187)(127 156 131 188)(128 157 132 189)
(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)
(1 166 11 170)(2 173 12 161)(3 164 13 176)(4 171 14 167)(5 162 15 174)(6 169 16 165)(7 168 9 172)(8 175 10 163)(17 111 44 137)(18 140 45 106)(19 109 46 143)(20 138 47 112)(21 107 48 141)(22 144 41 110)(23 105 42 139)(24 142 43 108)(25 180 81 150)(26 145 82 183)(27 178 83 148)(28 151 84 181)(29 184 85 146)(30 149 86 179)(31 182 87 152)(32 147 88 177)(33 124 59 136)(34 131 60 127)(35 122 61 134)(36 129 62 125)(37 128 63 132)(38 135 64 123)(39 126 57 130)(40 133 58 121)(49 99 117 77)(50 80 118 102)(51 97 119 75)(52 78 120 100)(53 103 113 73)(54 76 114 98)(55 101 115 79)(56 74 116 104)(65 157 92 189)(66 192 93 160)(67 155 94 187)(68 190 95 158)(69 153 96 185)(70 188 89 156)(71 159 90 191)(72 186 91 154)
G:=sub<Sym(192)| (1,83,39)(2,84,40)(3,85,33)(4,86,34)(5,87,35)(6,88,36)(7,81,37)(8,82,38)(9,25,63)(10,26,64)(11,27,57)(12,28,58)(13,29,59)(14,30,60)(15,31,61)(16,32,62)(17,74,69)(18,75,70)(19,76,71)(20,77,72)(21,78,65)(22,79,66)(23,80,67)(24,73,68)(41,101,93)(42,102,94)(43,103,95)(44,104,96)(45,97,89)(46,98,90)(47,99,91)(48,100,92)(49,186,138)(50,187,139)(51,188,140)(52,189,141)(53,190,142)(54,191,143)(55,192,144)(56,185,137)(105,118,155)(106,119,156)(107,120,157)(108,113,158)(109,114,159)(110,115,160)(111,116,153)(112,117,154)(121,161,151)(122,162,152)(123,163,145)(124,164,146)(125,165,147)(126,166,148)(127,167,149)(128,168,150)(129,169,177)(130,170,178)(131,171,179)(132,172,180)(133,173,181)(134,174,182)(135,175,183)(136,176,184), (1,42,11,23)(2,43,12,24)(3,44,13,17)(4,45,14,18)(5,46,15,19)(6,47,16,20)(7,48,9,21)(8,41,10,22)(25,78,81,100)(26,79,82,101)(27,80,83,102)(28,73,84,103)(29,74,85,104)(30,75,86,97)(31,76,87,98)(32,77,88,99)(33,96,59,69)(34,89,60,70)(35,90,61,71)(36,91,62,72)(37,92,63,65)(38,93,64,66)(39,94,57,67)(40,95,58,68)(49,147,117,177)(50,148,118,178)(51,149,119,179)(52,150,120,180)(53,151,113,181)(54,152,114,182)(55,145,115,183)(56,146,116,184)(105,170,139,166)(106,171,140,167)(107,172,141,168)(108,173,142,161)(109,174,143,162)(110,175,144,163)(111,176,137,164)(112,169,138,165)(121,158,133,190)(122,159,134,191)(123,160,135,192)(124,153,136,185)(125,154,129,186)(126,155,130,187)(127,156,131,188)(128,157,132,189), (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), (1,166,11,170)(2,173,12,161)(3,164,13,176)(4,171,14,167)(5,162,15,174)(6,169,16,165)(7,168,9,172)(8,175,10,163)(17,111,44,137)(18,140,45,106)(19,109,46,143)(20,138,47,112)(21,107,48,141)(22,144,41,110)(23,105,42,139)(24,142,43,108)(25,180,81,150)(26,145,82,183)(27,178,83,148)(28,151,84,181)(29,184,85,146)(30,149,86,179)(31,182,87,152)(32,147,88,177)(33,124,59,136)(34,131,60,127)(35,122,61,134)(36,129,62,125)(37,128,63,132)(38,135,64,123)(39,126,57,130)(40,133,58,121)(49,99,117,77)(50,80,118,102)(51,97,119,75)(52,78,120,100)(53,103,113,73)(54,76,114,98)(55,101,115,79)(56,74,116,104)(65,157,92,189)(66,192,93,160)(67,155,94,187)(68,190,95,158)(69,153,96,185)(70,188,89,156)(71,159,90,191)(72,186,91,154)>;
