direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C4.D4, C23.C28, C28.58D4, M4(2)⋊3C14, C4.9(C7×D4), (D4×C14).8C2, (C2×D4).2C14, (C7×M4(2))⋊9C2, C22.3(C2×C28), (C22×C14).1C4, (C2×C28).59C22, C14.22(C22⋊C4), (C2×C4).1(C2×C14), C2.4(C7×C22⋊C4), (C2×C14).20(C2×C4), SmallGroup(224,49)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4.D4
G = < a,b,c,d | a7=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c3 >
Smallest permutation representation of C7×C4.D4
►On 56 points
Generators in S56
(1 27 17 53 16 45 35)(2 28 18 54 9 46 36)(3 29 19 55 10 47 37)(4 30 20 56 11 48 38)(5 31 21 49 12 41 39)(6 32 22 50 13 42 40)(7 25 23 51 14 43 33)(8 26 24 52 15 44 34)
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)(17 19 21 23)(18 24 22 20)(25 27 29 31)(26 32 30 28)(33 35 37 39)(34 40 38 36)(41 43 45 47)(42 48 46 44)(49 51 53 55)(50 56 54 52)
(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 4 3 2 5 8 7 6)(9 12 15 14 13 16 11 10)(17 20 19 18 21 24 23 22)(25 32 27 30 29 28 31 26)(33 40 35 38 37 36 39 34)(41 44 43 42 45 48 47 46)(49 52 51 50 53 56 55 54)
G:=sub<Sym(56)| (1,27,17,53,16,45,35)(2,28,18,54,9,46,36)(3,29,19,55,10,47,37)(4,30,20,56,11,48,38)(5,31,21,49,12,41,39)(6,32,22,50,13,42,40)(7,25,23,51,14,43,33)(8,26,24,52,15,44,34), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,19,21,23)(18,24,22,20)(25,27,29,31)(26,32,30,28)(33,35,37,39)(34,40,38,36)(41,43,45,47)(42,48,46,44)(49,51,53,55)(50,56,54,52), (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), (1,4,3,2,5,8,7,6)(9,12,15,14,13,16,11,10)(17,20,19,18,21,24,23,22)(25,32,27,30,29,28,31,26)(33,40,35,38,37,36,39,34)(41,44,43,42,45,48,47,46)(49,52,51,50,53,56,55,54)>;
G:=Group( (1,27,17,53,16,45,35)(2,28,18,54,9,46,36)(3,29,19,55,10,47,37)(4,30,20,56,11,48,38)(5,31,21,49,12,41,39)(6,32,22,50,13,42,40)(7,25,23,51,14,43,33)(8,26,24,52,15,44,34), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,19,21,23)(18,24,22,20)(25,27,29,31)(26,32,30,28)(33,35,37,39)(34,40,38,36)(41,43,45,47)(42,48,46,44)(49,51,53,55)(50,56,54,52), (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), (1,4,3,2,5,8,7,6)(9,12,15,14,13,16,11,10)(17,20,19,18,21,24,23,22)(25,32,27,30,29,28,31,26)(33,40,35,38,37,36,39,34)(41,44,43,42,45,48,47,46)(49,52,51,50,53,56,55,54) );
G=PermutationGroup([[(1,27,17,53,16,45,35),(2,28,18,54,9,46,36),(3,29,19,55,10,47,37),(4,30,20,56,11,48,38),(5,31,21,49,12,41,39),(6,32,22,50,13,42,40),(7,25,23,51,14,43,33),(8,26,24,52,15,44,34)], [(1,3,5,7),(2,8,6,4),(9,15,13,11),(10,12,14,16),(17,19,21,23),(18,24,22,20),(25,27,29,31),(26,32,30,28),(33,35,37,39),(34,40,38,36),(41,43,45,47),(42,48,46,44),(49,51,53,55),(50,56,54,52)], [(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)], [(1,4,3,2,5,8,7,6),(9,12,15,14,13,16,11,10),(17,20,19,18,21,24,23,22),(25,32,27,30,29,28,31,26),(33,40,35,38,37,36,39,34),(41,44,43,42,45,48,47,46),(49,52,51,50,53,56,55,54)]])
C7×C4.D4 is a maximal subgroup of
C23.3D28 C23.4D28 M4(2).19D14 D28.1D4 D28⋊1D4 D28.2D4 D28.3D4
77 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14X | 28A | ··· | 28L | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 | D4 | C7×D4 | C4.D4 | C7×C4.D4 |
| kernel | C7×C4.D4 | C7×M4(2) | D4×C14 | C22×C14 | C4.D4 | M4(2) | C2×D4 | C23 | C28 | C4 | C7 | C1 |
| # reps | 1 | 2 | 1 | 4 | 6 | 12 | 6 | 24 | 2 | 12 | 1 | 6 |
Matrix representation of C7×C4.D4 ►in GL6(𝔽113)
| 106 | 0 | 0 | 0 | 0 | 0 |
| 0 | 106 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 112 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 112 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
G:=sub<GL(6,GF(113))| [106,0,0,0,0,0,0,106,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0],[1,112,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,1,0,0,0],[1,112,0,0,0,0,2,112,0,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
C7×C4.D4 in GAP, Magma, Sage, TeX
C_7\times C_4.D_4
% in TeX
G:=Group("C7xC4.D4"); // GroupNames label
G:=SmallGroup(224,49);
// by ID
G=gap.SmallGroup(224,49);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-2,336,361,3363,2530,88]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=1,c^4=b^2,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations
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