metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.8D4, C23.Dic7, (C2×D4).2D7, (C2×C4).3D14, (D4×C14).2C2, C7⋊2(C4.D4), C4.Dic7⋊3C2, C4.13(C7⋊D4), (C22×C14).2C4, (C2×C28).17C22, C2.4(C23.D7), C22.2(C2×Dic7), C14.14(C22⋊C4), (C2×C14).28(C2×C4), SmallGroup(224,39)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.D4
G = < a,b,c | a28=1, b4=a14, c2=a21, bab-1=a-1, cac-1=a13, cbc-1=a7b3 >
Smallest permutation representation of C28.D4
►On 56 points
(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)
(1 34 22 41 15 48 8 55)(2 33 23 40 16 47 9 54)(3 32 24 39 17 46 10 53)(4 31 25 38 18 45 11 52)(5 30 26 37 19 44 12 51)(6 29 27 36 20 43 13 50)(7 56 28 35 21 42 14 49)
(1 48 22 41 15 34 8 55)(2 33 23 54 16 47 9 40)(3 46 24 39 17 32 10 53)(4 31 25 52 18 45 11 38)(5 44 26 37 19 30 12 51)(6 29 27 50 20 43 13 36)(7 42 28 35 21 56 14 49)
G:=sub<Sym(56)| (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), (1,34,22,41,15,48,8,55)(2,33,23,40,16,47,9,54)(3,32,24,39,17,46,10,53)(4,31,25,38,18,45,11,52)(5,30,26,37,19,44,12,51)(6,29,27,36,20,43,13,50)(7,56,28,35,21,42,14,49), (1,48,22,41,15,34,8,55)(2,33,23,54,16,47,9,40)(3,46,24,39,17,32,10,53)(4,31,25,52,18,45,11,38)(5,44,26,37,19,30,12,51)(6,29,27,50,20,43,13,36)(7,42,28,35,21,56,14,49)>;
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), (1,34,22,41,15,48,8,55)(2,33,23,40,16,47,9,54)(3,32,24,39,17,46,10,53)(4,31,25,38,18,45,11,52)(5,30,26,37,19,44,12,51)(6,29,27,36,20,43,13,50)(7,56,28,35,21,42,14,49), (1,48,22,41,15,34,8,55)(2,33,23,54,16,47,9,40)(3,46,24,39,17,32,10,53)(4,31,25,52,18,45,11,38)(5,44,26,37,19,30,12,51)(6,29,27,50,20,43,13,36)(7,42,28,35,21,56,14,49) );
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)], [(1,34,22,41,15,48,8,55),(2,33,23,40,16,47,9,54),(3,32,24,39,17,46,10,53),(4,31,25,38,18,45,11,52),(5,30,26,37,19,44,12,51),(6,29,27,36,20,43,13,50),(7,56,28,35,21,42,14,49)], [(1,48,22,41,15,34,8,55),(2,33,23,54,16,47,9,40),(3,46,24,39,17,32,10,53),(4,31,25,52,18,45,11,38),(5,44,26,37,19,30,12,51),(6,29,27,50,20,43,13,36),(7,42,28,35,21,56,14,49)]])
C28.D4 is a maximal subgroup of
C7⋊C2≀C4 (C2×C28).D4 C24⋊Dic7 (C22×C28)⋊C4 D7×C4.D4 M4(2).19D14 C42⋊5D14 D28⋊5D4 C56.23D4 C56.44D4 M4(2).D14 M4(2).13D14 (D4×C14).16C4 2+ 1+4⋊D7 2+ 1+4.D7
C28.D4 is a maximal quotient of
C24.Dic7 C28.(C4⋊C4) C42.7D14 C28.9D8 C28.5Q16
41 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 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 | D14 | Dic7 | C7⋊D4 | C4.D4 | C28.D4 |
| kernel | C28.D4 | C4.Dic7 | D4×C14 | C22×C14 | C28 | C2×D4 | C2×C4 | C23 | C4 | C7 | C1 |
| # reps | 1 | 2 | 1 | 4 | 2 | 3 | 3 | 6 | 12 | 1 | 6 |
Matrix representation of C28.D4 ►in GL4(𝔽113) generated by
| 7 | 14 | 0 | 0 |
| 106 | 106 | 0 | 0 |
| 14 | 38 | 0 | 16 |
| 89 | 75 | 97 | 0 |
| 70 | 0 | 111 | 0 |
| 43 | 0 | 1 | 112 |
| 21 | 1 | 43 | 0 |
| 92 | 0 | 70 | 0 |
| 70 | 0 | 111 | 0 |
| 0 | 0 | 1 | 1 |
| 21 | 1 | 43 | 0 |
| 93 | 0 | 70 | 0 |
G:=sub<GL(4,GF(113))| [7,106,14,89,14,106,38,75,0,0,0,97,0,0,16,0],[70,43,21,92,0,0,1,0,111,1,43,70,0,112,0,0],[70,0,21,93,0,0,1,0,111,1,43,70,0,1,0,0] >;
C28.D4 in GAP, Magma, Sage, TeX
C_{28}.D_4 % in TeX
G:=Group("C28.D4"); // GroupNames label
G:=SmallGroup(224,39);
// by ID
G=gap.SmallGroup(224,39);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,121,188,86,579,6917]);
// 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^7*b^3>;
// generators/relations
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