metacyclic, supersoluble, monomial
Aliases: C28.1C12, C7⋊C8⋊5C6, C7⋊C24⋊5C2, C4.(C7⋊C12), C4.Dic7⋊C3, C7⋊C3⋊2M4(2), (C2×C4).2F7, (C2×C28).1C6, C22.(C7⋊C12), C7⋊2(C3×M4(2)), C4.15(C2×F7), (C2×C14).2C12, C14.7(C2×C12), C28.16(C2×C6), (C4×C7⋊C3).1C4, C2.3(C2×C7⋊C12), (C22×C7⋊C3).2C4, (C4×C7⋊C3).16C22, (C2×C4×C7⋊C3).1C2, (C2×C7⋊C3).6(C2×C4), SmallGroup(336,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.C12
G = < a,b | a28=1, b12=a14, bab-1=a19 >
Smallest permutation representation of C28.C12
►On 56 points
Generators in S56
(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)
(1 51 22 30 15 37 8 44)(2 54 3 29 12 56 9 47 10 50 19 49 16 40 17 43 26 42 23 33 24 36 5 35)(4 32 21 55 6 38 11 53 28 48 13 31 18 46 7 41 20 52 25 39 14 34 27 45)
G:=sub<Sym(56)| (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), (1,51,22,30,15,37,8,44)(2,54,3,29,12,56,9,47,10,50,19,49,16,40,17,43,26,42,23,33,24,36,5,35)(4,32,21,55,6,38,11,53,28,48,13,31,18,46,7,41,20,52,25,39,14,34,27,45)>;
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), (1,51,22,30,15,37,8,44)(2,54,3,29,12,56,9,47,10,50,19,49,16,40,17,43,26,42,23,33,24,36,5,35)(4,32,21,55,6,38,11,53,28,48,13,31,18,46,7,41,20,52,25,39,14,34,27,45) );
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)], [(1,51,22,30,15,37,8,44),(2,54,3,29,12,56,9,47,10,50,19,49,16,40,17,43,26,42,23,33,24,36,5,35),(4,32,21,55,6,38,11,53,28,48,13,31,18,46,7,41,20,52,25,39,14,34,27,45)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 7 | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 14A | 14B | 14C | 24A | ··· | 24H | 28A | 28B | 28C | 28D |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 7 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 14 | 14 | 14 | 24 | ··· | 24 | 28 | 28 | 28 | 28 |
| size | 1 | 1 | 2 | 7 | 7 | 1 | 1 | 2 | 7 | 7 | 14 | 14 | 6 | 14 | 14 | 14 | 14 | 7 | 7 | 7 | 7 | 14 | 14 | 6 | 6 | 6 | 14 | ··· | 14 | 6 | 6 | 6 | 6 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | - | + | - | ||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | M4(2) | C3×M4(2) | F7 | C7⋊C12 | C2×F7 | C7⋊C12 | C28.C12 |
| kernel | C28.C12 | C7⋊C24 | C2×C4×C7⋊C3 | C4.Dic7 | C4×C7⋊C3 | C22×C7⋊C3 | C7⋊C8 | C2×C28 | C28 | C2×C14 | C7⋊C3 | C7 | C2×C4 | C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 2 | 4 | 1 | 1 | 1 | 1 | 4 |
Matrix representation of C28.C12 ►in GL6(𝔽337)
| 0 | 148 | 189 | 0 | 0 | 0 |
| 154 | 0 | 35 | 0 | 0 | 0 |
| 189 | 0 | 183 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 154 | 35 |
| 0 | 0 | 0 | 189 | 189 | 183 |
| 0 | 0 | 0 | 0 | 189 | 183 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 213 | 1 | 125 |
| 0 | 0 | 0 | 1 | 125 | 336 |
| 0 | 189 | 0 | 0 | 0 | 0 |
| 154 | 189 | 35 | 0 | 0 | 0 |
| 189 | 35 | 148 | 0 | 0 | 0 |
G:=sub<GL(6,GF(337))| [0,154,189,0,0,0,148,0,0,0,0,0,189,35,183,0,0,0,0,0,0,0,189,0,0,0,0,154,189,189,0,0,0,35,183,183],[0,0,0,0,154,189,0,0,0,189,189,35,0,0,0,0,35,148,0,213,1,0,0,0,1,1,125,0,0,0,0,125,336,0,0,0] >;
C28.C12 in GAP, Magma, Sage, TeX
C_{28}.C_{12} % in TeX
G:=Group("C28.C12"); // GroupNames label
G:=SmallGroup(336,13);
// by ID
G=gap.SmallGroup(336,13);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,72,313,69,10373,1745]);
// Polycyclic
G:=Group<a,b|a^28=1,b^12=a^14,b*a*b^-1=a^19>;
// generators/relations
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