metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.9Q16, C12.16SD16, C42.221D6, C4⋊Q8.5S3, (C2×Q8).66D6, C6.41(C2×Q16), (C2×C12).152D4, C6.74(C2×SD16), C4.4(C3⋊Q16), C3⋊3(C4.SD16), C12.78(C4○D4), C12⋊2Q8.19C2, C4.3(Q8⋊2S3), (C6×Q8).60C22, C4.24(D4⋊2S3), (C2×C12).397C23, (C4×C12).126C22, Q8⋊2Dic3.11C2, C6.45(C4.4D4), C4⋊Dic3.157C22, C2.12(C23.12D6), (C4×C3⋊C8).12C2, (C3×C4⋊Q8).5C2, (C2×C6).528(C2×D4), C2.12(C2×C3⋊Q16), (C2×C3⋊C8).260C22, C2.11(C2×Q8⋊2S3), (C2×C4).134(C3⋊D4), (C2×C4).494(C22×S3), C22.200(C2×C3⋊D4), SmallGroup(192,638)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.9Q16
G = < a,b,c | a12=b8=1, c2=b4, bab-1=a5, cac-1=a7, cbc-1=a6b-1 >
Subgroups: 240 in 98 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C2×C8, C2×Q8, C2×Q8, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C4×C8, Q8⋊C4, C4⋊Q8, C4⋊Q8, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3, C4×C12, C3×C4⋊C4, C2×Dic6, C6×Q8, C4.SD16, C4×C3⋊C8, Q8⋊2Dic3, C12⋊2Q8, C3×C4⋊Q8, C12.9Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, Q16, C2×D4, C4○D4, C3⋊D4, C22×S3, C4.4D4, C2×SD16, C2×Q16, Q8⋊2S3, C3⋊Q16, D4⋊2S3, C2×C3⋊D4, C4.SD16, C23.12D6, C2×Q8⋊2S3, C2×C3⋊Q16, C12.9Q16
►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 67 174 104 121 59 185 79)(2 72 175 97 122 52 186 84)(3 65 176 102 123 57 187 77)(4 70 177 107 124 50 188 82)(5 63 178 100 125 55 189 75)(6 68 179 105 126 60 190 80)(7 61 180 98 127 53 191 73)(8 66 169 103 128 58 192 78)(9 71 170 108 129 51 181 83)(10 64 171 101 130 56 182 76)(11 69 172 106 131 49 183 81)(12 62 173 99 132 54 184 74)(13 116 47 94 36 147 143 166)(14 109 48 87 25 152 144 159)(15 114 37 92 26 145 133 164)(16 119 38 85 27 150 134 157)(17 112 39 90 28 155 135 162)(18 117 40 95 29 148 136 167)(19 110 41 88 30 153 137 160)(20 115 42 93 31 146 138 165)(21 120 43 86 32 151 139 158)(22 113 44 91 33 156 140 163)(23 118 45 96 34 149 141 168)(24 111 46 89 35 154 142 161)
(1 42 121 138)(2 37 122 133)(3 44 123 140)(4 39 124 135)(5 46 125 142)(6 41 126 137)(7 48 127 144)(8 43 128 139)(9 38 129 134)(10 45 130 141)(11 40 131 136)(12 47 132 143)(13 184 36 173)(14 191 25 180)(15 186 26 175)(16 181 27 170)(17 188 28 177)(18 183 29 172)(19 190 30 179)(20 185 31 174)(21 192 32 169)(22 187 33 176)(23 182 34 171)(24 189 35 178)(49 154 69 111)(50 149 70 118)(51 156 71 113)(52 151 72 120)(53 146 61 115)(54 153 62 110)(55 148 63 117)(56 155 64 112)(57 150 65 119)(58 145 66 114)(59 152 67 109)(60 147 68 116)(73 93 98 165)(74 88 99 160)(75 95 100 167)(76 90 101 162)(77 85 102 157)(78 92 103 164)(79 87 104 159)(80 94 105 166)(81 89 106 161)(82 96 107 168)(83 91 108 163)(84 86 97 158)
