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## G = D4⋊6D14order 224 = 25·7

### 2nd semidirect product of D4 and D14 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D4⋊6D14
 Chief series C1 — C7 — C14 — D14 — C22×D7 — D4×D7 — D4⋊6D14
 Lower central C7 — C14 — D4⋊6D14
 Upper central C1 — C2 — C2×D4

Generators and relations for D46D14
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, D42D7, C2×C7⋊D4, D4×C14, D46D14
Quotients: C1, C2, C22, C23, D7, C24, D14, 2+ 1+4, C22×D7, C23×D7, D46D14

Smallest permutation representation of D46D14
On 56 points
Generators in S56
```(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)]])`

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 D46D14 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] >;`

D46D14 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`

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