metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C56.1C8, C8.29D28, C56.65D4, C56.16Q8, C8.14Dic14, C42.9Dic7, C8.1(C7⋊C8), (C4×C8).11D7, C14.4(C4⋊C8), C7⋊1(C8.C8), C28.37(C2×C8), (C4×C28).14C4, (C2×C56).18C4, (C4×C56).13C2, C28.39(C4⋊C4), (C2×C8).9Dic7, (C2×C8).321D14, C2.5(C28⋊C8), C28.C8.5C2, C4.19(C4⋊Dic7), (C2×C56).397C22, (C2×C14).19M4(2), C22.2(C4.Dic7), C4.8(C2×C7⋊C8), (C2×C28).301(C2×C4), (C2×C4).69(C2×Dic7), SmallGroup(448,20)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.16Q8
G = < a,b,c | a56=1, b4=a28, c2=a21b2, ab=ba, cac-1=a13, cbc-1=b3 >
Smallest permutation representation of C56.16Q8
►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 50 43 36 29 22 15 8)(2 51 44 37 30 23 16 9)(3 52 45 38 31 24 17 10)(4 53 46 39 32 25 18 11)(5 54 47 40 33 26 19 12)(6 55 48 41 34 27 20 13)(7 56 49 42 35 28 21 14)(57 64 71 78 85 92 99 106)(58 65 72 79 86 93 100 107)(59 66 73 80 87 94 101 108)(60 67 74 81 88 95 102 109)(61 68 75 82 89 96 103 110)(62 69 76 83 90 97 104 111)(63 70 77 84 91 98 105 112)
(1 64 8 99 15 78 22 57 29 92 36 71 43 106 50 85)(2 77 9 112 16 91 23 70 30 105 37 84 44 63 51 98)(3 90 10 69 17 104 24 83 31 62 38 97 45 76 52 111)(4 103 11 82 18 61 25 96 32 75 39 110 46 89 53 68)(5 60 12 95 19 74 26 109 33 88 40 67 47 102 54 81)(6 73 13 108 20 87 27 66 34 101 41 80 48 59 55 94)(7 86 14 65 21 100 28 79 35 58 42 93 49 72 56 107)
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), (1,50,43,36,29,22,15,8)(2,51,44,37,30,23,16,9)(3,52,45,38,31,24,17,10)(4,53,46,39,32,25,18,11)(5,54,47,40,33,26,19,12)(6,55,48,41,34,27,20,13)(7,56,49,42,35,28,21,14)(57,64,71,78,85,92,99,106)(58,65,72,79,86,93,100,107)(59,66,73,80,87,94,101,108)(60,67,74,81,88,95,102,109)(61,68,75,82,89,96,103,110)(62,69,76,83,90,97,104,111)(63,70,77,84,91,98,105,112), (1,64,8,99,15,78,22,57,29,92,36,71,43,106,50,85)(2,77,9,112,16,91,23,70,30,105,37,84,44,63,51,98)(3,90,10,69,17,104,24,83,31,62,38,97,45,76,52,111)(4,103,11,82,18,61,25,96,32,75,39,110,46,89,53,68)(5,60,12,95,19,74,26,109,33,88,40,67,47,102,54,81)(6,73,13,108,20,87,27,66,34,101,41,80,48,59,55,94)(7,86,14,65,21,100,28,79,35,58,42,93,49,72,56,107)>;
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), (1,50,43,36,29,22,15,8)(2,51,44,37,30,23,16,9)(3,52,45,38,31,24,17,10)(4,53,46,39,32,25,18,11)(5,54,47,40,33,26,19,12)(6,55,48,41,34,27,20,13)(7,56,49,42,35,28,21,14)(57,64,71,78,85,92,99,106)(58,65,72,79,86,93,100,107)(59,66,73,80,87,94,101,108)(60,67,74,81,88,95,102,109)(61,68,75,82,89,96,103,110)(62,69,76,83,90,97,104,111)(63,70,77,84,91,98,105,112), (1,64,8,99,15,78,22,57,29,92,36,71,43,106,50,85)(2,77,9,112,16,91,23,70,30,105,37,84,44,63,51,98)(3,90,10,69,17,104,24,83,31,62,38,97,45,76,52,111)(4,103,11,82,18,61,25,96,32,75,39,110,46,89,53,68)(5,60,12,95,19,74,26,109,33,88,40,67,47,102,54,81)(6,73,13,108,20,87,27,66,34,101,41,80,48,59,55,94)(7,86,14,65,21,100,28,79,35,58,42,93,49,72,56,107) );
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)], [(1,50,43,36,29,22,15,8),(2,51,44,37,30,23,16,9),(3,52,45,38,31,24,17,10),(4,53,46,39,32,25,18,11),(5,54,47,40,33,26,19,12),(6,55,48,41,34,27,20,13),(7,56,49,42,35,28,21,14),(57,64,71,78,85,92,99,106),(58,65,72,79,86,93,100,107),(59,66,73,80,87,94,101,108),(60,67,74,81,88,95,102,109),(61,68,75,82,89,96,103,110),(62,69,76,83,90,97,104,111),(63,70,77,84,91,98,105,112)], [(1,64,8,99,15,78,22,57,29,92,36,71,43,106,50,85),(2,77,9,112,16,91,23,70,30,105,37,84,44,63,51,98),(3,90,10,69,17,104,24,83,31,62,38,97,45,76,52,111),(4,103,11,82,18,61,25,96,32,75,39,110,46,89,53,68),(5,60,12,95,19,74,26,109,33,88,40,67,47,102,54,81),(6,73,13,108,20,87,27,66,34,101,41,80,48,59,55,94),(7,86,14,65,21,100,28,79,35,58,42,93,49,72,56,107)]])
124 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | ··· | 4G | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 14A | ··· | 14I | 16A | ··· | 16H | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 28 | ··· | 28 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | - | + | - | + | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | D7 | M4(2) | Dic7 | Dic7 | D14 | C7⋊C8 | Dic14 | D28 | C8.C8 | C4.Dic7 | C56.16Q8 |
| kernel | C56.16Q8 | C28.C8 | C4×C56 | C4×C28 | C2×C56 | C56 | C56 | C56 | C4×C8 | C2×C14 | C42 | C2×C8 | C2×C8 | C8 | C8 | C8 | C7 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 1 | 1 | 3 | 2 | 3 | 3 | 3 | 12 | 6 | 6 | 8 | 12 | 48 |
Matrix representation of C56.16Q8 ►in GL2(𝔽113) generated by
| 51 | 0 |
| 0 | 100 |
| 95 | 0 |
| 0 | 44 |
| 0 | 1 |
| 69 | 0 |
G:=sub<GL(2,GF(113))| [51,0,0,100],[95,0,0,44],[0,69,1,0] >;
C56.16Q8 in GAP, Magma, Sage, TeX
C_{56}._{16}Q_8 % in TeX
G:=Group("C56.16Q8"); // GroupNames label
G:=SmallGroup(448,20);
// by ID
G=gap.SmallGroup(448,20);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,64,100,1123,136,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=1,b^4=a^28,c^2=a^21*b^2,a*b=b*a,c*a*c^-1=a^13,c*b*c^-1=b^3>;
// generators/relations
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