metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊1C4, C42⋊3D7, C4.17D28, C28.33D4, Dic14⋊1C4, C7⋊1C4≀C2, (C4×C28)⋊6C2, C4.6(C4×D7), C28.16(C2×C4), C4○D28.1C2, (C2×C14).26D4, (C2×C4).66D14, C4.Dic7⋊1C2, C2.3(D14⋊C4), C14.1(C22⋊C4), (C2×C28).96C22, C22.7(C7⋊D4), SmallGroup(224,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic14⋊C4
G = < a,b,c | a28=c4=1, b2=a14, bab-1=a-1, ac=ca, cbc-1=a7b >
Smallest permutation representation of Dic14⋊C4
►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 36 15 50)(2 35 16 49)(3 34 17 48)(4 33 18 47)(5 32 19 46)(6 31 20 45)(7 30 21 44)(8 29 22 43)(9 56 23 42)(10 55 24 41)(11 54 25 40)(12 53 26 39)(13 52 27 38)(14 51 28 37)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 50 43 36)(30 51 44 37)(31 52 45 38)(32 53 46 39)(33 54 47 40)(34 55 48 41)(35 56 49 42)
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), (1,36,15,50)(2,35,16,49)(3,34,17,48)(4,33,18,47)(5,32,19,46)(6,31,20,45)(7,30,21,44)(8,29,22,43)(9,56,23,42)(10,55,24,41)(11,54,25,40)(12,53,26,39)(13,52,27,38)(14,51,28,37), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,50,43,36)(30,51,44,37)(31,52,45,38)(32,53,46,39)(33,54,47,40)(34,55,48,41)(35,56,49,42)>;
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), (1,36,15,50)(2,35,16,49)(3,34,17,48)(4,33,18,47)(5,32,19,46)(6,31,20,45)(7,30,21,44)(8,29,22,43)(9,56,23,42)(10,55,24,41)(11,54,25,40)(12,53,26,39)(13,52,27,38)(14,51,28,37), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,50,43,36)(30,51,44,37)(31,52,45,38)(32,53,46,39)(33,54,47,40)(34,55,48,41)(35,56,49,42) );
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)], [(1,36,15,50),(2,35,16,49),(3,34,17,48),(4,33,18,47),(5,32,19,46),(6,31,20,45),(7,30,21,44),(8,29,22,43),(9,56,23,42),(10,55,24,41),(11,54,25,40),(12,53,26,39),(13,52,27,38),(14,51,28,37)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,50,43,36),(30,51,44,37),(31,52,45,38),(32,53,46,39),(33,54,47,40),(34,55,48,41),(35,56,49,42)]])
Dic14⋊C4 is a maximal subgroup of
D56⋊11C4 D56⋊4C4 D7×C4≀C2 C42⋊D14 D4⋊4D28 M4(2).22D14 C42.196D14 M4(2)⋊D14 D4.9D28 D4.10D28 C42⋊4D14 C42⋊5D14 D28.14D4 D28⋊5D4 D28.15D4
Dic14⋊C4 is a maximal quotient of
C14.C4≀C2 C4⋊Dic7⋊C4 C4.8Dic28 C4.17D56 C42.D14 C42.2D14 C28.8C42
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | ··· | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 14A | ··· | 14I | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 28 | 1 | 1 | 2 | ··· | 2 | 28 | 2 | 2 | 2 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D7 | D14 | C4≀C2 | C4×D7 | D28 | C7⋊D4 | Dic14⋊C4 |
| kernel | Dic14⋊C4 | C4.Dic7 | C4×C28 | C4○D28 | Dic14 | D28 | C28 | C2×C14 | C42 | C2×C4 | C7 | C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 3 | 3 | 4 | 6 | 6 | 6 | 24 |
Matrix representation of Dic14⋊C4 ►in GL2(𝔽29) generated by
| 2 | 27 |
| 5 | 10 |
| 15 | 2 |
| 3 | 14 |
| 18 | 5 |
| 2 | 27 |
G:=sub<GL(2,GF(29))| [2,5,27,10],[15,3,2,14],[18,2,5,27] >;
Dic14⋊C4 in GAP, Magma, Sage, TeX
{\rm Dic}_{14}\rtimes C_4 % in TeX
G:=Group("Dic14:C4"); // GroupNames label
G:=SmallGroup(224,11);
// by ID
G=gap.SmallGroup(224,11);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,121,31,362,579,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^28=c^4=1,b^2=a^14,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^7*b>;
// generators/relations
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