Copied to
clipboard

## G = D4.D7order 112 = 24·7

### The non-split extension by D4 of D7 acting via D7/C7=C2

Aliases: D4.D7, C72SD16, C4.2D14, C14.8D4, Dic142C2, C28.2C22, C7⋊C82C2, (C7×D4).1C2, C2.5(C7⋊D4), SmallGroup(112,15)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D4.D7
 Chief series C1 — C7 — C14 — C28 — Dic14 — D4.D7
 Lower central C7 — C14 — C28 — D4.D7
 Upper central C1 — C2 — C4 — D4

Generators and relations for D4.D7
G = < a,b,c,d | a4=b2=c7=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

Character table of D4.D7

 class 1 2A 2B 4A 4B 7A 7B 7C 8A 8B 14A 14B 14C 14D 14E 14F 14G 14H 14I 28A 28B 28C size 1 1 4 2 28 2 2 2 14 14 2 2 2 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ3 1 1 1 1 -1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 -1 1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ5 2 2 0 -2 0 2 2 2 0 0 2 2 2 0 0 0 0 0 0 -2 -2 -2 orthogonal lifted from D4 ρ6 2 2 -2 2 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 -ζ75-ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D14 ρ7 2 2 -2 2 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 -ζ76-ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D14 ρ8 2 2 2 2 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D7 ρ9 2 2 -2 2 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 -ζ74-ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D14 ρ10 2 2 2 2 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D7 ρ11 2 2 2 2 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D7 ρ12 2 2 0 -2 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74-ζ73 ζ76-ζ7 -ζ76+ζ7 -ζ74+ζ73 ζ75-ζ72 -ζ75+ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 complex lifted from C7⋊D4 ρ13 2 2 0 -2 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74+ζ73 -ζ76+ζ7 ζ76-ζ7 ζ74-ζ73 -ζ75+ζ72 ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 complex lifted from C7⋊D4 ρ14 2 2 0 -2 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75-ζ72 ζ74-ζ73 -ζ74+ζ73 -ζ75+ζ72 -ζ76+ζ7 ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 complex lifted from C7⋊D4 ρ15 2 2 0 -2 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76-ζ7 -ζ75+ζ72 ζ75-ζ72 -ζ76+ζ7 ζ74-ζ73 -ζ74+ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 complex lifted from C7⋊D4 ρ16 2 2 0 -2 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75+ζ72 -ζ74+ζ73 ζ74-ζ73 ζ75-ζ72 ζ76-ζ7 -ζ76+ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 complex lifted from C7⋊D4 ρ17 2 2 0 -2 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76+ζ7 ζ75-ζ72 -ζ75+ζ72 ζ76-ζ7 -ζ74+ζ73 ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 complex lifted from C7⋊D4 ρ18 2 -2 0 0 0 2 2 2 √-2 -√-2 -2 -2 -2 0 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ19 2 -2 0 0 0 2 2 2 -√-2 √-2 -2 -2 -2 0 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ20 4 -4 0 0 0 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 0 0 -2ζ76-2ζ7 -2ζ75-2ζ72 -2ζ74-2ζ73 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ21 4 -4 0 0 0 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 0 0 -2ζ75-2ζ72 -2ζ74-2ζ73 -2ζ76-2ζ7 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ22 4 -4 0 0 0 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 0 0 -2ζ74-2ζ73 -2ζ76-2ζ7 -2ζ75-2ζ72 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of D4.D7
On 56 points
Generators in S56
```(1 27 13 20)(2 28 14 21)(3 22 8 15)(4 23 9 16)(5 24 10 17)(6 25 11 18)(7 26 12 19)(29 43 36 50)(30 44 37 51)(31 45 38 52)(32 46 39 53)(33 47 40 54)(34 48 41 55)(35 49 42 56)
(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)
(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 42 13 35)(2 41 14 34)(3 40 8 33)(4 39 9 32)(5 38 10 31)(6 37 11 30)(7 36 12 29)(15 54 22 47)(16 53 23 46)(17 52 24 45)(18 51 25 44)(19 50 26 43)(20 56 27 49)(21 55 28 48)```

`G:=sub<Sym(56)| (1,27,13,20)(2,28,14,21)(3,22,8,15)(4,23,9,16)(5,24,10,17)(6,25,11,18)(7,26,12,19)(29,43,36,50)(30,44,37,51)(31,45,38,52)(32,46,39,53)(33,47,40,54)(34,48,41,55)(35,49,42,56), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (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), (1,42,13,35)(2,41,14,34)(3,40,8,33)(4,39,9,32)(5,38,10,31)(6,37,11,30)(7,36,12,29)(15,54,22,47)(16,53,23,46)(17,52,24,45)(18,51,25,44)(19,50,26,43)(20,56,27,49)(21,55,28,48)>;`

`G:=Group( (1,27,13,20)(2,28,14,21)(3,22,8,15)(4,23,9,16)(5,24,10,17)(6,25,11,18)(7,26,12,19)(29,43,36,50)(30,44,37,51)(31,45,38,52)(32,46,39,53)(33,47,40,54)(34,48,41,55)(35,49,42,56), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (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), (1,42,13,35)(2,41,14,34)(3,40,8,33)(4,39,9,32)(5,38,10,31)(6,37,11,30)(7,36,12,29)(15,54,22,47)(16,53,23,46)(17,52,24,45)(18,51,25,44)(19,50,26,43)(20,56,27,49)(21,55,28,48) );`

`G=PermutationGroup([[(1,27,13,20),(2,28,14,21),(3,22,8,15),(4,23,9,16),(5,24,10,17),(6,25,11,18),(7,26,12,19),(29,43,36,50),(30,44,37,51),(31,45,38,52),(32,46,39,53),(33,47,40,54),(34,48,41,55),(35,49,42,56)], [(1,20),(2,21),(3,15),(4,16),(5,17),(6,18),(7,19),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42)], [(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)], [(1,42,13,35),(2,41,14,34),(3,40,8,33),(4,39,9,32),(5,38,10,31),(6,37,11,30),(7,36,12,29),(15,54,22,47),(16,53,23,46),(17,52,24,45),(18,51,25,44),(19,50,26,43),(20,56,27,49),(21,55,28,48)]])`

D4.D7 is a maximal subgroup of
D8⋊D7  D83D7  D7×SD16  SD16⋊D7  D4.D14  D4.8D14  D4.9D14  D4.F7  C42.D4  D12.D7  D4.D21
D4.D7 is a maximal quotient of
C4.Dic14  C14.Q16  D4⋊Dic7  C42.D4  D12.D7  D4.D21

Matrix representation of D4.D7 in GL4(𝔽113) generated by

 1 0 0 0 0 1 0 0 0 0 112 84 0 0 78 1
,
 1 0 0 0 0 1 0 0 0 0 112 84 0 0 0 1
,
 0 1 0 0 112 9 0 0 0 0 1 0 0 0 0 1
,
 88 90 0 0 91 25 0 0 0 0 87 75 0 0 110 26
`G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,112,78,0,0,84,1],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,84,1],[0,112,0,0,1,9,0,0,0,0,1,0,0,0,0,1],[88,91,0,0,90,25,0,0,0,0,87,110,0,0,75,26] >;`

D4.D7 in GAP, Magma, Sage, TeX

`D_4.D_7`
`% in TeX`

`G:=Group("D4.D7");`
`// GroupNames label`

`G:=SmallGroup(112,15);`
`// by ID`

`G=gap.SmallGroup(112,15);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-7,40,61,182,97,42,2404]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=c^7=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;`
`// generators/relations`

Export

׿
×
𝔽