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## G = D28⋊4C4order 224 = 25·7

### 4th semidirect product of D28 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D28⋊4C4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4○D28 — D28⋊4C4
 Lower central C7 — C14 — C28 — D28⋊4C4
 Upper central C1 — C4 — C2×C4 — M4(2)

Generators and relations for D284C4
G = < a,b,c | a28=b2=c4=1, bab=a-1, cac-1=a13, cbc-1=a19b >

Smallest permutation representation of D284C4
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 54)(2 53)(3 52)(4 51)(5 50)(6 49)(7 48)(8 47)(9 46)(10 45)(11 44)(12 43)(13 42)(14 41)(15 40)(16 39)(17 38)(18 37)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 56)(28 55)
(1 15)(2 28)(3 13)(4 26)(5 11)(6 24)(7 9)(8 22)(10 20)(12 18)(14 16)(17 27)(19 25)(21 23)(29 46 43 32)(30 31 44 45)(33 42 47 56)(34 55 48 41)(35 40 49 54)(36 53 50 39)(37 38 51 52)```

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

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

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

D284C4 is a maximal subgroup of
D28.1D4  D281D4  D28.4D4  D28.5D4  D7×C4≀C2  C42⋊D14  D5610C4  D567C4  C23.20D28  C56.93D4  C56.50D4  D2818D4  D28.38D4  D28.39D4  D28.40D4
D284C4 is a maximal quotient of
C42.D14  C42.2D14  C23.30D28  C22.2D56  D282C8  Dic142C8  C28.3C42

44 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 8A 8B 14A 14B 14C 14D 14E 14F 28A ··· 28F 28G 28H 28I 56A ··· 56L order 1 2 2 2 4 4 4 4 4 4 4 4 7 7 7 8 8 14 14 14 14 14 14 28 ··· 28 28 28 28 56 ··· 56 size 1 1 2 28 1 1 2 14 14 14 14 28 2 2 2 4 4 2 2 2 4 4 4 2 ··· 2 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C4 C4 D4 D4 D7 D14 C4≀C2 C4×D7 C7⋊D4 D28 D28⋊4C4 kernel D28⋊4C4 C4×Dic7 C7×M4(2) C4○D28 Dic14 D28 C28 C2×C14 M4(2) C2×C4 C7 C4 C4 C22 C1 # reps 1 1 1 1 2 2 1 1 3 3 4 6 6 6 6

Matrix representation of D284C4 in GL4(𝔽113) generated by

 79 104 0 0 88 0 0 0 0 0 98 0 0 0 0 15
,
 0 9 0 0 88 0 0 0 0 0 0 98 0 0 15 0
,
 98 43 0 0 0 15 0 0 0 0 112 0 0 0 0 15
`G:=sub<GL(4,GF(113))| [79,88,0,0,104,0,0,0,0,0,98,0,0,0,0,15],[0,88,0,0,9,0,0,0,0,0,0,15,0,0,98,0],[98,0,0,0,43,15,0,0,0,0,112,0,0,0,0,15] >;`

D284C4 in GAP, Magma, Sage, TeX

`D_{28}\rtimes_4C_4`
`% in TeX`

`G:=Group("D28:4C4");`
`// GroupNames label`

`G:=SmallGroup(224,31);`
`// by ID`

`G=gap.SmallGroup(224,31);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,121,31,86,579,297,69,6917]);`
`// Polycyclic`

`G:=Group<a,b,c|a^28=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^19*b>;`
`// generators/relations`

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