metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.39SD16, C42.191D6, C12.2M4(2), C3⋊C8⋊2C8, C4⋊C8.2S3, C3⋊2(C8⋊2C8), C6.7(C4⋊C8), C4.12(S3×C8), C12.2(C2×C8), (C2×C12).32Q8, C12⋊C8.7C2, C6.1(C4.Q8), C4.8(C8⋊S3), (C2×C12).486D4, (C2×C4).19Dic6, C6.5(C8.C4), C2.4(Dic3⋊C8), C4.14(D4.S3), (C4×C12).38C22, C4.14(Q8⋊2S3), C2.1(C12.Q8), C2.2(C12.53D4), C22.19(Dic3⋊C4), (C4×C3⋊C8).2C2, (C2×C3⋊C8).5C4, (C3×C4⋊C8).2C2, (C2×C6).32(C4⋊C4), (C2×C4).134(C4×S3), (C2×C12).47(C2×C4), (C2×C4).264(C3⋊D4), SmallGroup(192,39)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.39SD16
G = < a,b,c | a12=b8=1, c2=a9, bab-1=a5, ac=ca, cbc-1=b3 >
Smallest permutation representation of C12.39SD16
►Regular action on 192 points
Generators in S192
(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 91 21 51 153 69 113 182)(2 96 22 56 154 62 114 187)(3 89 23 49 155 67 115 192)(4 94 24 54 156 72 116 185)(5 87 13 59 145 65 117 190)(6 92 14 52 146 70 118 183)(7 85 15 57 147 63 119 188)(8 90 16 50 148 68 120 181)(9 95 17 55 149 61 109 186)(10 88 18 60 150 66 110 191)(11 93 19 53 151 71 111 184)(12 86 20 58 152 64 112 189)(25 144 176 97 123 46 165 75)(26 137 177 102 124 39 166 80)(27 142 178 107 125 44 167 73)(28 135 179 100 126 37 168 78)(29 140 180 105 127 42 157 83)(30 133 169 98 128 47 158 76)(31 138 170 103 129 40 159 81)(32 143 171 108 130 45 160 74)(33 136 172 101 131 38 161 79)(34 141 173 106 132 43 162 84)(35 134 174 99 121 48 163 77)(36 139 175 104 122 41 164 82)
(1 81 10 78 7 75 4 84)(2 82 11 79 8 76 5 73)(3 83 12 80 9 77 6 74)(13 44 22 41 19 38 16 47)(14 45 23 42 20 39 17 48)(15 46 24 43 21 40 18 37)(25 54 34 51 31 60 28 57)(26 55 35 52 32 49 29 58)(27 56 36 53 33 50 30 59)(61 163 70 160 67 157 64 166)(62 164 71 161 68 158 65 167)(63 165 72 162 69 159 66 168)(85 176 94 173 91 170 88 179)(86 177 95 174 92 171 89 180)(87 178 96 175 93 172 90 169)(97 156 106 153 103 150 100 147)(98 145 107 154 104 151 101 148)(99 146 108 155 105 152 102 149)(109 134 118 143 115 140 112 137)(110 135 119 144 116 141 113 138)(111 136 120 133 117 142 114 139)(121 183 130 192 127 189 124 186)(122 184 131 181 128 190 125 187)(123 185 132 182 129 191 126 188)
G:=sub<Sym(192)| (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), (1,91,21,51,153,69,113,182)(2,96,22,56,154,62,114,187)(3,89,23,49,155,67,115,192)(4,94,24,54,156,72,116,185)(5,87,13,59,145,65,117,190)(6,92,14,52,146,70,118,183)(7,85,15,57,147,63,119,188)(8,90,16,50,148,68,120,181)(9,95,17,55,149,61,109,186)(10,88,18,60,150,66,110,191)(11,93,19,53,151,71,111,184)(12,86,20,58,152,64,112,189)(25,144,176,97,123,46,165,75)(26,137,177,102,124,39,166,80)(27,142,178,107,125,44,167,73)(28,135,179,100,126,37,168,78)(29,140,180,105,127,42,157,83)(30,133,169,98,128,47,158,76)(31,138,170,103,129,40,159,81)(32,143,171,108,130,45,160,74)(33,136,172,101,131,38,161,79)(34,141,173,106,132,43,162,84)(35,134,174,99,121,48,163,77)(36,139,175,104,122,41,164,82), (1,81,10,78,7,75,4,84)(2,82,11,79,8,76,5,73)(3,83,12,80,9,77,6,74)(13,44,22,41,19,38,16,47)(14,45,23,42,20,39,17,48)(15,46,24,43,21,40,18,37)(25,54,34,51,31,60,28,57)(26,55,35,52,32,49,29,58)(27,56,36,53,33,50,30,59)(61,163,70,160,67,157,64,166)(62,164,71,161,68,158,65,167)(63,165,72,162,69,159,66,168)(85,176,94,173,91,170,88,179)(86,177,95,174,92,171,89,180)(87,178,96,175,93,172,90,169)(97,156,106,153,103,150,100,147)(98,145,107,154,104,151,101,148)(99,146,108,155,105,152,102,149)(109,134,118,143,115,140,112,137)(110,135,119,144,116,141,113,138)(111,136,120,133,117,142,114,139)(121,183,130,192,127,189,124,186)(122,184,131,181,128,190,125,187)(123,185,132,182,129,191,126,188)>;
