metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.10D4, (C2×C4).Dic7, (C2×C28).1C4, (C2×C4).4D14, (C2×Q8).2D7, (Q8×C14).2C2, C4.15(C7⋊D4), C7⋊2(C4.10D4), C4.Dic7.4C2, (C2×C28).19C22, C2.7(C23.D7), C22.4(C2×Dic7), C14.17(C22⋊C4), (C2×C14).30(C2×C4), SmallGroup(224,42)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.10D4
G = < a,b,c | a28=1, b4=a14, c2=a21, bab-1=a-1, cac-1=a13, cbc-1=a21b3 >
Smallest permutation representation of C28.10D4
►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 88 8 109 15 102 22 95)(2 87 9 108 16 101 23 94)(3 86 10 107 17 100 24 93)(4 85 11 106 18 99 25 92)(5 112 12 105 19 98 26 91)(6 111 13 104 20 97 27 90)(7 110 14 103 21 96 28 89)(29 69 36 62 43 83 50 76)(30 68 37 61 44 82 51 75)(31 67 38 60 45 81 52 74)(32 66 39 59 46 80 53 73)(33 65 40 58 47 79 54 72)(34 64 41 57 48 78 55 71)(35 63 42 84 49 77 56 70)
(1 65 22 58 15 79 8 72)(2 78 23 71 16 64 9 57)(3 63 24 84 17 77 10 70)(4 76 25 69 18 62 11 83)(5 61 26 82 19 75 12 68)(6 74 27 67 20 60 13 81)(7 59 28 80 21 73 14 66)(29 106 50 99 43 92 36 85)(30 91 51 112 44 105 37 98)(31 104 52 97 45 90 38 111)(32 89 53 110 46 103 39 96)(33 102 54 95 47 88 40 109)(34 87 55 108 48 101 41 94)(35 100 56 93 49 86 42 107)
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), (1,88,8,109,15,102,22,95)(2,87,9,108,16,101,23,94)(3,86,10,107,17,100,24,93)(4,85,11,106,18,99,25,92)(5,112,12,105,19,98,26,91)(6,111,13,104,20,97,27,90)(7,110,14,103,21,96,28,89)(29,69,36,62,43,83,50,76)(30,68,37,61,44,82,51,75)(31,67,38,60,45,81,52,74)(32,66,39,59,46,80,53,73)(33,65,40,58,47,79,54,72)(34,64,41,57,48,78,55,71)(35,63,42,84,49,77,56,70), (1,65,22,58,15,79,8,72)(2,78,23,71,16,64,9,57)(3,63,24,84,17,77,10,70)(4,76,25,69,18,62,11,83)(5,61,26,82,19,75,12,68)(6,74,27,67,20,60,13,81)(7,59,28,80,21,73,14,66)(29,106,50,99,43,92,36,85)(30,91,51,112,44,105,37,98)(31,104,52,97,45,90,38,111)(32,89,53,110,46,103,39,96)(33,102,54,95,47,88,40,109)(34,87,55,108,48,101,41,94)(35,100,56,93,49,86,42,107)>;
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), (1,88,8,109,15,102,22,95)(2,87,9,108,16,101,23,94)(3,86,10,107,17,100,24,93)(4,85,11,106,18,99,25,92)(5,112,12,105,19,98,26,91)(6,111,13,104,20,97,27,90)(7,110,14,103,21,96,28,89)(29,69,36,62,43,83,50,76)(30,68,37,61,44,82,51,75)(31,67,38,60,45,81,52,74)(32,66,39,59,46,80,53,73)(33,65,40,58,47,79,54,72)(34,64,41,57,48,78,55,71)(35,63,42,84,49,77,56,70), (1,65,22,58,15,79,8,72)(2,78,23,71,16,64,9,57)(3,63,24,84,17,77,10,70)(4,76,25,69,18,62,11,83)(5,61,26,82,19,75,12,68)(6,74,27,67,20,60,13,81)(7,59,28,80,21,73,14,66)(29,106,50,99,43,92,36,85)(30,91,51,112,44,105,37,98)(31,104,52,97,45,90,38,111)(32,89,53,110,46,103,39,96)(33,102,54,95,47,88,40,109)(34,87,55,108,48,101,41,94)(35,100,56,93,49,86,42,107) );
