metabelian, supersoluble, monomial
Aliases: C24.26D6, (C3×D8)⋊3S3, D8⋊3(C3⋊S3), (C3×D4).15D6, C6.120(S3×D4), (C32×D8)⋊6C2, C3⋊4(D8⋊3S3), C32⋊5Q16⋊8C2, C32⋊19(C4○D8), C3⋊Dic3.69D4, C12.D6⋊4C2, C32⋊9SD16⋊6C2, (C3×C24).29C22, C12.89(C22×S3), (C3×C12).93C23, C32⋊4C8.26C22, (D4×C32).16C22, C32⋊4Q8.16C22, (C8×C3⋊S3)⋊4C2, C8.8(C2×C3⋊S3), D4.1(C2×C3⋊S3), C2.17(D4×C3⋊S3), (C2×C3⋊S3).46D4, C4.3(C22×C3⋊S3), (C3×C6).241(C2×D4), (C4×C3⋊S3).71C22, SmallGroup(288,769)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.26D6
G = < a,b,c | a24=b6=1, c2=a12, bab-1=a7, cac-1=a-1, cbc-1=a12b-1 >
Subgroups: 708 in 186 conjugacy classes, 55 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C32, Dic3, C12, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C4○D8, C3⋊Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, S3×C8, Dic12, D4.S3, C3×D8, D4⋊2S3, C32⋊4C8, C3×C24, C32⋊4Q8, C4×C3⋊S3, C2×C3⋊Dic3, C32⋊7D4, D4×C32, D8⋊3S3, C8×C3⋊S3, C32⋊5Q16, C32⋊9SD16, C32×D8, C12.D6, C24.26D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C22×S3, C4○D8, C2×C3⋊S3, S3×D4, C22×C3⋊S3, D8⋊3S3, D4×C3⋊S3, C24.26D6
►On 144 points
Generators in S144
(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)
(1 130 62 34 80 110)(2 137 63 41 81 117)(3 144 64 48 82 100)(4 127 65 31 83 107)(5 134 66 38 84 114)(6 141 67 45 85 97)(7 124 68 28 86 104)(8 131 69 35 87 111)(9 138 70 42 88 118)(10 121 71 25 89 101)(11 128 72 32 90 108)(12 135 49 39 91 115)(13 142 50 46 92 98)(14 125 51 29 93 105)(15 132 52 36 94 112)(16 139 53 43 95 119)(17 122 54 26 96 102)(18 129 55 33 73 109)(19 136 56 40 74 116)(20 143 57 47 75 99)(21 126 58 30 76 106)(22 133 59 37 77 113)(23 140 60 44 78 120)(24 123 61 27 79 103)
(1 110 13 98)(2 109 14 97)(3 108 15 120)(4 107 16 119)(5 106 17 118)(6 105 18 117)(7 104 19 116)(8 103 20 115)(9 102 21 114)(10 101 22 113)(11 100 23 112)(12 99 24 111)(25 59 37 71)(26 58 38 70)(27 57 39 69)(28 56 40 68)(29 55 41 67)(30 54 42 66)(31 53 43 65)(32 52 44 64)(33 51 45 63)(34 50 46 62)(35 49 47 61)(36 72 48 60)(73 137 85 125)(74 136 86 124)(75 135 87 123)(76 134 88 122)(77 133 89 121)(78 132 90 144)(79 131 91 143)(80 130 92 142)(81 129 93 141)(82 128 94 140)(83 127 95 139)(84 126 96 138)
G:=sub<Sym(144)| (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), (1,130,62,34,80,110)(2,137,63,41,81,117)(3,144,64,48,82,100)(4,127,65,31,83,107)(5,134,66,38,84,114)(6,141,67,45,85,97)(7,124,68,28,86,104)(8,131,69,35,87,111)(9,138,70,42,88,118)(10,121,71,25,89,101)(11,128,72,32,90,108)(12,135,49,39,91,115)(13,142,50,46,92,98)(14,125,51,29,93,105)(15,132,52,36,94,112)(16,139,53,43,95,119)(17,122,54,26,96,102)(18,129,55,33,73,109)(19,136,56,40,74,116)(20,143,57,47,75,99)(21,126,58,30,76,106)(22,133,59,37,77,113)(23,140,60,44,78,120)(24,123,61,27,79,103), (1,110,13,98)(2,109,14,97)(3,108,15,120)(4,107,16,119)(5,106,17,118)(6,105,18,117)(7,104,19,116)(8,103,20,115)(9,102,21,114)(10,101,22,113)(11,100,23,112)(12,99,24,111)(25,59,37,71)(26,58,38,70)(27,57,39,69)(28,56,40,68)(29,55,41,67)(30,54,42,66)(31,53,43,65)(32,52,44,64)(33,51,45,63)(34,50,46,62)(35,49,47,61)(36,72,48,60)(73,137,85,125)(74,136,86,124)(75,135,87,123)(76,134,88,122)(77,133,89,121)(78,132,90,144)(79,131,91,143)(80,130,92,142)(81,129,93,141)(82,128,94,140)(83,127,95,139)(84,126,96,138)>;
