metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic28, C8.D7, C7⋊1Q16, C56.1C2, C14.3D4, C2.5D28, C4.10D14, C28.10C22, Dic14.1C2, SmallGroup(112,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic28
G = < a,b | a56=1, b2=a28, bab-1=a-1 >
Smallest permutation representation of Dic28
►Regular action 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 85 29 57)(2 84 30 112)(3 83 31 111)(4 82 32 110)(5 81 33 109)(6 80 34 108)(7 79 35 107)(8 78 36 106)(9 77 37 105)(10 76 38 104)(11 75 39 103)(12 74 40 102)(13 73 41 101)(14 72 42 100)(15 71 43 99)(16 70 44 98)(17 69 45 97)(18 68 46 96)(19 67 47 95)(20 66 48 94)(21 65 49 93)(22 64 50 92)(23 63 51 91)(24 62 52 90)(25 61 53 89)(26 60 54 88)(27 59 55 87)(28 58 56 86)
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), (1,85,29,57)(2,84,30,112)(3,83,31,111)(4,82,32,110)(5,81,33,109)(6,80,34,108)(7,79,35,107)(8,78,36,106)(9,77,37,105)(10,76,38,104)(11,75,39,103)(12,74,40,102)(13,73,41,101)(14,72,42,100)(15,71,43,99)(16,70,44,98)(17,69,45,97)(18,68,46,96)(19,67,47,95)(20,66,48,94)(21,65,49,93)(22,64,50,92)(23,63,51,91)(24,62,52,90)(25,61,53,89)(26,60,54,88)(27,59,55,87)(28,58,56,86)>;
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), (1,85,29,57)(2,84,30,112)(3,83,31,111)(4,82,32,110)(5,81,33,109)(6,80,34,108)(7,79,35,107)(8,78,36,106)(9,77,37,105)(10,76,38,104)(11,75,39,103)(12,74,40,102)(13,73,41,101)(14,72,42,100)(15,71,43,99)(16,70,44,98)(17,69,45,97)(18,68,46,96)(19,67,47,95)(20,66,48,94)(21,65,49,93)(22,64,50,92)(23,63,51,91)(24,62,52,90)(25,61,53,89)(26,60,54,88)(27,59,55,87)(28,58,56,86) );
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)], [(1,85,29,57),(2,84,30,112),(3,83,31,111),(4,82,32,110),(5,81,33,109),(6,80,34,108),(7,79,35,107),(8,78,36,106),(9,77,37,105),(10,76,38,104),(11,75,39,103),(12,74,40,102),(13,73,41,101),(14,72,42,100),(15,71,43,99),(16,70,44,98),(17,69,45,97),(18,68,46,96),(19,67,47,95),(20,66,48,94),(21,65,49,93),(22,64,50,92),(23,63,51,91),(24,62,52,90),(25,61,53,89),(26,60,54,88),(27,59,55,87),(28,58,56,86)]])
Dic28 is a maximal subgroup of
C112⋊C2 Dic56 D8.D7 C7⋊Q32 D56⋊7C2 C8.D14 D8⋊3D7 SD16⋊D7 D7×Q16 C8.F7 C3⋊Dic28 Dic84
Dic28 is a maximal quotient of
C28.44D4 C56⋊1C4 C3⋊Dic28 Dic84
31 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
31 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | D4 | D7 | Q16 | D14 | D28 | Dic28 |
| kernel | Dic28 | C56 | Dic14 | C14 | C8 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 3 | 2 | 3 | 6 | 12 |
Matrix representation of Dic28 ►in GL2(𝔽113) generated by
| 40 | 106 |
| 69 | 84 |
| 55 | 37 |
| 19 | 58 |
G:=sub<GL(2,GF(113))| [40,69,106,84],[55,19,37,58] >;
Dic28 in GAP, Magma, Sage, TeX
{\rm Dic}_{28} % in TeX
G:=Group("Dic28"); // GroupNames label
G:=SmallGroup(112,7);
// by ID
G=gap.SmallGroup(112,7);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-7,40,61,66,182,42,2404]);
// Polycyclic
G:=Group<a,b|a^56=1,b^2=a^28,b*a*b^-1=a^-1>;
// generators/relations
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