metacyclic, supersoluble, monomial
Aliases: C28⋊1C12, C4⋊(C7⋊C12), C4⋊Dic7⋊C3, (C2×C4).3F7, (C2×C28).2C6, C14.4(C3×D4), C2.1(C4⋊F7), C14.2(C3×Q8), C14.8(C2×C12), C2.2(C4.F7), C22.5(C2×F7), (C2×Dic7).2C6, C7⋊2(C3×C4⋊C4), C7⋊C3⋊2(C4⋊C4), (C4×C7⋊C3)⋊1C4, C2.4(C2×C7⋊C12), (C2×C7⋊C3).4D4, (C2×C7⋊C3).2Q8, (C2×C7⋊C12).2C2, (C2×C14).4(C2×C6), (C22×C7⋊C3).4C22, (C2×C4×C7⋊C3).2C2, (C2×C7⋊C3).8(C2×C4), SmallGroup(336,16)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28⋊C12
G = < a,b | a28=b12=1, bab-1=a19 >
Smallest permutation representation of C28⋊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)
(1 106 46 74)(2 109 55 73 26 97 47 77 10 105 43 65)(3 112 36 72 23 88 48 80 19 104 40 84)(4 87 45 71 20 107 49 83 28 103 37 75)(5 90 54 70 17 98 50 58 9 102 34 66)(6 93 35 69 14 89 51 61 18 101 31 57)(7 96 44 68 11 108 52 64 27 100 56 76)(8 99 53 67)(12 111 33 63 24 91 29 79 16 95 41 59)(13 86 42 62 21 110 30 82 25 94 38 78)(15 92 32 60)(22 85 39 81)
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), (1,106,46,74)(2,109,55,73,26,97,47,77,10,105,43,65)(3,112,36,72,23,88,48,80,19,104,40,84)(4,87,45,71,20,107,49,83,28,103,37,75)(5,90,54,70,17,98,50,58,9,102,34,66)(6,93,35,69,14,89,51,61,18,101,31,57)(7,96,44,68,11,108,52,64,27,100,56,76)(8,99,53,67)(12,111,33,63,24,91,29,79,16,95,41,59)(13,86,42,62,21,110,30,82,25,94,38,78)(15,92,32,60)(22,85,39,81)>;
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), (1,106,46,74)(2,109,55,73,26,97,47,77,10,105,43,65)(3,112,36,72,23,88,48,80,19,104,40,84)(4,87,45,71,20,107,49,83,28,103,37,75)(5,90,54,70,17,98,50,58,9,102,34,66)(6,93,35,69,14,89,51,61,18,101,31,57)(7,96,44,68,11,108,52,64,27,100,56,76)(8,99,53,67)(12,111,33,63,24,91,29,79,16,95,41,59)(13,86,42,62,21,110,30,82,25,94,38,78)(15,92,32,60)(22,85,39,81) );
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)], [(1,106,46,74),(2,109,55,73,26,97,47,77,10,105,43,65),(3,112,36,72,23,88,48,80,19,104,40,84),(4,87,45,71,20,107,49,83,28,103,37,75),(5,90,54,70,17,98,50,58,9,102,34,66),(6,93,35,69,14,89,51,61,18,101,31,57),(7,96,44,68,11,108,52,64,27,100,56,76),(8,99,53,67),(12,111,33,63,24,91,29,79,16,95,41,59),(13,86,42,62,21,110,30,82,25,94,38,78),(15,92,32,60),(22,85,39,81)]])
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 | C7⋊C12 | C2×F7 | C4.F7 | C4⋊F7 |
| kernel | C28⋊C12 | C2×C7⋊C12 | C2×C4×C7⋊C3 | C4⋊Dic7 | C4×C7⋊C3 | C2×Dic7 | C2×C28 | C28 | C2×C7⋊C3 | C2×C7⋊C3 | C14 | C14 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 2 |
Matrix representation of C28⋊C12 ►in GL8(𝔽337)
| 0 | 336 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 336 | 336 | 336 | 336 | 336 | 336 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 15 | 232 | 0 | 0 | 0 | 0 | 0 | 0 |
| 232 | 322 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 66 | 108 | 108 | 0 | 108 | 0 |
| 0 | 0 | 0 | 229 | 0 | 229 | 229 | 295 |
| 0 | 0 | 108 | 0 | 0 | 66 | 108 | 108 |
| 0 | 0 | 0 | 66 | 108 | 108 | 0 | 108 |
| 0 | 0 | 42 | 42 | 271 | 42 | 271 | 271 |
| 0 | 0 | 229 | 0 | 229 | 229 | 295 | 0 |
G:=sub<GL(8,GF(337))| [0,1,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,0,0,0,0,0,336,1,0,0,0,0,0,0,336,0,0,0,1,0,0,0,336,0,0,0,0,1,0,0,336,0,0,0,0,0,1,0,336,0,0,0,0,0,0,1,336,0],[15,232,0,0,0,0,0,0,232,322,0,0,0,0,0,0,0,0,66,0,108,0,42,229,0,0,108,229,0,66,42,0,0,0,108,0,0,108,271,229,0,0,0,229,66,108,42,229,0,0,108,229,108,0,271,295,0,0,0,295,108,108,271,0] >;
C28⋊C12 in GAP, Magma, Sage, TeX
C_{28}\rtimes C_{12} % in TeX
G:=Group("C28:C12"); // GroupNames label
G:=SmallGroup(336,16);
// by ID
G=gap.SmallGroup(336,16);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,72,313,151,10373,1745]);
// Polycyclic
G:=Group<a,b|a^28=b^12=1,b*a*b^-1=a^19>;
// generators/relations
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