metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.53D4, M4(2).1D7, C22.1Dic14, C7⋊C8.1C4, (C2×C14).Q8, C28.5(C2×C4), C4.13(C4×D7), C14.8(C4⋊C4), C7⋊2(C8.C4), (C2×C4).37D14, C4.28(C7⋊D4), C4.Dic7.2C2, C2.5(Dic7⋊C4), (C2×C28).12C22, (C7×M4(2)).1C2, (C2×C7⋊C8).4C2, SmallGroup(224,28)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.53D4
G = < a,b,c | a28=1, b4=a14, c2=a21, bab-1=cac-1=a13, cbc-1=a14b3 >
Smallest permutation representation of C28.53D4
►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 67 8 74 15 81 22 60)(2 80 9 59 16 66 23 73)(3 65 10 72 17 79 24 58)(4 78 11 57 18 64 25 71)(5 63 12 70 19 77 26 84)(6 76 13 83 20 62 27 69)(7 61 14 68 21 75 28 82)(29 99 50 92 43 85 36 106)(30 112 51 105 44 98 37 91)(31 97 52 90 45 111 38 104)(32 110 53 103 46 96 39 89)(33 95 54 88 47 109 40 102)(34 108 55 101 48 94 41 87)(35 93 56 86 49 107 42 100)
(1 109 22 102 15 95 8 88)(2 94 23 87 16 108 9 101)(3 107 24 100 17 93 10 86)(4 92 25 85 18 106 11 99)(5 105 26 98 19 91 12 112)(6 90 27 111 20 104 13 97)(7 103 28 96 21 89 14 110)(29 78 50 71 43 64 36 57)(30 63 51 84 44 77 37 70)(31 76 52 69 45 62 38 83)(32 61 53 82 46 75 39 68)(33 74 54 67 47 60 40 81)(34 59 55 80 48 73 41 66)(35 72 56 65 49 58 42 79)
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), (1,67,8,74,15,81,22,60)(2,80,9,59,16,66,23,73)(3,65,10,72,17,79,24,58)(4,78,11,57,18,64,25,71)(5,63,12,70,19,77,26,84)(6,76,13,83,20,62,27,69)(7,61,14,68,21,75,28,82)(29,99,50,92,43,85,36,106)(30,112,51,105,44,98,37,91)(31,97,52,90,45,111,38,104)(32,110,53,103,46,96,39,89)(33,95,54,88,47,109,40,102)(34,108,55,101,48,94,41,87)(35,93,56,86,49,107,42,100), (1,109,22,102,15,95,8,88)(2,94,23,87,16,108,9,101)(3,107,24,100,17,93,10,86)(4,92,25,85,18,106,11,99)(5,105,26,98,19,91,12,112)(6,90,27,111,20,104,13,97)(7,103,28,96,21,89,14,110)(29,78,50,71,43,64,36,57)(30,63,51,84,44,77,37,70)(31,76,52,69,45,62,38,83)(32,61,53,82,46,75,39,68)(33,74,54,67,47,60,40,81)(34,59,55,80,48,73,41,66)(35,72,56,65,49,58,42,79)>;
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), (1,67,8,74,15,81,22,60)(2,80,9,59,16,66,23,73)(3,65,10,72,17,79,24,58)(4,78,11,57,18,64,25,71)(5,63,12,70,19,77,26,84)(6,76,13,83,20,62,27,69)(7,61,14,68,21,75,28,82)(29,99,50,92,43,85,36,106)(30,112,51,105,44,98,37,91)(31,97,52,90,45,111,38,104)(32,110,53,103,46,96,39,89)(33,95,54,88,47,109,40,102)(34,108,55,101,48,94,41,87)(35,93,56,86,49,107,42,100), (1,109,22,102,15,95,8,88)(2,94,23,87,16,108,9,101)(3,107,24,100,17,93,10,86)(4,92,25,85,18,106,11,99)(5,105,26,98,19,91,12,112)(6,90,27,111,20,104,13,97)(7,103,28,96,21,89,14,110)(29,78,50,71,43,64,36,57)(30,63,51,84,44,77,37,70)(31,76,52,69,45,62,38,83)(32,61,53,82,46,75,39,68)(33,74,54,67,47,60,40,81)(34,59,55,80,48,73,41,66)(35,72,56,65,49,58,42,79) );
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)], [(1,67,8,74,15,81,22,60),(2,80,9,59,16,66,23,73),(3,65,10,72,17,79,24,58),(4,78,11,57,18,64,25,71),(5,63,12,70,19,77,26,84),(6,76,13,83,20,62,27,69),(7,61,14,68,21,75,28,82),(29,99,50,92,43,85,36,106),(30,112,51,105,44,98,37,91),(31,97,52,90,45,111,38,104),(32,110,53,103,46,96,39,89),(33,95,54,88,47,109,40,102),(34,108,55,101,48,94,41,87),(35,93,56,86,49,107,42,100)], [(1,109,22,102,15,95,8,88),(2,94,23,87,16,108,9,101),(3,107,24,100,17,93,10,86),(4,92,25,85,18,106,11,99),(5,105,26,98,19,91,12,112),(6,90,27,111,20,104,13,97),(7,103,28,96,21,89,14,110),(29,78,50,71,43,64,36,57),(30,63,51,84,44,77,37,70),(31,76,52,69,45,62,38,83),(32,61,53,82,46,75,39,68),(33,74,54,67,47,60,40,81),(34,59,55,80,48,73,41,66),(35,72,56,65,49,58,42,79)]])
C28.53D4 is a maximal subgroup of
D28.2D4 D28.3D4 D28.6D4 D28.7D4 M4(2).22D14 C42.196D14 D7×C8.C4 M4(2).25D14 C23.Dic14 C56.93D4 C56.50D4 M4(2).D14 M4(2).13D14 M4(2).15D14 M4(2).16D14
C28.53D4 is a maximal quotient of
C28.53D8 C28.39SD16 C28.4C42
44 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | 14B | 14C | 14D | 14E | 14F | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | D7 | D14 | C8.C4 | C4×D7 | C7⋊D4 | Dic14 | C28.53D4 |
| kernel | C28.53D4 | C2×C7⋊C8 | C4.Dic7 | C7×M4(2) | C7⋊C8 | C28 | C2×C14 | M4(2) | C2×C4 | C7 | C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 3 | 4 | 6 | 6 | 6 | 6 |
Matrix representation of C28.53D4 ►in GL4(𝔽113) generated by
| 10 | 104 | 0 | 0 |
| 99 | 24 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 61 | 110 | 0 | 0 |
| 73 | 52 | 0 | 0 |
| 0 | 0 | 69 | 0 |
| 0 | 0 | 47 | 95 |
| 82 | 4 | 0 | 0 |
| 42 | 31 | 0 | 0 |
| 0 | 0 | 39 | 36 |
| 0 | 0 | 108 | 74 |
G:=sub<GL(4,GF(113))| [10,99,0,0,104,24,0,0,0,0,98,0,0,0,0,98],[61,73,0,0,110,52,0,0,0,0,69,47,0,0,0,95],[82,42,0,0,4,31,0,0,0,0,39,108,0,0,36,74] >;
C28.53D4 in GAP, Magma, Sage, TeX
C_{28}._{53}D_4 % in TeX
G:=Group("C28.53D4"); // GroupNames label
G:=SmallGroup(224,28);
// by ID
G=gap.SmallGroup(224,28);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,48,121,31,86,297,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^28=1,b^4=a^14,c^2=a^21,b*a*b^-1=c*a*c^-1=a^13,c*b*c^-1=a^14*b^3>;
// generators/relations
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