metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.47D4, C4.12D28, M4(2).2D7, (C2×C4).2D14, (C2×Dic7).C4, C22.5(C4×D7), C4.22(C7⋊D4), C7⋊1(C4.10D4), C4.Dic7.3C2, C2.10(D14⋊C4), C14.9(C22⋊C4), (C2×C28).14C22, (C2×Dic14).6C2, (C7×M4(2)).2C2, (C2×C14).3(C2×C4), SmallGroup(224,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.47D4
G = < a,b,c | a28=1, b4=c2=a14, bab-1=cac-1=a-1, cbc-1=a21b3 >
Smallest permutation representation of C28.47D4
►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)
(1 101 22 108 15 87 8 94)(2 100 23 107 16 86 9 93)(3 99 24 106 17 85 10 92)(4 98 25 105 18 112 11 91)(5 97 26 104 19 111 12 90)(6 96 27 103 20 110 13 89)(7 95 28 102 21 109 14 88)(29 79 36 72 43 65 50 58)(30 78 37 71 44 64 51 57)(31 77 38 70 45 63 52 84)(32 76 39 69 46 62 53 83)(33 75 40 68 47 61 54 82)(34 74 41 67 48 60 55 81)(35 73 42 66 49 59 56 80)
(1 74 15 60)(2 73 16 59)(3 72 17 58)(4 71 18 57)(5 70 19 84)(6 69 20 83)(7 68 21 82)(8 67 22 81)(9 66 23 80)(10 65 24 79)(11 64 25 78)(12 63 26 77)(13 62 27 76)(14 61 28 75)(29 85 43 99)(30 112 44 98)(31 111 45 97)(32 110 46 96)(33 109 47 95)(34 108 48 94)(35 107 49 93)(36 106 50 92)(37 105 51 91)(38 104 52 90)(39 103 53 89)(40 102 54 88)(41 101 55 87)(42 100 56 86)
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), (1,101,22,108,15,87,8,94)(2,100,23,107,16,86,9,93)(3,99,24,106,17,85,10,92)(4,98,25,105,18,112,11,91)(5,97,26,104,19,111,12,90)(6,96,27,103,20,110,13,89)(7,95,28,102,21,109,14,88)(29,79,36,72,43,65,50,58)(30,78,37,71,44,64,51,57)(31,77,38,70,45,63,52,84)(32,76,39,69,46,62,53,83)(33,75,40,68,47,61,54,82)(34,74,41,67,48,60,55,81)(35,73,42,66,49,59,56,80), (1,74,15,60)(2,73,16,59)(3,72,17,58)(4,71,18,57)(5,70,19,84)(6,69,20,83)(7,68,21,82)(8,67,22,81)(9,66,23,80)(10,65,24,79)(11,64,25,78)(12,63,26,77)(13,62,27,76)(14,61,28,75)(29,85,43,99)(30,112,44,98)(31,111,45,97)(32,110,46,96)(33,109,47,95)(34,108,48,94)(35,107,49,93)(36,106,50,92)(37,105,51,91)(38,104,52,90)(39,103,53,89)(40,102,54,88)(41,101,55,87)(42,100,56,86)>;
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), (1,101,22,108,15,87,8,94)(2,100,23,107,16,86,9,93)(3,99,24,106,17,85,10,92)(4,98,25,105,18,112,11,91)(5,97,26,104,19,111,12,90)(6,96,27,103,20,110,13,89)(7,95,28,102,21,109,14,88)(29,79,36,72,43,65,50,58)(30,78,37,71,44,64,51,57)(31,77,38,70,45,63,52,84)(32,76,39,69,46,62,53,83)(33,75,40,68,47,61,54,82)(34,74,41,67,48,60,55,81)(35,73,42,66,49,59,56,80), (1,74,15,60)(2,73,16,59)(3,72,17,58)(4,71,18,57)(5,70,19,84)(6,69,20,83)(7,68,21,82)(8,67,22,81)(9,66,23,80)(10,65,24,79)(11,64,25,78)(12,63,26,77)(13,62,27,76)(14,61,28,75)(29,85,43,99)(30,112,44,98)(31,111,45,97)(32,110,46,96)(33,109,47,95)(34,108,48,94)(35,107,49,93)(36,106,50,92)(37,105,51,91)(38,104,52,90)(39,103,53,89)(40,102,54,88)(41,101,55,87)(42,100,56,86) );
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)], [(1,101,22,108,15,87,8,94),(2,100,23,107,16,86,9,93),(3,99,24,106,17,85,10,92),(4,98,25,105,18,112,11,91),(5,97,26,104,19,111,12,90),(6,96,27,103,20,110,13,89),(7,95,28,102,21,109,14,88),(29,79,36,72,43,65,50,58),(30,78,37,71,44,64,51,57),(31,77,38,70,45,63,52,84),(32,76,39,69,46,62,53,83),(33,75,40,68,47,61,54,82),(34,74,41,67,48,60,55,81),(35,73,42,66,49,59,56,80)], [(1,74,15,60),(2,73,16,59),(3,72,17,58),(4,71,18,57),(5,70,19,84),(6,69,20,83),(7,68,21,82),(8,67,22,81),(9,66,23,80),(10,65,24,79),(11,64,25,78),(12,63,26,77),(13,62,27,76),(14,61,28,75),(29,85,43,99),(30,112,44,98),(31,111,45,97),(32,110,46,96),(33,109,47,95),(34,108,48,94),(35,107,49,93),(36,106,50,92),(37,105,51,91),(38,104,52,90),(39,103,53,89),(40,102,54,88),(41,101,55,87),(42,100,56,86)])
41 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 14D | 14E | 14F | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D7 | D14 | D28 | C7⋊D4 | C4×D7 | C4.10D4 | C28.47D4 |
| kernel | C28.47D4 | C4.Dic7 | C7×M4(2) | C2×Dic14 | C2×Dic7 | C28 | M4(2) | C2×C4 | C4 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 3 | 3 | 6 | 6 | 6 | 1 | 6 |
Matrix representation of C28.47D4 ►in GL4(𝔽113) generated by
| 67 | 81 | 0 | 0 |
| 112 | 100 | 0 | 0 |
| 0 | 0 | 67 | 81 |
| 0 | 0 | 112 | 100 |
| 0 | 0 | 80 | 40 |
| 0 | 0 | 1 | 33 |
| 33 | 81 | 0 | 0 |
| 27 | 80 | 0 | 0 |
| 80 | 40 | 0 | 0 |
| 1 | 33 | 0 | 0 |
| 0 | 0 | 80 | 40 |
| 0 | 0 | 1 | 33 |
G:=sub<GL(4,GF(113))| [67,112,0,0,81,100,0,0,0,0,67,112,0,0,81,100],[0,0,33,27,0,0,81,80,80,1,0,0,40,33,0,0],[80,1,0,0,40,33,0,0,0,0,80,1,0,0,40,33] >;
C28.47D4 in GAP, Magma, Sage, TeX
C_{28}._{47}D_4 % in TeX
G:=Group("C28.47D4"); // GroupNames label
G:=SmallGroup(224,30);
// by ID
G=gap.SmallGroup(224,30);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,121,31,362,86,297,6917]);
// Polycyclic
G:=Group<a,b,c|a^28=1,b^4=c^2=a^14,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^21*b^3>;
// generators/relations