metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.3D8, D56.2C4, C4.11D56, C56.80D4, M5(2)⋊5D7, C8.4(C4×D7), C56.1(C2×C4), (C2×C4).9D28, (C2×D56).6C2, (C2×C28).99D4, (C2×C8).46D14, C7⋊2(M5(2)⋊C2), C8.37(C7⋊D4), (C7×M5(2))⋊9C2, C56.C4⋊11C2, (C2×C14).8SD16, C4.18(D14⋊C4), (C2×C56).50C22, C2.9(C2.D56), C28.42(C22⋊C4), C22.6(C56⋊C2), C14.17(D4⋊C4), SmallGroup(448,73)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.3D8
G = < a,b,c | a28=1, b8=a14, c2=a7, bab-1=a15, cac-1=a13, cbc-1=a21b7 >
Subgroups: 564 in 62 conjugacy classes, 25 normal (23 characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C8, C2×C4, D4, C23, D7, C14, C14, C16, C2×C8, M4(2), D8, C2×D4, C28, D14, C2×C14, C8.C4, M5(2), C2×D8, C7⋊C8, C56, D28, C2×C28, C22×D7, M5(2)⋊C2, C112, D56, D56, C4.Dic7, C2×C56, C2×D28, C56.C4, C7×M5(2), C2×D56, C28.3D8
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, D14, D4⋊C4, C4×D7, D28, C7⋊D4, M5(2)⋊C2, C56⋊C2, D56, D14⋊C4, C2.D56, C28.3D8
►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)
(1 111 48 69 8 104 55 62 15 97 34 83 22 90 41 76)(2 98 49 84 9 91 56 77 16 112 35 70 23 105 42 63)(3 85 50 71 10 106 29 64 17 99 36 57 24 92 43 78)(4 100 51 58 11 93 30 79 18 86 37 72 25 107 44 65)(5 87 52 73 12 108 31 66 19 101 38 59 26 94 45 80)(6 102 53 60 13 95 32 81 20 88 39 74 27 109 46 67)(7 89 54 75 14 110 33 68 21 103 40 61 28 96 47 82)
(1 97 8 104 15 111 22 90)(2 110 9 89 16 96 23 103)(3 95 10 102 17 109 24 88)(4 108 11 87 18 94 25 101)(5 93 12 100 19 107 26 86)(6 106 13 85 20 92 27 99)(7 91 14 98 21 105 28 112)(29 67 36 74 43 81 50 60)(30 80 37 59 44 66 51 73)(31 65 38 72 45 79 52 58)(32 78 39 57 46 64 53 71)(33 63 40 70 47 77 54 84)(34 76 41 83 48 62 55 69)(35 61 42 68 49 75 56 82)
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), (1,111,48,69,8,104,55,62,15,97,34,83,22,90,41,76)(2,98,49,84,9,91,56,77,16,112,35,70,23,105,42,63)(3,85,50,71,10,106,29,64,17,99,36,57,24,92,43,78)(4,100,51,58,11,93,30,79,18,86,37,72,25,107,44,65)(5,87,52,73,12,108,31,66,19,101,38,59,26,94,45,80)(6,102,53,60,13,95,32,81,20,88,39,74,27,109,46,67)(7,89,54,75,14,110,33,68,21,103,40,61,28,96,47,82), (1,97,8,104,15,111,22,90)(2,110,9,89,16,96,23,103)(3,95,10,102,17,109,24,88)(4,108,11,87,18,94,25,101)(5,93,12,100,19,107,26,86)(6,106,13,85,20,92,27,99)(7,91,14,98,21,105,28,112)(29,67,36,74,43,81,50,60)(30,80,37,59,44,66,51,73)(31,65,38,72,45,79,52,58)(32,78,39,57,46,64,53,71)(33,63,40,70,47,77,54,84)(34,76,41,83,48,62,55,69)(35,61,42,68,49,75,56,82)>;