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), (1,67,174,104,121,59,185,79)(2,72,175,97,122,52,186,84)(3,65,176,102,123,57,187,77)(4,70,177,107,124,50,188,82)(5,63,178,100,125,55,189,75)(6,68,179,105,126,60,190,80)(7,61,180,98,127,53,191,73)(8,66,169,103,128,58,192,78)(9,71,170,108,129,51,181,83)(10,64,171,101,130,56,182,76)(11,69,172,106,131,49,183,81)(12,62,173,99,132,54,184,74)(13,116,47,94,36,147,143,166)(14,109,48,87,25,152,144,159)(15,114,37,92,26,145,133,164)(16,119,38,85,27,150,134,157)(17,112,39,90,28,155,135,162)(18,117,40,95,29,148,136,167)(19,110,41,88,30,153,137,160)(20,115,42,93,31,146,138,165)(21,120,43,86,32,151,139,158)(22,113,44,91,33,156,140,163)(23,118,45,96,34,149,141,168)(24,111,46,89,35,154,142,161), (1,42,121,138)(2,37,122,133)(3,44,123,140)(4,39,124,135)(5,46,125,142)(6,41,126,137)(7,48,127,144)(8,43,128,139)(9,38,129,134)(10,45,130,141)(11,40,131,136)(12,47,132,143)(13,184,36,173)(14,191,25,180)(15,186,26,175)(16,181,27,170)(17,188,28,177)(18,183,29,172)(19,190,30,179)(20,185,31,174)(21,192,32,169)(22,187,33,176)(23,182,34,171)(24,189,35,178)(49,154,69,111)(50,149,70,118)(51,156,71,113)(52,151,72,120)(53,146,61,115)(54,153,62,110)(55,148,63,117)(56,155,64,112)(57,150,65,119)(58,145,66,114)(59,152,67,109)(60,147,68,116)(73,93,98,165)(74,88,99,160)(75,95,100,167)(76,90,101,162)(77,85,102,157)(78,92,103,164)(79,87,104,159)(80,94,105,166)(81,89,106,161)(82,96,107,168)(83,91,108,163)(84,86,97,158)>;
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), (1,67,174,104,121,59,185,79)(2,72,175,97,122,52,186,84)(3,65,176,102,123,57,187,77)(4,70,177,107,124,50,188,82)(5,63,178,100,125,55,189,75)(6,68,179,105,126,60,190,80)(7,61,180,98,127,53,191,73)(8,66,169,103,128,58,192,78)(9,71,170,108,129,51,181,83)(10,64,171,101,130,56,182,76)(11,69,172,106,131,49,183,81)(12,62,173,99,132,54,184,74)(13,116,47,94,36,147,143,166)(14,109,48,87,25,152,144,159)(15,114,37,92,26,145,133,164)(16,119,38,85,27,150,134,157)(17,112,39,90,28,155,135,162)(18,117,40,95,29,148,136,167)(19,110,41,88,30,153,137,160)(20,115,42,93,31,146,138,165)(21,120,43,86,32,151,139,158)(22,113,44,91,33,156,140,163)(23,118,45,96,34,149,141,168)(24,111,46,89,35,154,142,161), (1,42,121,138)(2,37,122,133)(3,44,123,140)(4,39,124,135)(5,46,125,142)(6,41,126,137)(7,48,127,144)(8,43,128,139)(9,38,129,134)(10,45,130,141)(11,40,131,136)(12,47,132,143)(13,184,36,173)(14,191,25,180)(15,186,26,175)(16,181,27,170)(17,188,28,177)(18,183,29,172)(19,190,30,179)(20,185,31,174)(21,192,32,169)(22,187,33,176)(23,182,34,171)(24,189,35,178)(49,154,69,111)(50,149,70,118)(51,156,71,113)(52,151,72,120)(53,146,61,115)(54,153,62,110)(55,148,63,117)(56,155,64,112)(57,150,65,119)(58,145,66,114)(59,152,67,109)(60,147,68,116)(73,93,98,165)(74,88,99,160)(75,95,100,167)(76,90,101,162)(77,85,102,157)(78,92,103,164)(79,87,104,159)(80,94,105,166)(81,89,106,161)(82,96,107,168)(83,91,108,163)(84,86,97,158) );
