metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊6D14, C23⋊2D14, D28⋊8C22, C14.7C24, C7⋊12+ 1+4, C28.21C23, D14.3C23, Dic14⋊8C22, Dic7.4C23, (D4×D7)⋊4C2, (C2×D4)⋊7D7, (C2×C4)⋊3D14, C4○D28⋊5C2, (D4×C14)⋊7C2, D4⋊2D7⋊4C2, (C2×C28)⋊3C22, (C7×D4)⋊7C22, (C4×D7)⋊1C22, C7⋊D4⋊3C22, C2.8(C23×D7), (C2×C14).2C23, C4.21(C22×D7), (C22×C14)⋊5C22, (C2×Dic7)⋊4C22, (C22×D7)⋊3C22, C22.6(C22×D7), (C2×C7⋊D4)⋊11C2, SmallGroup(224,180)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊6D14
G = < a,b,c,d | a4=b2=c14=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
Subgroups: 670 in 166 conjugacy classes, 85 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, D4, Q8, C23, C23, D7, C14, C14, C2×D4, C2×D4, C4○D4, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, 2+ 1+4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×C14, C4○D28, D4×D7, D4⋊2D7, C2×C7⋊D4, D4×C14, D4⋊6D14
Quotients: C1, C2, C22, C23, D7, C24, D14, 2+ 1+4, C22×D7, C23×D7, D4⋊6D14
►On 56 points
(1 51 12 44)(2 45 13 52)(3 53 14 46)(4 47 8 54)(5 55 9 48)(6 49 10 56)(7 43 11 50)(15 35 26 42)(16 29 27 36)(17 37 28 30)(18 31 22 38)(19 39 23 32)(20 33 24 40)(21 41 25 34)
(1 38)(2 32)(3 40)(4 34)(5 42)(6 36)(7 30)(8 41)(9 35)(10 29)(11 37)(12 31)(13 39)(14 33)(15 48)(16 56)(17 50)(18 44)(19 52)(20 46)(21 54)(22 51)(23 45)(24 53)(25 47)(26 55)(27 49)(28 43)
(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 7)(2 6)(3 5)(9 14)(10 13)(11 12)(15 24)(16 23)(17 22)(18 28)(19 27)(20 26)(21 25)(29 32)(30 31)(33 42)(34 41)(35 40)(36 39)(37 38)(43 51)(44 50)(45 49)(46 48)(52 56)(53 55)
G:=sub<Sym(56)| (1,51,12,44)(2,45,13,52)(3,53,14,46)(4,47,8,54)(5,55,9,48)(6,49,10,56)(7,43,11,50)(15,35,26,42)(16,29,27,36)(17,37,28,30)(18,31,22,38)(19,39,23,32)(20,33,24,40)(21,41,25,34), (1,38)(2,32)(3,40)(4,34)(5,42)(6,36)(7,30)(8,41)(9,35)(10,29)(11,37)(12,31)(13,39)(14,33)(15,48)(16,56)(17,50)(18,44)(19,52)(20,46)(21,54)(22,51)(23,45)(24,53)(25,47)(26,55)(27,49)(28,43), (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), (1,7)(2,6)(3,5)(9,14)(10,13)(11,12)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(29,32)(30,31)(33,42)(34,41)(35,40)(36,39)(37,38)(43,51)(44,50)(45,49)(46,48)(52,56)(53,55)>;
G:=Group( (1,51,12,44)(2,45,13,52)(3,53,14,46)(4,47,8,54)(5,55,9,48)(6,49,10,56)(7,43,11,50)(15,35,26,42)(16,29,27,36)(17,37,28,30)(18,31,22,38)(19,39,23,32)(20,33,24,40)(21,41,25,34), (1,38)(2,32)(3,40)(4,34)(5,42)(6,36)(7,30)(8,41)(9,35)(10,29)(11,37)(12,31)(13,39)(14,33)(15,48)(16,56)(17,50)(18,44)(19,52)(20,46)(21,54)(22,51)(23,45)(24,53)(25,47)(26,55)(27,49)(28,43), (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), (1,7)(2,6)(3,5)(9,14)(10,13)(11,12)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(29,32)(30,31)(33,42)(34,41)(35,40)(36,39)(37,38)(43,51)(44,50)(45,49)(46,48)(52,56)(53,55) );
