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## G = D14⋊C4order 112 = 24·7

### The semidirect product of D14 and C4 acting via C4/C2=C2

Aliases: D14⋊C4, C14.6D4, C2.2D28, C22.6D14, (C2×C4)⋊1D7, (C2×C28)⋊1C2, C2.5(C4×D7), C71(C22⋊C4), C14.5(C2×C4), (C22×D7).C2, (C2×Dic7)⋊1C2, C2.2(C7⋊D4), (C2×C14).6C22, SmallGroup(112,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D14⋊C4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — D14⋊C4
 Lower central C7 — C14 — D14⋊C4
 Upper central C1 — C22 — C2×C4

Generators and relations for D14⋊C4
G = < a,b,c | a14=b2=c4=1, bab=a-1, ac=ca, cbc-1=a7b >

Smallest permutation representation of D14⋊C4
On 56 points
Generators in S56
```(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 28)(11 27)(12 26)(13 25)(14 24)(29 51)(30 50)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 56)(39 55)(40 54)(41 53)(42 52)
(1 55 24 33)(2 56 25 34)(3 43 26 35)(4 44 27 36)(5 45 28 37)(6 46 15 38)(7 47 16 39)(8 48 17 40)(9 49 18 41)(10 50 19 42)(11 51 20 29)(12 52 21 30)(13 53 22 31)(14 54 23 32)```

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 7A 7B 7C 14A ··· 14I 28A ··· 28L order 1 2 2 2 2 2 4 4 4 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 1 1 14 14 2 2 14 14 2 2 2 2 ··· 2 2 ··· 2

34 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C4 D4 D7 D14 C4×D7 D28 C7⋊D4 kernel D14⋊C4 C2×Dic7 C2×C28 C22×D7 D14 C14 C2×C4 C22 C2 C2 C2 # reps 1 1 1 1 4 2 3 3 6 6 6

Matrix representation of D14⋊C4 in GL4(𝔽29) generated by

 25 25 0 0 4 11 0 0 0 0 11 8 0 0 18 0
,
 4 4 0 0 18 25 0 0 0 0 1 0 0 0 24 28
,
 17 0 0 0 0 17 0 0 0 0 16 18 0 0 26 13
`G:=sub<GL(4,GF(29))| [25,4,0,0,25,11,0,0,0,0,11,18,0,0,8,0],[4,18,0,0,4,25,0,0,0,0,1,24,0,0,0,28],[17,0,0,0,0,17,0,0,0,0,16,26,0,0,18,13] >;`

D14⋊C4 in GAP, Magma, Sage, TeX

`D_{14}\rtimes C_4`
`% in TeX`

`G:=Group("D14:C4");`
`// GroupNames label`

`G:=SmallGroup(112,13);`
`// by ID`

`G=gap.SmallGroup(112,13);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-7,101,26,2404]);`
`// Polycyclic`

`G:=Group<a,b,c|a^14=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^7*b>;`
`// generators/relations`

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