metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.6D14, C28.15D4, D28⋊6C22, C28.12C23, Dic14⋊5C22, D4⋊D7⋊5C2, (C2×D4)⋊2D7, C7⋊C8⋊3C22, C4○D28⋊3C2, (D4×C14)⋊2C2, C7⋊4(C8⋊C22), D4.D7⋊5C2, C14.45(C2×D4), (C2×C14).39D4, (C2×C4).17D14, C4.Dic7⋊6C2, C4.16(C7⋊D4), (C7×D4).6C22, C4.12(C22×D7), (C2×C28).30C22, C22.10(C7⋊D4), C2.9(C2×C7⋊D4), SmallGroup(224,127)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D14
G = < a,b,c,d | a4=b2=1, c14=d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=a-1b, dcd-1=c13 >
Subgroups: 254 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D7, C14, C14, M4(2), D8, SD16, C2×D4, C4○D4, Dic7, C28, D14, C2×C14, C2×C14, C8⋊C22, C7⋊C8, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C7×D4, C7×D4, C22×C14, C4.Dic7, D4⋊D7, D4.D7, C4○D28, D4×C14, D4.D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C8⋊C22, C7⋊D4, C22×D7, C2×C7⋊D4, D4.D14
►On 56 points
(1 22 15 8)(2 23 16 9)(3 24 17 10)(4 25 18 11)(5 26 19 12)(6 27 20 13)(7 28 21 14)(29 36 43 50)(30 37 44 51)(31 38 45 52)(32 39 46 53)(33 40 47 54)(34 41 48 55)(35 42 49 56)
(1 8)(2 23)(3 10)(4 25)(5 12)(6 27)(7 14)(9 16)(11 18)(13 20)(15 22)(17 24)(19 26)(21 28)(30 44)(32 46)(34 48)(36 50)(38 52)(40 54)(42 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 31 15 45)(2 44 16 30)(3 29 17 43)(4 42 18 56)(5 55 19 41)(6 40 20 54)(7 53 21 39)(8 38 22 52)(9 51 23 37)(10 36 24 50)(11 49 25 35)(12 34 26 48)(13 47 27 33)(14 32 28 46)
G:=sub<Sym(56)| (1,22,15,8)(2,23,16,9)(3,24,17,10)(4,25,18,11)(5,26,19,12)(6,27,20,13)(7,28,21,14)(29,36,43,50)(30,37,44,51)(31,38,45,52)(32,39,46,53)(33,40,47,54)(34,41,48,55)(35,42,49,56), (1,8)(2,23)(3,10)(4,25)(5,12)(6,27)(7,14)(9,16)(11,18)(13,20)(15,22)(17,24)(19,26)(21,28)(30,44)(32,46)(34,48)(36,50)(38,52)(40,54)(42,56), (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), (1,31,15,45)(2,44,16,30)(3,29,17,43)(4,42,18,56)(5,55,19,41)(6,40,20,54)(7,53,21,39)(8,38,22,52)(9,51,23,37)(10,36,24,50)(11,49,25,35)(12,34,26,48)(13,47,27,33)(14,32,28,46)>;
G:=Group( (1,22,15,8)(2,23,16,9)(3,24,17,10)(4,25,18,11)(5,26,19,12)(6,27,20,13)(7,28,21,14)(29,36,43,50)(30,37,44,51)(31,38,45,52)(32,39,46,53)(33,40,47,54)(34,41,48,55)(35,42,49,56), (1,8)(2,23)(3,10)(4,25)(5,12)(6,27)(7,14)(9,16)(11,18)(13,20)(15,22)(17,24)(19,26)(21,28)(30,44)(32,46)(34,48)(36,50)(38,52)(40,54)(42,56), (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), (1,31,15,45)(2,44,16,30)(3,29,17,43)(4,42,18,56)(5,55,19,41)(6,40,20,54)(7,53,21,39)(8,38,22,52)(9,51,23,37)(10,36,24,50)(11,49,25,35)(12,34,26,48)(13,47,27,33)(14,32,28,46) );
