metabelian, supersoluble, monomial
Aliases: Dic7⋊C12, C7⋊C12⋊C4, Dic7⋊C4⋊C3, (C2×C4).1F7, C2.4(C4×F7), (C2×C28).8C6, C14.5(C3×D4), C14.1(C3×Q8), C14.4(C2×C12), C2.1(C4.F7), C22.4(C2×F7), (C2×Dic7).1C6, C2.1(Dic7⋊C6), C7⋊1(C3×C4⋊C4), C7⋊C3⋊1(C4⋊C4), (C2×C7⋊C3).5D4, (C2×C7⋊C3).1Q8, (C2×C7⋊C12).1C2, (C2×C14).3(C2×C6), (C22×C7⋊C3).3C22, (C2×C4×C7⋊C3).9C2, (C2×C7⋊C3).3(C2×C4), SmallGroup(336,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic7⋊C12
G = < a,b,c | a14=c12=1, b2=a7, bab-1=a-1, cac-1=a9, cbc-1=a7b >
Smallest permutation representation of Dic7⋊C12
►On 112 points
Generators in S112
(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)
(1 64 8 57)(2 63 9 70)(3 62 10 69)(4 61 11 68)(5 60 12 67)(6 59 13 66)(7 58 14 65)(15 81 22 74)(16 80 23 73)(17 79 24 72)(18 78 25 71)(19 77 26 84)(20 76 27 83)(21 75 28 82)(29 94 36 87)(30 93 37 86)(31 92 38 85)(32 91 39 98)(33 90 40 97)(34 89 41 96)(35 88 42 95)(43 105 50 112)(44 104 51 111)(45 103 52 110)(46 102 53 109)(47 101 54 108)(48 100 55 107)(49 99 56 106)
(1 56 18 38)(2 53 27 39 12 51 19 35 10 43 15 33)(3 50 22 40 9 46 20 32 5 44 26 42)(4 47 17 41 6 55 21 29 14 45 23 37)(7 52 16 30 11 54 24 34 13 48 28 36)(8 49 25 31)(57 106 71 85)(58 103 80 86 68 101 72 96 66 107 82 94)(59 100 75 87 65 110 73 93 61 108 79 89)(60 111 84 88 62 105 74 90 70 109 76 98)(63 102 83 91 67 104 77 95 69 112 81 97)(64 99 78 92)
G:=sub<Sym(112)| (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), (1,64,8,57)(2,63,9,70)(3,62,10,69)(4,61,11,68)(5,60,12,67)(6,59,13,66)(7,58,14,65)(15,81,22,74)(16,80,23,73)(17,79,24,72)(18,78,25,71)(19,77,26,84)(20,76,27,83)(21,75,28,82)(29,94,36,87)(30,93,37,86)(31,92,38,85)(32,91,39,98)(33,90,40,97)(34,89,41,96)(35,88,42,95)(43,105,50,112)(44,104,51,111)(45,103,52,110)(46,102,53,109)(47,101,54,108)(48,100,55,107)(49,99,56,106), (1,56,18,38)(2,53,27,39,12,51,19,35,10,43,15,33)(3,50,22,40,9,46,20,32,5,44,26,42)(4,47,17,41,6,55,21,29,14,45,23,37)(7,52,16,30,11,54,24,34,13,48,28,36)(8,49,25,31)(57,106,71,85)(58,103,80,86,68,101,72,96,66,107,82,94)(59,100,75,87,65,110,73,93,61,108,79,89)(60,111,84,88,62,105,74,90,70,109,76,98)(63,102,83,91,67,104,77,95,69,112,81,97)(64,99,78,92)>;
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), (1,64,8,57)(2,63,9,70)(3,62,10,69)(4,61,11,68)(5,60,12,67)(6,59,13,66)(7,58,14,65)(15,81,22,74)(16,80,23,73)(17,79,24,72)(18,78,25,71)(19,77,26,84)(20,76,27,83)(21,75,28,82)(29,94,36,87)(30,93,37,86)(31,92,38,85)(32,91,39,98)(33,90,40,97)(34,89,41,96)(35,88,42,95)(43,105,50,112)(44,104,51,111)(45,103,52,110)(46,102,53,109)(47,101,54,108)(48,100,55,107)(49,99,56,106), (1,56,18,38)(2,53,27,39,12,51,19,35,10,43,15,33)(3,50,22,40,9,46,20,32,5,44,26,42)(4,47,17,41,6,55,21,29,14,45,23,37)(7,52,16,30,11,54,24,34,13,48,28,36)(8,49,25,31)(57,106,71,85)(58,103,80,86,68,101,72,96,66,107,82,94)(59,100,75,87,65,110,73,93,61,108,79,89)(60,111,84,88,62,105,74,90,70,109,76,98)(63,102,83,91,67,104,77,95,69,112,81,97)(64,99,78,92) );
