metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C56.1C4, C4.18D28, C28.34D4, C8.1Dic7, C22.2Dic14, (C2×C8).5D7, (C2×C56).7C2, C14.7(C4⋊C4), (C2×C14).3Q8, C7⋊1(C8.C4), C28.35(C2×C4), (C2×C4).70D14, C4.8(C2×Dic7), C2.5(C4⋊Dic7), C4.Dic7.1C2, (C2×C28).97C22, SmallGroup(224,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.C4
G = < a,b,c | a8=1, b14=a4, c2=a4b7, ab=ba, cac-1=a-1, cbc-1=b13 >
Smallest permutation representation of C56.C4
►On 112 points
Generators in S112
(1 52 22 45 15 38 8 31)(2 53 23 46 16 39 9 32)(3 54 24 47 17 40 10 33)(4 55 25 48 18 41 11 34)(5 56 26 49 19 42 12 35)(6 29 27 50 20 43 13 36)(7 30 28 51 21 44 14 37)(57 85 64 92 71 99 78 106)(58 86 65 93 72 100 79 107)(59 87 66 94 73 101 80 108)(60 88 67 95 74 102 81 109)(61 89 68 96 75 103 82 110)(62 90 69 97 76 104 83 111)(63 91 70 98 77 105 84 112)
(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 64 22 57 15 78 8 71)(2 77 23 70 16 63 9 84)(3 62 24 83 17 76 10 69)(4 75 25 68 18 61 11 82)(5 60 26 81 19 74 12 67)(6 73 27 66 20 59 13 80)(7 58 28 79 21 72 14 65)(29 94 50 87 43 108 36 101)(30 107 51 100 44 93 37 86)(31 92 52 85 45 106 38 99)(32 105 53 98 46 91 39 112)(33 90 54 111 47 104 40 97)(34 103 55 96 48 89 41 110)(35 88 56 109 49 102 42 95)
G:=sub<Sym(112)| (1,52,22,45,15,38,8,31)(2,53,23,46,16,39,9,32)(3,54,24,47,17,40,10,33)(4,55,25,48,18,41,11,34)(5,56,26,49,19,42,12,35)(6,29,27,50,20,43,13,36)(7,30,28,51,21,44,14,37)(57,85,64,92,71,99,78,106)(58,86,65,93,72,100,79,107)(59,87,66,94,73,101,80,108)(60,88,67,95,74,102,81,109)(61,89,68,96,75,103,82,110)(62,90,69,97,76,104,83,111)(63,91,70,98,77,105,84,112), (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), (1,64,22,57,15,78,8,71)(2,77,23,70,16,63,9,84)(3,62,24,83,17,76,10,69)(4,75,25,68,18,61,11,82)(5,60,26,81,19,74,12,67)(6,73,27,66,20,59,13,80)(7,58,28,79,21,72,14,65)(29,94,50,87,43,108,36,101)(30,107,51,100,44,93,37,86)(31,92,52,85,45,106,38,99)(32,105,53,98,46,91,39,112)(33,90,54,111,47,104,40,97)(34,103,55,96,48,89,41,110)(35,88,56,109,49,102,42,95)>;
G:=Group( (1,52,22,45,15,38,8,31)(2,53,23,46,16,39,9,32)(3,54,24,47,17,40,10,33)(4,55,25,48,18,41,11,34)(5,56,26,49,19,42,12,35)(6,29,27,50,20,43,13,36)(7,30,28,51,21,44,14,37)(57,85,64,92,71,99,78,106)(58,86,65,93,72,100,79,107)(59,87,66,94,73,101,80,108)(60,88,67,95,74,102,81,109)(61,89,68,96,75,103,82,110)(62,90,69,97,76,104,83,111)(63,91,70,98,77,105,84,112), (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), (1,64,22,57,15,78,8,71)(2,77,23,70,16,63,9,84)(3,62,24,83,17,76,10,69)(4,75,25,68,18,61,11,82)(5,60,26,81,19,74,12,67)(6,73,27,66,20,59,13,80)(7,58,28,79,21,72,14,65)(29,94,50,87,43,108,36,101)(30,107,51,100,44,93,37,86)(31,92,52,85,45,106,38,99)(32,105,53,98,46,91,39,112)(33,90,54,111,47,104,40,97)(34,103,55,96,48,89,41,110)(35,88,56,109,49,102,42,95) );
G=PermutationGroup([[(1,52,22,45,15,38,8,31),(2,53,23,46,16,39,9,32),(3,54,24,47,17,40,10,33),(4,55,25,48,18,41,11,34),(5,56,26,49,19,42,12,35),(6,29,27,50,20,43,13,36),(7,30,28,51,21,44,14,37),(57,85,64,92,71,99,78,106),(58,86,65,93,72,100,79,107),(59,87,66,94,73,101,80,108),(60,88,67,95,74,102,81,109),(61,89,68,96,75,103,82,110),(62,90,69,97,76,104,83,111),(63,91,70,98,77,105,84,112)], [(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)], [(1,64,22,57,15,78,8,71),(2,77,23,70,16,63,9,84),(3,62,24,83,17,76,10,69),(4,75,25,68,18,61,11,82),(5,60,26,81,19,74,12,67),(6,73,27,66,20,59,13,80),(7,58,28,79,21,72,14,65),(29,94,50,87,43,108,36,101),(30,107,51,100,44,93,37,86),(31,92,52,85,45,106,38,99),(32,105,53,98,46,91,39,112),(33,90,54,111,47,104,40,97),(34,103,55,96,48,89,41,110),(35,88,56,109,49,102,42,95)]])
C56.C4 is a maximal subgroup of
C56.Q8 C8.7Dic14 C112.C4 D56.1C4 C16⋊Dic7 C28.3D8 C28.4D8 D8.Dic7 Q16.Dic7 C28.58D8 D56⋊11C4 D56⋊4C4 D7×C8.C4 M4(2).25D14 M4(2).Dic7 D4.3D28 D4.4D28 D4.5D28 C56.23D4 C56.44D4 C56.29D4 D8⋊5Dic7 D8⋊4Dic7
C56.C4 is a maximal quotient of
C56⋊2C8 C56⋊1C8 C28.10C42
62 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 28A | ··· | 28L | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C4 | D4 | Q8 | D7 | Dic7 | D14 | C8.C4 | D28 | Dic14 | C56.C4 |
| kernel | C56.C4 | C4.Dic7 | C2×C56 | C56 | C28 | C2×C14 | C2×C8 | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 3 | 6 | 3 | 4 | 6 | 6 | 24 |
Matrix representation of C56.C4 ►in GL4(𝔽113) generated by
| 18 | 0 | 0 | 0 |
| 0 | 44 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 5 | 1 |
| 0 | 0 | 19 | 4 |
| 0 | 1 | 0 | 0 |
| 98 | 0 | 0 | 0 |
| 0 | 0 | 85 | 33 |
| 0 | 0 | 79 | 28 |
G:=sub<GL(4,GF(113))| [18,0,0,0,0,44,0,0,0,0,1,0,0,0,0,1],[98,0,0,0,0,98,0,0,0,0,5,19,0,0,1,4],[0,98,0,0,1,0,0,0,0,0,85,79,0,0,33,28] >;
C56.C4 in GAP, Magma, Sage, TeX
C_{56}.C_4 % in TeX
G:=Group("C56.C4"); // GroupNames label
G:=SmallGroup(224,25);
// by ID
G=gap.SmallGroup(224,25);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,121,55,86,579,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^14=a^4,c^2=a^4*b^7,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^13>;
// generators/relations
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