G=PermutationGroup([[(1,22,15,8),(2,23,16,9),(3,24,17,10),(4,25,18,11),(5,26,19,12),(6,27,20,13),(7,28,21,14),(29,36,43,50),(30,37,44,51),(31,38,45,52),(32,39,46,53),(33,40,47,54),(34,41,48,55),(35,42,49,56)], [(1,8),(2,23),(3,10),(4,25),(5,12),(6,27),(7,14),(9,16),(11,18),(13,20),(15,22),(17,24),(19,26),(21,28),(30,44),(32,46),(34,48),(36,50),(38,52),(40,54),(42,56)], [(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)], [(1,31,15,45),(2,44,16,30),(3,29,17,43),(4,42,18,56),(5,55,19,41),(6,40,20,54),(7,53,21,39),(8,38,22,52),(9,51,23,37),(10,36,24,50),(11,49,25,35),(12,34,26,48),(13,47,27,33),(14,32,28,46)]])
D4.D14 is a maximal subgroup of
D28.2D4 D28.3D4 D28.14D4 D28⋊5D4 C56.23D4 C56.44D4 D28⋊18D4 D28.38D4 D8⋊13D14 D28.29D4 D7×C8⋊C22 SD16⋊D14 C28.C24 D28.32C23 D28.33C23
D4.D14 is a maximal quotient of
C4.Dic7⋊C4 C4○D28⋊C4 C4⋊C4.228D14 C4⋊C4.230D14 D4.3Dic14 C42.48D14 D4.1D28 C42.51D14 (C2×D4).D14 D28⋊17D4 C4⋊D4⋊D7 C7⋊C8⋊5D4 C42.72D14 C28⋊2D8 C42.74D14 Dic14⋊9D4 C42.76D14 D28⋊5Q8 C42.82D14 Dic14⋊5Q8 (D4×C14)⋊6C4 (C2×C14)⋊8D8 (C7×D4).31D4
41 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 4 | 4 | 28 | 2 | 2 | 28 | 2 | 2 | 2 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | C7⋊D4 | C7⋊D4 | C8⋊C22 | D4.D14 |
| kernel | D4.D14 | C4.Dic7 | D4⋊D7 | D4.D7 | C4○D28 | D4×C14 | C28 | C2×C14 | C2×D4 | C2×C4 | D4 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 6 | 6 | 6 | 1 | 6 |
Matrix representation of D4.D14 ►in GL4(𝔽113) generated by
| 1 | 99 | 0 | 0 |
| 97 | 112 | 0 | 0 |
| 0 | 0 | 112 | 14 |
| 0 | 0 | 16 | 1 |
| 1 | 99 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 16 | 1 |
| 109 | 56 | 0 | 0 |
| 64 | 4 | 0 | 0 |
| 0 | 0 | 28 | 60 |
| 0 | 0 | 4 | 85 |
| 0 | 0 | 28 | 60 |
| 0 | 0 | 4 | 85 |
| 109 | 56 | 0 | 0 |
| 64 | 4 | 0 | 0 |
G:=sub<GL(4,GF(113))| [1,97,0,0,99,112,0,0,0,0,112,16,0,0,14,1],[1,0,0,0,99,112,0,0,0,0,112,16,0,0,0,1],[109,64,0,0,56,4,0,0,0,0,28,4,0,0,60,85],[0,0,109,64,0,0,56,4,28,4,0,0,60,85,0,0] >;
D4.D14 in GAP, Magma, Sage, TeX
D_4.D_{14} % in TeX
G:=Group("D4.D14"); // GroupNames label
G:=SmallGroup(224,127);
// by ID
G=gap.SmallGroup(224,127);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,218,188,579,159,69,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^14=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^13>;
// generators/relations