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)], [(1,64,8,57),(2,63,9,70),(3,62,10,69),(4,61,11,68),(5,60,12,67),(6,59,13,66),(7,58,14,65),(15,81,22,74),(16,80,23,73),(17,79,24,72),(18,78,25,71),(19,77,26,84),(20,76,27,83),(21,75,28,82),(29,94,36,87),(30,93,37,86),(31,92,38,85),(32,91,39,98),(33,90,40,97),(34,89,41,96),(35,88,42,95),(43,105,50,112),(44,104,51,111),(45,103,52,110),(46,102,53,109),(47,101,54,108),(48,100,55,107),(49,99,56,106)], [(1,56,18,38),(2,53,27,39,12,51,19,35,10,43,15,33),(3,50,22,40,9,46,20,32,5,44,26,42),(4,47,17,41,6,55,21,29,14,45,23,37),(7,52,16,30,11,54,24,34,13,48,28,36),(8,49,25,31),(57,106,71,85),(58,103,80,86,68,101,72,96,66,107,82,94),(59,100,75,87,65,110,73,93,61,108,79,89),(60,111,84,88,62,105,74,90,70,109,76,98),(63,102,83,91,67,104,77,95,69,112,81,97),(64,99,78,92)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 7 | 12A | ··· | 12L | 14A | 14B | 14C | 28A | 28B | 28C | 28D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 7 | 12 | ··· | 12 | 14 | 14 | 14 | 28 | 28 | 28 | 28 |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 2 | 2 | 14 | 14 | 14 | 14 | 7 | ··· | 7 | 6 | 14 | ··· | 14 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | - | + | + | - | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D4 | Q8 | C3×D4 | C3×Q8 | F7 | C2×F7 | C4.F7 | C4×F7 | Dic7⋊C6 |
| kernel | Dic7⋊C12 | C2×C7⋊C12 | C2×C4×C7⋊C3 | Dic7⋊C4 | C7⋊C12 | C2×Dic7 | C2×C28 | Dic7 | C2×C7⋊C3 | C2×C7⋊C3 | C14 | C14 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of Dic7⋊C12 ►in GL8(𝔽337)
| 336 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 336 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 336 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 336 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 336 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 336 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 49 | 235 | 0 | 0 | 0 | 0 | 0 | 0 |
| 235 | 288 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 266 | 71 | 266 | 0 | 181 |
| 0 | 0 | 266 | 0 | 0 | 266 | 181 | 71 |
| 0 | 0 | 0 | 266 | 0 | 110 | 71 | 71 |
| 0 | 0 | 266 | 266 | 181 | 0 | 71 | 0 |
| 0 | 0 | 266 | 110 | 71 | 0 | 0 | 71 |
| 0 | 0 | 110 | 0 | 71 | 266 | 71 | 0 |
| 0 | 208 | 0 | 0 | 0 | 0 | 0 | 0 |
| 129 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 180 | 314 | 0 | 0 | 314 |
| 0 | 0 | 0 | 23 | 0 | 180 | 314 | 314 |
| 0 | 0 | 0 | 23 | 314 | 23 | 0 | 157 |
| 0 | 0 | 180 | 0 | 314 | 23 | 314 | 0 |
| 0 | 0 | 23 | 23 | 157 | 0 | 314 | 0 |
| 0 | 0 | 23 | 0 | 0 | 23 | 157 | 314 |
G:=sub<GL(8,GF(337))| [336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,1,1,1,1,1,1,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0],[49,235,0,0,0,0,0,0,235,288,0,0,0,0,0,0,0,0,0,266,0,266,266,110,0,0,266,0,266,266,110,0,0,0,71,0,0,181,71,71,0,0,266,266,110,0,0,266,0,0,0,181,71,71,0,71,0,0,181,71,71,0,71,0],[0,129,0,0,0,0,0,0,208,0,0,0,0,0,0,0,0,0,23,0,0,180,23,23,0,0,180,23,23,0,23,0,0,0,314,0,314,314,157,0,0,0,0,180,23,23,0,23,0,0,0,314,0,314,314,157,0,0,314,314,157,0,0,314] >;
Dic7⋊C12 in GAP, Magma, Sage, TeX
{\rm Dic}_7\rtimes C_{12} % in TeX
G:=Group("Dic7:C12"); // GroupNames label
G:=SmallGroup(336,15);
// by ID
G=gap.SmallGroup(336,15);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,144,313,79,10373,1745]);
// Polycyclic
G:=Group<a,b,c|a^14=c^12=1,b^2=a^7,b*a*b^-1=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^7*b>;
// generators/relations
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