G:=Group( (1,83,39)(2,84,40)(3,85,33)(4,86,34)(5,87,35)(6,88,36)(7,81,37)(8,82,38)(9,25,63)(10,26,64)(11,27,57)(12,28,58)(13,29,59)(14,30,60)(15,31,61)(16,32,62)(17,74,69)(18,75,70)(19,76,71)(20,77,72)(21,78,65)(22,79,66)(23,80,67)(24,73,68)(41,101,93)(42,102,94)(43,103,95)(44,104,96)(45,97,89)(46,98,90)(47,99,91)(48,100,92)(49,186,138)(50,187,139)(51,188,140)(52,189,141)(53,190,142)(54,191,143)(55,192,144)(56,185,137)(105,118,155)(106,119,156)(107,120,157)(108,113,158)(109,114,159)(110,115,160)(111,116,153)(112,117,154)(121,161,151)(122,162,152)(123,163,145)(124,164,146)(125,165,147)(126,166,148)(127,167,149)(128,168,150)(129,169,177)(130,170,178)(131,171,179)(132,172,180)(133,173,181)(134,174,182)(135,175,183)(136,176,184), (1,42,11,23)(2,43,12,24)(3,44,13,17)(4,45,14,18)(5,46,15,19)(6,47,16,20)(7,48,9,21)(8,41,10,22)(25,78,81,100)(26,79,82,101)(27,80,83,102)(28,73,84,103)(29,74,85,104)(30,75,86,97)(31,76,87,98)(32,77,88,99)(33,96,59,69)(34,89,60,70)(35,90,61,71)(36,91,62,72)(37,92,63,65)(38,93,64,66)(39,94,57,67)(40,95,58,68)(49,147,117,177)(50,148,118,178)(51,149,119,179)(52,150,120,180)(53,151,113,181)(54,152,114,182)(55,145,115,183)(56,146,116,184)(105,170,139,166)(106,171,140,167)(107,172,141,168)(108,173,142,161)(109,174,143,162)(110,175,144,163)(111,176,137,164)(112,169,138,165)(121,158,133,190)(122,159,134,191)(123,160,135,192)(124,153,136,185)(125,154,129,186)(126,155,130,187)(127,156,131,188)(128,157,132,189), (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), (1,166,11,170)(2,173,12,161)(3,164,13,176)(4,171,14,167)(5,162,15,174)(6,169,16,165)(7,168,9,172)(8,175,10,163)(17,111,44,137)(18,140,45,106)(19,109,46,143)(20,138,47,112)(21,107,48,141)(22,144,41,110)(23,105,42,139)(24,142,43,108)(25,180,81,150)(26,145,82,183)(27,178,83,148)(28,151,84,181)(29,184,85,146)(30,149,86,179)(31,182,87,152)(32,147,88,177)(33,124,59,136)(34,131,60,127)(35,122,61,134)(36,129,62,125)(37,128,63,132)(38,135,64,123)(39,126,57,130)(40,133,58,121)(49,99,117,77)(50,80,118,102)(51,97,119,75)(52,78,120,100)(53,103,113,73)(54,76,114,98)(55,101,115,79)(56,74,116,104)(65,157,92,189)(66,192,93,160)(67,155,94,187)(68,190,95,158)(69,153,96,185)(70,188,89,156)(71,159,90,191)(72,186,91,154) );
G=PermutationGroup([[(1,83,39),(2,84,40),(3,85,33),(4,86,34),(5,87,35),(6,88,36),(7,81,37),(8,82,38),(9,25,63),(10,26,64),(11,27,57),(12,28,58),(13,29,59),(14,30,60),(15,31,61),(16,32,62),(17,74,69),(18,75,70),(19,76,71),(20,77,72),(21,78,65),(22,79,66),(23,80,67),(24,73,68),(41,101,93),(42,102,94),(43,103,95),(44,104,96),(45,97,89),(46,98,90),(47,99,91),(48,100,92),(49,186,138),(50,187,139),(51,188,140),(52,189,141),(53,190,142),(54,191,143),(55,192,144),(56,185,137),(105,118,155),(106,119,156),(107,120,157),(108,113,158),(109,114,159),(110,115,160),(111,116,153),(112,117,154),(121,161,151),(122,162,152),(123,163,145),(124,164,146),(125,165,147),(126,166,148),(127,167,149),(128,168,150),(129,