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)], [(1,67,174,104,121,59,185,79),(2,72,175,97,122,52,186,84),(3,65,176,102,123,57,187,77),(4,70,177,107,124,50,188,82),(5,63,178,100,125,55,189,75),(6,68,179,105,126,60,190,80),(7,61,180,98,127,53,191,73),(8,66,169,103,128,58,192,78),(9,71,170,108,129,51,181,83),(10,64,171,101,130,56,182,76),(11,69,172,106,131,49,183,81),(12,62,173,99,132,54,184,74),(13,116,47,94,36,147,143,166),(14,109,48,87,25,152,144,159),(15,114,37,92,26,145,133,164),(16,119,38,85,27,150,134,157),(17,112,39,90,28,155,135,162),(18,117,40,95,29,148,136,167),(19,110,41,88,30,153,137,160),(20,115,42,93,31,146,138,165),(21,120,43,86,32,151,139,158),(22,113,44,91,33,156,140,163),(23,118,45,96,34,149,141,168),(24,111,46,89,35,154,142,161)], [(1,42,121,138),(2,37,122,133),(3,44,123,140),(4,39,124,135),(5,46,125,142),(6,41,126,137),(7,48,127,144),(8,43,128,139),(9,38,129,134),(10,45,130,141),(11,40,131,136),(12,47,132,143),(13,184,36,173),(14,191,25,180),(15,186,26,175),(16,181,27,170),(17,188,28,177),(18,183,29,172),(19,190,30,179),(20,185,31,174),(21,192,32,169),(22,187,33,176),(23,182,34,171),(24,189,35,178),(49,154,69,111),(50,149,70,118),(51,156,71,113),(52,151,72,120),(53,146,61,115),(54,153,62,110),(55,148,63,117),(56,155,64,112),(57,150,65,119),(58,145,66,114),(59,152,67,109),(60,147,68,116),(73,93,98,165),(74,88,99,160),(75,95,100,167),(76,90,101,162),(77,85,102,157),(78,92,103,164),(79,87,104,159),(80,94,105,166),(81,89,106,161),(82,96,107,168),(83,91,108,163),(84,86,97,158)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | ··· | 2 | 8 | 8 | 24 | 24 | 2 | 2 | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | SD16 | Q16 | C4○D4 | C3⋊D4 | Q8⋊2S3 | C3⋊Q16 | D4⋊2S3 |
| kernel | C12.9Q16 | C4×C3⋊C8 | Q8⋊2Dic3 | C12⋊2Q8 | C3×C4⋊Q8 | C4⋊Q8 | C2×C12 | C42 | C2×Q8 | C12 | C12 | C12 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 |
Matrix representation of C12.9Q16 ►in GL6(𝔽73)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 28 |
| 0 | 0 | 0 | 0 | 30 | 53 |
| 67 | 6 | 0 | 0 | 0 | 0 |
| 67 | 67 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 2 |
| 0 | 0 | 0 | 0 | 1 | 46 |
G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,20,30,0,0,0,0,28,53],[67,67,0,0,0,0,6,67,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,27,0,0,0,0,0,0,27],[0,46,0,0,0,0,46,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,1,0,0,0,0,2,46] >;
C12.9Q16 in GAP, Magma, Sage, TeX
C_{12}._9Q_{16} % in TeX
G:=Group("C12.9Q16"); // GroupNames label
G:=SmallGroup(192,638);
// by ID
G=gap.SmallGroup(192,638);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,64,590,135,184,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=1,c^2=b^4,b*a*b^-1=a^5,c*a*c^-1=a^7,c*b*c^-1=a^6*b^-1>;
// generators/relations