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), (1,130,62,34,80,110)(2,137,63,41,81,117)(3,144,64,48,82,100)(4,127,65,31,83,107)(5,134,66,38,84,114)(6,141,67,45,85,97)(7,124,68,28,86,104)(8,131,69,35,87,111)(9,138,70,42,88,118)(10,121,71,25,89,101)(11,128,72,32,90,108)(12,135,49,39,91,115)(13,142,50,46,92,98)(14,125,51,29,93,105)(15,132,52,36,94,112)(16,139,53,43,95,119)(17,122,54,26,96,102)(18,129,55,33,73,109)(19,136,56,40,74,116)(20,143,57,47,75,99)(21,126,58,30,76,106)(22,133,59,37,77,113)(23,140,60,44,78,120)(24,123,61,27,79,103), (1,110,13,98)(2,109,14,97)(3,108,15,120)(4,107,16,119)(5,106,17,118)(6,105,18,117)(7,104,19,116)(8,103,20,115)(9,102,21,114)(10,101,22,113)(11,100,23,112)(12,99,24,111)(25,59,37,71)(26,58,38,70)(27,57,39,69)(28,56,40,68)(29,55,41,67)(30,54,42,66)(31,53,43,65)(32,52,44,64)(33,51,45,63)(34,50,46,62)(35,49,47,61)(36,72,48,60)(73,137,85,125)(74,136,86,124)(75,135,87,123)(76,134,88,122)(77,133,89,121)(78,132,90,144)(79,131,91,143)(80,130,92,142)(81,129,93,141)(82,128,94,140)(83,127,95,139)(84,126,96,138) );
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)], [(1,130,62,34,80,110),(2,137,63,41,81,117),(3,144,64,48,82,100),(4,127,65,31,83,107),(5,134,66,38,84,114),(6,141,67,45,85,97),(7,124,68,28,86,104),(8,131,69,35,87,111),(9,138,70,42,88,118),(10,121,71,25,89,101),(11,128,72,32,90,108),(12,135,49,39,91,115),(13,142,50,46,92,98),(14,125,51,29,93,105),(15,132,52,36,94,112),(16,139,53,43,95,119),(17,122,54,26,96,102),(18,129,55,33,73,109),(19,136,56,40,74,116),(20,143,57,47,75,99),(21,126,58,30,76,106),(22,133,59,37,77,113),(23,140,60,44,78,120),(24,123,61,27,79,103)], [(1,110,13,98),(2,109,14,97),(3,108,15,120),(4,107,16,119),(5,106,17,118),(6,105,18,117),(7,104,19,116),(8,103,20,115),(9,102,21,114),(10,101,22,113),(11,100,23,112),(12,99,24,111),(25,59,37,71),(26,58,38,70),(27,57,39,69),(28,56,40,68),(29,55,41,67),(30,54,42,66),(31,53,43,65),(32,52,44,64),(33,51,45,63),(34,50,46,62),(35,49,47,61),(36,72,48,60),(73,137,85,125),(74,136,86,124),(75,135,87,123),(76,134,88,122),(77,133,89,121),(78,132,90,144),(79,131,91,143),(80,130,92,142),(81,129,93,141),(82,128,94,140),(83,127,95,139),(84,126,96,138)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | ··· | 6L | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 4 | 4 | 18 | 2 | 2 | 2 | 2 | 2 | 9 | 9 | 36 | 36 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | 2 | 18 | 18 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D8 | S3×D4 | D8⋊3S3 |
| kernel | C24.26D6 | C8×C3⋊S3 | C32⋊5Q16 | C32⋊9SD16 | C32×D8 | C12.D6 | C3×D8 | C3⋊Dic3 | C2×C3⋊S3 | C24 | C3×D4 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 1 | 1 | 4 | 8 | 4 | 4 | 8 |
Matrix representation of C24.26D6 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 3 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 12 | 22 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 22 |
| 0 | 0 | 0 | 0 | 61 | 51 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 | 51 |
| 0 | 0 | 0 | 0 | 32 | 22 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,72,0,0,0,0,3,72,0,0,0,0,0,0,10,12,0,0,0,0,0,22],[1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,22,61,0,0,0,0,22,51],[1,0,0,0,0,0,1,72,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,51,32,0,0,0,0,51,22] >;
C24.26D6 in GAP, Magma, Sage, TeX
C_{24}._{26}D_6 % in TeX
G:=Group("C24.26D6"); // GroupNames label
G:=SmallGroup(288,769);
// by ID
G=gap.SmallGroup(288,769);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,422,135,346,185,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=b^6=1,c^2=a^12,b*a*b^-1=a^7,c*a*c^-1=a^-1,c*b*c^-1=a^12*b^-1>;
// generators/relations