169,177),(130,170,178),(131,171,179),(132,172,180),(133,173,181),(134,174,182),(135,175,183),(136,176,184)], [(1,42,11,23),(2,43,12,24),(3,44,13,17),(4,45,14,18),(5,46,15,19),(6,47,16,20),(7,48,9,21),(8,41,10,22),(25,78,81,100),(26,79,82,101),(27,80,83,102),(28,73,84,103),(29,74,85,104),(30,75,86,97),(31,76,87,98),(32,77,88,99),(33,96,59,69),(34,89,60,70),(35,90,61,71),(36,91,62,72),(37,92,63,65),(38,93,64,66),(39,94,57,67),(40,95,58,68),(49,147,117,177),(50,148,118,178),(51,149,119,179),(52,150,120,180),(53,151,113,181),(54,152,114,182),(55,145,115,183),(56,146,116,184),(105,170,139,166),(106,171,140,167),(107,172,141,168),(108,173,142,161),(109,174,143,162),(110,175,144,163),(111,176,137,164),(112,169,138,165),(121,158,133,190),(122,159,134,191),(123,160,135,192),(124,153,136,185),(125,154,129,186),(126,155,130,187),(127,156,131,188),(128,157,132,189)], [(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)], [(1,166,11,170),(2,173,12,161),(3,164,13,176),(4,171,14,167),(5,162,15,174),(6,169,16,165),(7,168,9,172),(8,175,10,163),(17,111,44,137),(18,140,45,106),(19,109,46,143),(20,138,47,112),(21,107,48,141),(22,144,41,110),(23,105,42,139),(24,142,43,108),(25,180,81,150),(26,145,82,183),(27,178,83,148),(28,151,84,181),(29,184,85,146),(30,149,86,179),(31,182,87,152),(32,147,88,177),(33,124,59,136),(34,131,60,127),(35,122,61,134),(36,129,62,125),(37,128,63,132),(38,135,64,123),(39,126,57,130),(40,133,58,121),(49,99,117,77),(50,80,118,102),(51,97,119,75),(52,78,120,100),(53,103,113,73),(54,76,114,98),(55,101,115,79),(56,74,116,104),(65,157,92,189),(66,192,93,160),(67,155,94,187),(68,190,95,158),(69,153,96,185),(70,188,89,156),(71,159,90,191),(72,186,91,154)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6F | 8A | ··· | 8H | 12A | ··· | 12L | 12M | ··· | 12T | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | SD16 | Q16 | C4○D4 | C3×D4 | C3×SD16 | C3×Q16 | C3×C4○D4 |
| kernel | C3×C4.SD16 | C4×C24 | C3×Q8⋊C4 | C3×C4⋊Q8 | C4.SD16 | C4×C8 | Q8⋊C4 | C4⋊Q8 | C2×C12 | C12 | C12 | C12 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 8 | 4 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
Matrix representation of C3×C4.SD16 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 0 |
| 67 | 67 | 0 | 0 |
| 6 | 67 | 0 | 0 |
| 0 | 0 | 57 | 16 |
| 0 | 0 | 57 | 57 |
| 46 | 41 | 0 | 0 |
| 41 | 27 | 0 | 0 |
| 0 | 0 | 21 | 54 |
| 0 | 0 | 54 | 52 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[72,0,0,0,0,72,0,0,0,0,0,1,0,0,72,0],[67,6,0,0,67,67,0,0,0,0,57,57,0,0,16,57],[46,41,0,0,41,27,0,0,0,0,21,54,0,0,54,52] >;
C3×C4.SD16 in GAP, Magma, Sage, TeX
C_3\times C_4.{\rm SD}_{16} % in TeX
G:=Group("C3xC4.SD16"); // GroupNames label
G:=SmallGroup(192,920);
// by ID
G=gap.SmallGroup(192,920);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,344,1094,142,4204,172,6053,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^8=1,d^2=b^2,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=b^2*c^3>;
// generators/relations