metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.5Q8, C8.1Dic14, C28.2SD16, C7⋊C16⋊1C4, C7⋊1(C8.Q8), C8.28(C4×D7), C28.1(C4⋊C4), C56.32(C2×C4), (C2×C28).87D4, C4.Q8.1D7, (C2×C8).40D14, C4.6(Q8⋊D7), C14.2(C4.Q8), C56.C4.4C2, C28.C8.2C2, C4.1(Dic7⋊C4), (C2×C56).46C22, (C2×C14).29SD16, C22.5(D4.D7), C2.3(C4.Dic14), (C7×C4.Q8).1C2, (C2×C4).15(C7⋊D4), SmallGroup(448,44)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.Q8
G = < a,b,c | a56=b4=1, c2=a49b2, bab-1=a43, cac-1=a13, cbc-1=a7b-1 >
Smallest permutation representation of C56.Q8
►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)
(2 44)(3 31)(4 18)(6 48)(7 35)(8 22)(10 52)(11 39)(12 26)(14 56)(15 43)(16 30)(19 47)(20 34)(23 51)(24 38)(27 55)(28 42)(32 46)(36 50)(40 54)(57 64 85 92)(58 107 86 79)(59 94 87 66)(60 81 88 109)(61 68 89 96)(62 111 90 83)(63 98 91 70)(65 72 93 100)(67 102 95 74)(69 76 97 104)(71 106 99 78)(73 80 101 108)(75 110 103 82)(77 84 105 112)
(1 106 50 71 43 92 36 57 29 78 22 99 15 64 8 85)(2 63 51 84 44 105 37 70 30 91 23 112 16 77 9 98)(3 76 52 97 45 62 38 83 31 104 24 69 17 90 10 111)(4 89 53 110 46 75 39 96 32 61 25 82 18 103 11 68)(5 102 54 67 47 88 40 109 33 74 26 95 19 60 12 81)(6 59 55 80 48 101 41 66 34 87 27 108 20 73 13 94)(7 72 56 93 49 58 42 79 35 100 28 65 21 86 14 107)
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), (2,44)(3,31)(4,18)(6,48)(7,35)(8,22)(10,52)(11,39)(12,26)(14,56)(15,43)(16,30)(19,47)(20,34)(23,51)(24,38)(27,55)(28,42)(32,46)(36,50)(40,54)(57,64,85,92)(58,107,86,79)(59,94,87,66)(60,81,88,109)(61,68,89,96)(62,111,90,83)(63,98,91,70)(65,72,93,100)(67,102,95,74)(69,76,97,104)(71,106,99,78)(73,80,101,108)(75,110,103,82)(77,84,105,112), (1,106,50,71,43,92,36,57,29,78,22,99,15,64,8,85)(2,63,51,84,44,105,37,70,30,91,23,112,16,77,9,98)(3,76,52,97,45,62,38,83,31,104,24,69,17,90,10,111)(4,89,53,110,46,75,39,96,32,61,25,82,18,103,11,68)(5,102,54,67,47,88,40,109,33,74,26,95,19,60,12,81)(6,59,55,80,48,101,41,66,34,87,27,108,20,73,13,94)(7,72,56,93,49,58,42,79,35,100,28,65,21,86,14,107)>;
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), (2,44)(3,31)(4,18)(6,48)(7,35)(8,22)(10,52)(11,39)(12,26)(14,56)(15,43)(16,30)(19,47)(20,34)(23,51)(24,38)(27,55)(28,42)(32,46)(36,50)(40,54)(57,64,85,92)(58,107,86,79)(59,94,87,66)(60,81,88,109)(61,68,89,96)(62,111,90,83)(63,98,91,70)(65,72,93,100)(67,102,95,74)(69,76,97,104)(71,106,99,78)(73,80,101,108)(75,110,103,82)(77,84,105,112), (1,106,50,71,43,92,36,57,29,78,22,99,15,64,8,85)(2,63,51,84,44,105,37,70,30,91,23,112,16,77,9,98)(3,76,52,97,45,62,38,83,31,104,24,69,17,90,10,111)(4,89,53,110,46,75,39,96,32,61,25,82,18,103,11,68)(5,102,54,67,47,88,40,109,33,74,26,95,19,60,12,81)(6,59,55,80,48,101,41,66,34,87,27,108,20,73,13,94)(7,72,56,93,49,58,42,79,35,100,28,65,21,86,14,107) );
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)], [(2,44),(3,31),(4,18),(6,48),(7,35),(8,22),(10,52),(11,39),(12,26),(14,56),(15,43),(16,30),(19,47),(20,34),(23,51),(24,38),(27,55),(28,42),(32,46),(36,50),(40,54),(57,64,85,92),(58,107,86,79),(59,94,87,66),(60,81,88,109),(61,68,89,96),(62,111,90,83),(63,98,91,70),(65,72,93,100),(67,102,95,74),(69,76,97,104),(71,106,99,78),(73,80,101,108),(75,110,103,82),(77,84,105,112)], [(1,106,50,71,43,92,36,57,29,78,22,99,15,64,8,85),(2,63,51,84,44,105,37,70,30,91,23,112,16,77,9,98),(3,76,52,97,45,62,38,83,31,104,24,69,17,90,10,111),(4,89,53,110,46,75,39,96,32,61,25,82,18,103,11,68),(5,102,54,67,47,88,40,109,33,74,26,95,19,60,12,81),(6,59,55,80,48,101,41,66,34,87,27,108,20,73,13,94),(7,72,56,93,49,58,42,79,35,100,28,65,21,86,14,107)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 14A | ··· | 14I | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 4 | 56 | 56 | 2 | ··· | 2 | 28 | 28 | 28 | 28 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
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 | Q8⋊D7 | D4.D7 | C56.Q8 |
| kernel | C56.Q8 | C28.C8 | C56.C4 | C7×C4.Q8 | C7⋊C16 | C56 | C2×C28 | C4.Q8 | 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 C56.Q8 ►in GL4(𝔽113) generated by
| 0 | 59 | 34 | 107 |
| 110 | 69 | 69 | 6 |
| 0 | 0 | 77 | 49 |
| 0 | 0 | 109 | 0 |
| 1 | 90 | 83 | 37 |
| 0 | 112 | 29 | 53 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 28 | 0 |
| 89 | 100 | 3 | 17 |
| 28 | 0 | 0 | 24 |
| 8 | 21 | 79 | 30 |
| 26 | 80 | 5 | 58 |
G:=sub<GL(4,GF(113))| [0,110,0,0,59,69,0,0,34,69,77,109,107,6,49,0],[1,0,0,0,90,112,0,0,83,29,0,28,37,53,4,0],[89,28,8,26,100,0,21,80,3,0,79,5,17,24,30,58] >;
C56.Q8 in GAP, Magma, Sage, TeX
C_{56}.Q_8 % in TeX
G:=Group("C56.Q8"); // GroupNames label
G:=SmallGroup(448,44);
// by ID
G=gap.SmallGroup(448,44);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,365,36,758,346,80,851,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=1,c^2=a^49*b^2,b*a*b^-1=a^43,c*a*c^-1=a^13,c*b*c^-1=a^7*b^-1>;
// generators/relations
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