G=PermutationGroup([[(1,51,12,44),(2,45,13,52),(3,53,14,46),(4,47,8,54),(5,55,9,48),(6,49,10,56),(7,43,11,50),(15,35,26,42),(16,29,27,36),(17,37,28,30),(18,31,22,38),(19,39,23,32),(20,33,24,40),(21,41,25,34)], [(1,38),(2,32),(3,40),(4,34),(5,42),(6,36),(7,30),(8,41),(9,35),(10,29),(11,37),(12,31),(13,39),(14,33),(15,48),(16,56),(17,50),(18,44),(19,52),(20,46),(21,54),(22,51),(23,45),(24,53),(25,47),(26,55),(27,49),(28,43)], [(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)], [(1,7),(2,6),(3,5),(9,14),(10,13),(11,12),(15,24),(16,23),(17,22),(18,28),(19,27),(20,26),(21,25),(29,32),(30,31),(33,42),(34,41),(35,40),(36,39),(37,38),(43,51),(44,50),(45,49),(46,48),(52,56),(53,55)]])
D4⋊6D14 is a maximal subgroup of
C23⋊D28 C23.5D28 D28.1D4 D28⋊1D4 C24⋊D14 C22⋊C4⋊D14 C42⋊5D14 D28⋊5D4 D8⋊13D14 D28.29D4 D8⋊5D14 D8⋊6D14 C14.C25 D7×2+ 1+4 D14.C24
D4⋊6D14 is a maximal quotient of
C23⋊2Dic14 C24.24D14 C24.27D14 C23⋊3D28 C24.30D14 C24.31D14 C14.72+ 1+4 C14.82+ 1+4 C14.2+ 1+4 C14.102+ 1+4 C14.112+ 1+4 C14.62- 1+4 D4⋊5Dic14 C42.104D14 C42⋊11D14 C42.108D14 D4⋊5D28 C42⋊16D14 C42.113D14 C42.114D14 C42⋊17D14 C42.115D14 C42.116D14 C42.118D14 C24.32D14 C24⋊2D14 C24⋊3D14 C24.33D14 C24.34D14 C24.35D14 C24⋊4D14 C24.36D14 C14.682- 1+4 Dic14⋊20D4 C14.342+ 1+4 C14.352+ 1+4 C14.712- 1+4 C14.372+ 1+4 C14.382+ 1+4 C14.722- 1+4 C14.402+ 1+4 D28⋊20D4 C14.422+ 1+4 C14.432+ 1+4 C14.442+ 1+4 C14.452+ 1+4 C14.462+ 1+4 C14.472+ 1+4 C14.482+ 1+4 C14.492+ 1+4 C14.752- 1+4 C14.512+ 1+4 C14.522+ 1+4 C14.532+ 1+4 C14.202- 1+4 C14.222- 1+4 C14.562+ 1+4 C14.572+ 1+4 C14.582+ 1+4 C14.262- 1+4 C14.602+ 1+4 C14.612+ 1+4 C14.622+ 1+4 C14.832- 1+4 C14.642+ 1+4 C14.842- 1+4 C14.662+ 1+4 C14.672+ 1+4 C14.682+ 1+4 C14.862- 1+4 C42.137D14 C42.138D14 C42.140D14 C42⋊20D14 C42⋊21D14 C42⋊22D14 C42.145D14 C42.166D14 C42⋊26D14 D28⋊11D4 Dic14⋊11D4 C42.168D14 C42⋊28D14 Dic14⋊9Q8 C42.174D14 D28⋊9Q8 C42.178D14 C42.179D14 C42.180D14 C24.38D14 D4×C7⋊D4 C24⋊7D14 C24.41D14 C24.42D14
47 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F |
| order | 1 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | ··· | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | D14 | 2+ 1+4 | D4⋊6D14 |
| kernel | D4⋊6D14 | C4○D28 | D4×D7 | D4⋊2D7 | C2×C7⋊D4 | D4×C14 | C2×D4 | C2×C4 | D4 | C23 | C7 | C1 |
| # reps | 1 | 2 | 4 | 4 | 4 | 1 | 3 | 3 | 12 | 6 | 1 | 6 |
Matrix representation of D4⋊6D14 ►in GL4(𝔽29) generated by
| 13 | 24 | 0 | 24 |
| 13 | 24 | 13 | 6 |
| 18 | 0 | 11 | 25 |
| 21 | 21 | 16 | 10 |
| 0 | 1 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 3 | 27 | 28 | 3 |
| 1 | 28 | 0 | 0 |
| 21 | 21 | 11 | 17 |
| 8 | 26 | 19 | 22 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 21 | 11 |
| 0 | 1 | 27 | 5 |
| 1 | 0 | 27 | 5 |
| 0 | 0 | 28 | 5 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(29))| [13,13,18,21,24,24,0,21,0,13,11,16,24,6,25,10],[0,0,3,1,1,1,27,28,0,0,28,0,1,0,3,0],[21,8,0,0,21,26,0,0,11,19,0,21,17,22,11,11],[0,1,0,0,1,0,0,0,27,27,28,0,5,5,5,1] >;
D4⋊6D14 in GAP, Magma, Sage, TeX
D_4\rtimes_6D_{14} % in TeX
G:=Group("D4:6D14"); // GroupNames label
G:=SmallGroup(224,180);
// by ID
G=gap.SmallGroup(224,180);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,188,579,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^14=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations