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 C8.Dic7
G = < a,b,c | a8=1, b14=a4, c2=a4b7, ab=ba, cac-1=a-1, cbc-1=b13 >
Smallest permutation representation of C8.Dic7
►On 112 points
Generators in S112
(1 51 22 44 15 37 8 30)(2 52 23 45 16 38 9 31)(3 53 24 46 17 39 10 32)(4 54 25 47 18 40 11 33)(5 55 26 48 19 41 12 34)(6 56 27 49 20 42 13 35)(7 29 28 50 21 43 14 36)(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)(29 107 50 100 43 93 36 86)(30 92 51 85 44 106 37 99)(31 105 52 98 45 91 38 112)(32 90 53 111 46 104 39 97)(33 103 54 96 47 89 40 110)(34 88 55 109 48 102 41 95)(35 101 56 94 49 87 42 108)
G:=sub<Sym(112)| (1,51,22,44,15,37,8,30)(2,52,23,45,16,38,9,31)(3,53,24,46,17,39,10,32)(4,54,25,47,18,40,11,33)(5,55,26,48,19,41,12,34)(6,56,27,49,20,42,13,35)(7,29,28,50,21,43,14,36)(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)(29,107,50,100,43,93,36,86)(30,92,51,85,44,106,37,99)(31,105,52,98,45,91,38,112)(32,90,53,111,46,104,39,97)(33,103,54,96,47,89,40,110)(34,88,55,109,48,102,41,95)(35,101,56,94,49,87,42,108)>;
G:=Group( (1,51,22,44,15,37,8,30)(2,52,23,45,16,38,9,31)(3,53,24,46,17,39,10,32)(4,54,25,47,18,40,11,33)(5,55,26,48,19,41,12,34)(6,56,27,49,20,42,13,35)(7,29,28,50,21,43,14,36)(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)(29,107,50,100,43,93,36,86)(30,92,51,85,44,106,37,99)(31,105,52,98,45,91,38,112)(32,90,53,111,46,104,39,97)(33,103,54,96,47,89,40,110)(34,88,55,109,48,102,41,95)(35,101,56,94,49,87,42,108) );
G=PermutationGroup([(1,51,22,44,15,37,8,30),(2,52,23,45,16,38,9,31),(3,53,24,46,17,39,10,32),(4,54,25,47,18,40,11,33),(5,55,26,48,19,41,12,34),(6,56,27,49,20,42,13,35),(7,29,28,50,21,43,14,36),(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),(29,107,50,100,43,93,36,86),(30,92,51,85,44,106,37,99),(31,105,52,98,45,91,38,112),(32,90,53,111,46,104,39,97),(33,103,54,96,47,89,40,110),(34,88,55,109,48,102,41,95),(35,101,56,94,49,87,42,108)])
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 | C8.Dic7 |
| kernel | C8.Dic7 | 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 C8.Dic7 ►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] >;
C8.Dic7 in GAP, Magma, Sage, TeX
C_8.{\rm Dic}_7 % in TeX
G:=Group("C8.Dic7"); // 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