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)], [(1,88,8,109,15,102,22,95),(2,87,9,108,16,101,23,94),(3,86,10,107,17,100,24,93),(4,85,11,106,18,99,25,92),(5,112,12,105,19,98,26,91),(6,111,13,104,20,97,27,90),(7,110,14,103,21,96,28,89),(29,69,36,62,43,83,50,76),(30,68,37,61,44,82,51,75),(31,67,38,60,45,81,52,74),(32,66,39,59,46,80,53,73),(33,65,40,58,47,79,54,72),(34,64,41,57,48,78,55,71),(35,63,42,84,49,77,56,70)], [(1,65,22,58,15,79,8,72),(2,78,23,71,16,64,9,57),(3,63,24,84,17,77,10,70),(4,76,25,69,18,62,11,83),(5,61,26,82,19,75,12,68),(6,74,27,67,20,60,13,81),(7,59,28,80,21,73,14,66),(29,106,50,99,43,92,36,85),(30,91,51,112,44,105,37,98),(31,104,52,97,45,90,38,111),(32,89,53,110,46,103,39,96),(33,102,54,95,47,88,40,109),(34,87,55,108,48,101,41,94),(35,100,56,93,49,86,42,107)]])
C28.10D4 is a maximal subgroup of
(C2×C4).D28 (C2×Q8).D14 C42.Dic7 C42.3Dic7 D7×C4.10D4 M4(2).21D14 D28.14D4 D28.15D4 C56.44D4 C56.29D4 M4(2).15D14 M4(2).16D14 (D4×C14).16C4 2- 1+4⋊D7 2- 1+4.D7
C28.10D4 is a maximal quotient of
(C2×C28)⋊C8 C28.(C4⋊C4) C42.8D14 C28.10D8
41 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C4 | D4 | D7 | Dic7 | D14 | C7⋊D4 | C4.10D4 | C28.10D4 |
| kernel | C28.10D4 | C4.Dic7 | Q8×C14 | C2×C28 | C28 | C2×Q8 | C2×C4 | C2×C4 | C4 | C7 | C1 |
| # reps | 1 | 2 | 1 | 4 | 2 | 3 | 6 | 3 | 12 | 1 | 6 |
Matrix representation of C28.10D4 ►in GL4(𝔽113) generated by
| 16 | 81 | 0 | 0 |
| 16 | 97 | 0 | 0 |
| 70 | 50 | 106 | 81 |
| 22 | 0 | 109 | 7 |
| 85 | 0 | 77 | 10 |
| 37 | 0 | 96 | 63 |
| 64 | 90 | 28 | 55 |
| 5 | 53 | 0 | 0 |
| 13 | 0 | 49 | 0 |
| 0 | 0 | 0 | 1 |
| 45 | 53 | 100 | 0 |
| 1 | 112 | 0 | 0 |
G:=sub<GL(4,GF(113))| [16,16,70,22,81,97,50,0,0,0,106,109,0,0,81,7],[85,37,64,5,0,0,90,53,77,96,28,0,10,63,55,0],[13,0,45,1,0,0,53,112,49,0,100,0,0,1,0,0] >;
C28.10D4 in GAP, Magma, Sage, TeX
C_{28}._{10}D_4 % in TeX
G:=Group("C28.10D4"); // GroupNames label
G:=SmallGroup(224,42);
// by ID
G=gap.SmallGroup(224,42);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,121,103,188,86,579,6917]);
// Polycyclic
G:=Group<a,b,c|a^28=1,b^4=a^14,c^2=a^21,b*a*b^-1=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^21*b^3>;
// generators/relations
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