metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊4D14, Q8⋊4D14, C28.49D4, C28.17C23, D28.11C22, D4⋊D7⋊6C2, C4○D4⋊1D7, C7⋊C8⋊4C22, Q8⋊D7⋊6C2, C7⋊5(C8⋊C22), (C2×C14).8D4, (C2×D28)⋊10C2, C14.59(C2×D4), (C2×C4).22D14, (C7×D4)⋊4C22, C4.Dic7⋊9C2, (C7×Q8)⋊4C22, C4.24(C7⋊D4), C4.17(C22×D7), (C2×C28).42C22, C22.5(C7⋊D4), (C7×C4○D4)⋊1C2, C2.23(C2×C7⋊D4), SmallGroup(224,144)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊D14
G = < a,b,c,d | a4=b2=c14=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, dbd=a-1b, dcd=c-1 >
Subgroups: 326 in 68 conjugacy classes, 29 normal (21 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, C28, C28, D14, C2×C14, C2×C14, C8⋊C22, C7⋊C8, D28, D28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, C4.Dic7, D4⋊D7, Q8⋊D7, C2×D28, C7×C4○D4, 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 14 20)(2 23 8 21)(3 24 9 15)(4 25 10 16)(5 26 11 17)(6 27 12 18)(7 28 13 19)(29 56 36 49)(30 43 37 50)(31 44 38 51)(32 45 39 52)(33 46 40 53)(34 47 41 54)(35 48 42 55)
(1 54)(2 48)(3 56)(4 50)(5 44)(6 52)(7 46)(8 55)(9 49)(10 43)(11 51)(12 45)(13 53)(14 47)(15 36)(16 30)(17 38)(18 32)(19 40)(20 34)(21 42)(22 41)(23 35)(24 29)(25 37)(26 31)(27 39)(28 33)
(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 17)(2 16)(3 15)(4 21)(5 20)(6 19)(7 18)(8 25)(9 24)(10 23)(11 22)(12 28)(13 27)(14 26)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(43 48)(44 47)(45 46)(49 56)(50 55)(51 54)(52 53)
G:=sub<Sym(56)| (1,22,14,20)(2,23,8,21)(3,24,9,15)(4,25,10,16)(5,26,11,17)(6,27,12,18)(7,28,13,19)(29,56,36,49)(30,43,37,50)(31,44,38,51)(32,45,39,52)(33,46,40,53)(34,47,41,54)(35,48,42,55), (1,54)(2,48)(3,56)(4,50)(5,44)(6,52)(7,46)(8,55)(9,49)(10,43)(11,51)(12,45)(13,53)(14,47)(15,36)(16,30)(17,38)(18,32)(19,40)(20,34)(21,42)(22,41)(23,35)(24,29)(25,37)(26,31)(27,39)(28,33), (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), (1,17)(2,16)(3,15)(4,21)(5,20)(6,19)(7,18)(8,25)(9,24)(10,23)(11,22)(12,28)(13,27)(14,26)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(43,48)(44,47)(45,46)(49,56)(50,55)(51,54)(52,53)>;
G:=Group( (1,22,14,20)(2,23,8,21)(3,24,9,15)(4,25,10,16)(5,26,11,17)(6,27,12,18)(7,28,13,19)(29,56,36,49)(30,43,37,50)(31,44,38,51)(32,45,39,52)(33,46,40,53)(34,47,41,54)(35,48,42,55), (1,54)(2,48)(3,56)(4,50)(5,44)(6,52)(7,46)(8,55)(9,49)(10,43)(11,51)(12,45)(13,53)(14,47)(15,36)(16,30)(17,38)(18,32)(19,40)(20,34)(21,42)(22,41)(23,35)(24,29)(25,37)(26,31)(27,39)(28,33), (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), (1,17)(2,16)(3,15)(4,21)(5,20)(6,19)(7,18)(8,25)(9,24)(10,23)(11,22)(12,28)(13,27)(14,26)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(43,48)(44,47)(45,46)(49,56)(50,55)(51,54)(52,53) );