G:=Group( (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), (1,91,21,51,153,69,113,182)(2,96,22,56,154,62,114,187)(3,89,23,49,155,67,115,192)(4,94,24,54,156,72,116,185)(5,87,13,59,145,65,117,190)(6,92,14,52,146,70,118,183)(7,85,15,57,147,63,119,188)(8,90,16,50,148,68,120,181)(9,95,17,55,149,61,109,186)(10,88,18,60,150,66,110,191)(11,93,19,53,151,71,111,184)(12,86,20,58,152,64,112,189)(25,144,176,97,123,46,165,75)(26,137,177,102,124,39,166,80)(27,142,178,107,125,44,167,73)(28,135,179,100,126,37,168,78)(29,140,180,105,127,42,157,83)(30,133,169,98,128,47,158,76)(31,138,170,103,129,40,159,81)(32,143,171,108,130,45,160,74)(33,136,172,101,131,38,161,79)(34,141,173,106,132,43,162,84)(35,134,174,99,121,48,163,77)(36,139,175,104,122,41,164,82), (1,81,10,78,7,75,4,84)(2,82,11,79,8,76,5,73)(3,83,12,80,9,77,6,74)(13,44,22,41,19,38,16,47)(14,45,23,42,20,39,17,48)(15,46,24,43,21,40,18,37)(25,54,34,51,31,60,28,57)(26,55,35,52,32,49,29,58)(27,56,36,53,33,50,30,59)(61,163,70,160,67,157,64,166)(62,164,71,161,68,158,65,167)(63,165,72,162,69,159,66,168)(85,176,94,173,91,170,88,179)(86,177,95,174,92,171,89,180)(87,178,96,175,93,172,90,169)(97,156,106,153,103,150,100,147)(98,145,107,154,104,151,101,148)(99,146,108,155,105,152,102,149)(109,134,118,143,115,140,112,137)(110,135,119,144,116,141,113,138)(111,136,120,133,117,142,114,139)(121,183,130,192,127,189,124,186)(122,184,131,181,128,190,125,187)(123,185,132,182,129,191,126,188) );
G=PermutationGroup([[(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)], [(1,91,21,51,153,69,113,182),(2,96,22,56,154,62,114,187),(3,89,23,49,155,67,115,192),(4,94,24,54,156,72,116,185),(5,87,13,59,145,65,117,190),(6,92,14,52,146,70,118,183),(7,85,15,57,147,63,119,188),(8,90,16,50,148,68,120,181),(9,95,17,55,149,61,109,186),(10,88,18,60,150,66,110,191),(11,93,19,53,151,71,111,184),(12,86,20,58,152,64,112,189),(25,144,176,97,123,46,165,75),(26,137,177,102,124,39,166,80),(27,142,178,107,125,44,167,73),(28,135,179,100,126,37,168,78),(29,140,180,105,127,42,157,83),(30,133,169,98,128,47,158,76),(31,138,170,103,129,40,159,81),(32,143,171,108,130,45,160,74),(33,136,172,101,131,38,161,79),(34,141,173,106,132,43,162,84),(35,134,174,99,121,48,163,77),(36,139,175,104,122,41,164,82)], [(1,81,10,78,7,75,4,84),(2,82,11,79,8,76,5,73),(3,83,12,80,9,77,6,74),(13,44,22,41,19,38,16,47),(14,45,23,42,20,39,17,48),(15,46,24,43,21,40,18,37),(25,54,34,51,31,60,28,57),(26,55,35,52,32,49,29,58),(27,56,36,53,33,50,30,59),(61,163,70,160,67,157,64,166),(62,164,71,161,68,158,65,167),(63,165,72,162,69,159,66,168),(85,176,94,173,91,170,88,179),(86,177,95,174,92,171,89,180),(87,178,96,175,93,172,90,169),(97,156,106,153,103,150,100,147),(98,145,107,154,104,151,101,148),(99,146,108,155,105,152,102,149),(109,134,118,143,115,140,112,137),(110,135,119,144,116,141,113,138),(111,136,120,133,117,142,114,139),(121,183,130,192,127,189,124,186),(122,184,131,181,128,190,125,187),(123,185,132,182,129,191,126,188)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 8M | 8N | 8O | 8P | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | - | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C8 | S3 | D4 | Q8 | D6 | M4(2) | SD16 | Dic6 | C4×S3 | C3⋊D4 | C8.C4 | S3×C8 | C8⋊S3 | D4.S3 | Q8⋊2S3 | C12.53D4 |
| kernel | C12.39SD16 | C4×C3⋊C8 | C12⋊C8 | C3×C4⋊C8 | C2×C3⋊C8 | C3⋊C8 | C4⋊C8 | C2×C12 | C2×C12 | C42 | C12 | C12 | C2×C4 | C2×C4 | C2×C4 | C6 | C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 2 |
Matrix representation of C12.39SD16 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 27 | 46 |
| 0 | 0 | 27 | 0 |
| 0 | 12 | 0 | 0 |
| 67 | 12 | 0 | 0 |
| 0 | 0 | 25 | 29 |
| 0 | 0 | 54 | 48 |
| 50 | 58 | 0 | 0 |
| 6 | 23 | 0 | 0 |
| 0 | 0 | 51 | 0 |
| 0 | 0 | 0 | 51 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,27,27,0,0,46,0],[0,67,0,0,12,12,0,0,0,0,25,54,0,0,29,48],[50,6,0,0,58,23,0,0,0,0,51,0,0,0,0,51] >;
C12.39SD16 in GAP, Magma, Sage, TeX
C_{12}._{39}{\rm SD}_{16} % in TeX
G:=Group("C12.39SD16"); // GroupNames label
G:=SmallGroup(192,39);
// by ID
G=gap.SmallGroup(192,39);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,589,36,100,570,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=1,c^2=a^9,b*a*b^-1=a^5,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
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