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), (1,111,48,69,8,104,55,62,15,97,34,83,22,90,41,76)(2,98,49,84,9,91,56,77,16,112,35,70,23,105,42,63)(3,85,50,71,10,106,29,64,17,99,36,57,24,92,43,78)(4,100,51,58,11,93,30,79,18,86,37,72,25,107,44,65)(5,87,52,73,12,108,31,66,19,101,38,59,26,94,45,80)(6,102,53,60,13,95,32,81,20,88,39,74,27,109,46,67)(7,89,54,75,14,110,33,68,21,103,40,61,28,96,47,82), (1,97,8,104,15,111,22,90)(2,110,9,89,16,96,23,103)(3,95,10,102,17,109,24,88)(4,108,11,87,18,94,25,101)(5,93,12,100,19,107,26,86)(6,106,13,85,20,92,27,99)(7,91,14,98,21,105,28,112)(29,67,36,74,43,81,50,60)(30,80,37,59,44,66,51,73)(31,65,38,72,45,79,52,58)(32,78,39,57,46,64,53,71)(33,63,40,70,47,77,54,84)(34,76,41,83,48,62,55,69)(35,61,42,68,49,75,56,82) );
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)], [(1,111,48,69,8,104,55,62,15,97,34,83,22,90,41,76),(2,98,49,84,9,91,56,77,16,112,35,70,23,105,42,63),(3,85,50,71,10,106,29,64,17,99,36,57,24,92,43,78),(4,100,51,58,11,93,30,79,18,86,37,72,25,107,44,65),(5,87,52,73,12,108,31,66,19,101,38,59,26,94,45,80),(6,102,53,60,13,95,32,81,20,88,39,74,27,109,46,67),(7,89,54,75,14,110,33,68,21,103,40,61,28,96,47,82)], [(1,97,8,104,15,111,22,90),(2,110,9,89,16,96,23,103),(3,95,10,102,17,109,24,88),(4,108,11,87,18,94,25,101),(5,93,12,100,19,107,26,86),(6,106,13,85,20,92,27,99),(7,91,14,98,21,105,28,112),(29,67,36,74,43,81,50,60),(30,80,37,59,44,66,51,73),(31,65,38,72,45,79,52,58),(32,78,39,57,46,64,53,71),(33,63,40,70,47,77,54,84),(34,76,41,83,48,62,55,69),(35,61,42,68,49,75,56,82)]])
76 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 14A | 14B | 14C | 14D | 14E | 14F | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L | 56M | ··· | 56R | 112A | ··· | 112X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
76 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | D8 | SD16 | D14 | C4×D7 | C7⋊D4 | D28 | D56 | C56⋊C2 | M5(2)⋊C2 | C28.3D8 |
| kernel | C28.3D8 | C56.C4 | C7×M5(2) | C2×D56 | D56 | C56 | C2×C28 | M5(2) | C28 | C2×C14 | C2×C8 | C8 | C8 | C2×C4 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 6 | 6 | 12 | 12 | 2 | 12 |
Matrix representation of C28.3D8 ►in GL4(𝔽113) generated by
| 13 | 32 | 0 | 0 |
| 1 | 46 | 0 | 0 |
| 32 | 58 | 58 | 81 |
| 7 | 51 | 32 | 109 |
| 51 | 33 | 0 | 43 |
| 47 | 88 | 70 | 55 |
| 14 | 26 | 104 | 80 |
| 0 | 42 | 33 | 96 |
| 107 | 72 | 74 | 1 |
| 76 | 86 | 72 | 5 |
| 46 | 54 | 17 | 108 |
| 24 | 37 | 104 | 16 |
G:=sub<GL(4,GF(113))| [13,1,32,7,32,46,58,51,0,0,58,32,0,0,81,109],[51,47,14,0,33,88,26,42,0,70,104,33,43,55,80,96],[107,76,46,24,72,86,54,37,74,72,17,104,1,5,108,16] >;
C28.3D8 in GAP, Magma, Sage, TeX
C_{28}._3D_8 % in TeX
G:=Group("C28.3D8"); // GroupNames label
G:=SmallGroup(448,73);
// by ID
G=gap.SmallGroup(448,73);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,85,92,422,387,268,570,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=1,b^8=a^14,c^2=a^7,b*a*b^-1=a^15,c*a*c^-1=a^13,c*b*c^-1=a^21*b^7>;
// generators/relations