G=PermutationGroup([[(1,22,14,20),(2,23,8,21),(3,24,9,15),(4,25,10,16),(5,26,11,17),(6,27,12,18),(7,28,13,19),(29,56,36,49),(30,43,37,50),(31,44,38,51),(32,45,39,52),(33,46,40,53),(34,47,41,54),(35,48,42,55)], [(1,54),(2,48),(3,56),(4,50),(5,44),(6,52),(7,46),(8,55),(9,49),(10,43),(11,51),(12,45),(13,53),(14,47),(15,36),(16,30),(17,38),(18,32),(19,40),(20,34),(21,42),(22,41),(23,35),(24,29),(25,37),(26,31),(27,39),(28,33)], [(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)], [(1,17),(2,16),(3,15),(4,21),(5,20),(6,19),(7,18),(8,25),(9,24),(10,23),(11,22),(12,28),(13,27),(14,26),(30,42),(31,41),(32,40),(33,39),(34,38),(35,37),(43,48),(44,47),(45,46),(49,56),(50,55),(51,54),(52,53)]])
D4⋊D14 is a maximal subgroup of
D4⋊4D28 D4.10D28 D4.3D28 D4.4D28 M4(2).D14 M4(2).15D14 2+ 1+4⋊D7 2- 1+4⋊D7 D8⋊15D14 D8⋊11D14 D7×C8⋊C22 D56⋊C22 C28.C24 D28.32C23 D28.34C23
D4⋊D14 is a maximal quotient of
C28.(C2×Q8) C4⋊C4⋊36D14 C4⋊C4.236D14 C28.38SD16 C42.48D14 C28⋊7D8 C28.23Q16 C42.56D14 Q8⋊D28 C4⋊D4.D7 D28⋊16D4 C4⋊D4⋊D7 (C2×C14).Q16 D28.36D4 C7⋊C8⋊6D4 C42.62D14 D28.23D4 C42.64D14 C42.68D14 D28.4Q8 C42.70D14 C4○D4⋊Dic7 (C7×D4)⋊14D4
41 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 14D | ··· | 14L | 28A | ··· | 28F | 28G | ··· | 28O |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 4 | 28 | 28 | 2 | 2 | 4 | 2 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | D14 | C7⋊D4 | C7⋊D4 | C8⋊C22 | D4⋊D14 |
| kernel | D4⋊D14 | C4.Dic7 | D4⋊D7 | Q8⋊D7 | C2×D28 | C7×C4○D4 | C28 | C2×C14 | C4○D4 | C2×C4 | D4 | Q8 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 6 | 6 | 1 | 6 |
Matrix representation of D4⋊D14 ►in GL4(𝔽113) generated by
| 58 | 75 | 0 | 0 |
| 38 | 55 | 0 | 0 |
| 17 | 23 | 66 | 38 |
| 94 | 30 | 37 | 47 |
| 82 | 36 | 8 | 43 |
| 87 | 46 | 65 | 106 |
| 4 | 30 | 112 | 5 |
| 46 | 108 | 33 | 99 |
| 80 | 80 | 0 | 0 |
| 33 | 9 | 0 | 0 |
| 103 | 111 | 105 | 33 |
| 67 | 53 | 47 | 32 |
| 80 | 80 | 0 | 0 |
| 9 | 33 | 0 | 0 |
| 29 | 87 | 68 | 58 |
| 82 | 91 | 82 | 45 |
G:=sub<GL(4,GF(113))| [58,38,17,94,75,55,23,30,0,0,66,37,0,0,38,47],[82,87,4,46,36,46,30,108,8,65,112,33,43,106,5,99],[80,33,103,67,80,9,111,53,0,0,105,47,0,0,33,32],[80,9,29,82,80,33,87,91,0,0,68,82,0,0,58,45] >;
D4⋊D14 in GAP, Magma, Sage, TeX
D_4\rtimes D_{14} % in TeX
G:=Group("D4:D14"); // GroupNames label
G:=SmallGroup(224,144);
// by ID
G=gap.SmallGroup(224,144);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,218,188,579,159,69,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^14=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations