metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.6Q8, C8.2Dic14, C28.36SD16, C7⋊C16⋊2C4, C7⋊2(C8.Q8), C8.29(C4×D7), C28.5(C4⋊C4), C56.33(C2×C4), (C2×C8).42D14, (C2×C28).94D4, C8⋊Dic7.12C2, (C2×C14).3SD16, C14.3(C4.Q8), C8.C4.2D7, C28.C8.4C2, C4.5(Dic7⋊C4), C4.11(D4.D7), C22.2(Q8⋊D7), (C2×C56).153C22, C2.4(C4.Dic14), (C7×C8.C4).3C2, (C2×C4).17(C7⋊D4), SmallGroup(448,51)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.Dic14
G = < a,b,c | a8=1, b28=a4, c2=ab14, bab-1=a-1, cac-1=a5, cbc-1=a-1b27 >
Smallest permutation representation of C8.Dic14
►On 112 points
Generators in S112
(1 110 43 96 29 82 15 68)(2 69 16 83 30 97 44 111)(3 112 45 98 31 84 17 70)(4 71 18 85 32 99 46 57)(5 58 47 100 33 86 19 72)(6 73 20 87 34 101 48 59)(7 60 49 102 35 88 21 74)(8 75 22 89 36 103 50 61)(9 62 51 104 37 90 23 76)(10 77 24 91 38 105 52 63)(11 64 53 106 39 92 25 78)(12 79 26 93 40 107 54 65)(13 66 55 108 41 94 27 80)(14 81 28 95 42 109 56 67)
(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 22 68 103 15 8 82 89 29 50 96 75 43 36 110 61)(2 88 83 35 44 102 69 49 30 60 111 7 16 74 97 21)(3 20 70 101 17 6 84 87 31 48 98 73 45 34 112 59)(4 86 85 33 46 100 71 47 32 58 57 5 18 72 99 19)(9 14 76 95 23 56 90 81 37 42 104 67 51 28 62 109)(10 80 91 27 52 94 77 41 38 108 63 55 24 66 105 13)(11 12 78 93 25 54 92 79 39 40 106 65 53 26 64 107)
G:=sub<Sym(112)| (1,110,43,96,29,82,15,68)(2,69,16,83,30,97,44,111)(3,112,45,98,31,84,17,70)(4,71,18,85,32,99,46,57)(5,58,47,100,33,86,19,72)(6,73,20,87,34,101,48,59)(7,60,49,102,35,88,21,74)(8,75,22,89,36,103,50,61)(9,62,51,104,37,90,23,76)(10,77,24,91,38,105,52,63)(11,64,53,106,39,92,25,78)(12,79,26,93,40,107,54,65)(13,66,55,108,41,94,27,80)(14,81,28,95,42,109,56,67), (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), (1,22,68,103,15,8,82,89,29,50,96,75,43,36,110,61)(2,88,83,35,44,102,69,49,30,60,111,7,16,74,97,21)(3,20,70,101,17,6,84,87,31,48,98,73,45,34,112,59)(4,86,85,33,46,100,71,47,32,58,57,5,18,72,99,19)(9,14,76,95,23,56,90,81,37,42,104,67,51,28,62,109)(10,80,91,27,52,94,77,41,38,108,63,55,24,66,105,13)(11,12,78,93,25,54,92,79,39,40,106,65,53,26,64,107)>;
G:=Group( (1,110,43,96,29,82,15,68)(2,69,16,83,30,97,44,111)(3,112,45,98,31,84,17,70)(4,71,18,85,32,99,46,57)(5,58,47,100,33,86,19,72)(6,73,20,87,34,101,48,59)(7,60,49,102,35,88,21,74)(8,75,22,89,36,103,50,61)(9,62,51,104,37,90,23,76)(10,77,24,91,38,105,52,63)(11,64,53,106,39,92,25,78)(12,79,26,93,40,107,54,65)(13,66,55,108,41,94,27,80)(14,81,28,95,42,109,56,67), (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), (1,22,68,103,15,8,82,89,29,50,96,75,43,36,110,61)(2,88,83,35,44,102,69,49,30,60,111,7,16,74,97,21)(3,20,70,101,17,6,84,87,31,48,98,73,45,34,112,59)(4,86,85,33,46,100,71,47,32,58,57,5,18,72,99,19)(9,14,76,95,23,56,90,81,37,42,104,67,51,28,62,109)(10,80,91,27,52,94,77,41,38,108,63,55,24,66,105,13)(11,12,78,93,25,54,92,79,39,40,106,65,53,26,64,107) );
G=PermutationGroup([[(1,110,43,96,29,82,15,68),(2,69,16,83,30,97,44,111),(3,112,45,98,31,84,17,70),(4,71,18,85,32,99,46,57),(5,58,47,100,33,86,19,72),(6,73,20,87,34,101,48,59),(7,60,49,102,35,88,21,74),(8,75,22,89,36,103,50,61),(9,62,51,104,37,90,23,76),(10,77,24,91,38,105,52,63),(11,64,53,106,39,92,25,78),(12,79,26,93,40,107,54,65),(13,66,55,108,41,94,27,80),(14,81,28,95,42,109,56,67)], [(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)], [(1,22,68,103,15,8,82,89,29,50,96,75,43,36,110,61),(2,88,83,35,44,102,69,49,30,60,111,7,16,74,97,21),(3,20,70,101,17,6,84,87,31,48,98,73,45,34,112,59),(4,86,85,33,46,100,71,47,32,58,57,5,18,72,99,19),(9,14,76,95,23,56,90,81,37,42,104,67,51,28,62,109),(10,80,91,27,52,94,77,41,38,108,63,55,24,66,105,13),(11,12,78,93,25,54,92,79,39,40,106,65,53,26,64,107)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 14A | 14B | 14C | 14D | 14E | 14F | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L | 56M | ··· | 56X |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | - | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | Q8 | D4 | D7 | SD16 | SD16 | D14 | Dic14 | C4×D7 | C7⋊D4 | C8.Q8 | D4.D7 | Q8⋊D7 | C8.Dic14 |
| kernel | C8.Dic14 | C28.C8 | C8⋊Dic7 | C7×C8.C4 | C7⋊C16 | C56 | C2×C28 | C8.C4 | C28 | C2×C14 | C2×C8 | C8 | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 6 | 6 | 2 | 3 | 3 | 12 |
Matrix representation of C8.Dic14 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 16 | 0 | 0 |
| 0 | 0 | 106 | 0 | 0 | 0 |
| 0 | 0 | 70 | 17 | 100 | 13 |
| 0 | 0 | 27 | 17 | 100 | 100 |
| 0 | 98 | 0 | 0 | 0 | 0 |
| 15 | 58 | 0 | 0 | 0 | 0 |
| 0 | 0 | 103 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 112 | 1 |
| 0 | 0 | 38 | 112 | 10 | 0 |
| 0 | 0 | 82 | 0 | 10 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 34 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 97 | 16 |
| 0 | 0 | 43 | 0 | 26 | 0 |
| 0 | 0 | 91 | 112 | 96 | 17 |
| 0 | 0 | 91 | 0 | 96 | 17 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,26,106,70,27,0,0,16,0,17,17,0,0,0,0,100,100,0,0,0,0,13,100],[0,15,0,0,0,0,98,58,0,0,0,0,0,0,103,0,38,82,0,0,0,0,112,0,0,0,36,112,10,10,0,0,0,1,0,0],[112,34,0,0,0,0,0,1,0,0,0,0,0,0,0,43,91,91,0,0,0,0,112,0,0,0,97,26,96,96,0,0,16,0,17,17] >;
C8.Dic14 in GAP, Magma, Sage, TeX
C_8.{\rm Dic}_{14} % in TeX
G:=Group("C8.Dic14"); // GroupNames label
G:=SmallGroup(448,51);
// by ID
G=gap.SmallGroup(448,51);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,365,36,758,184,346,80,851,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^28=a^4,c^2=a*b^14,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^-1*b^27>;
// generators/relations
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