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## G = D28⋊5D4order 448 = 26·7

### 5th semidirect product of D28 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — D28⋊5D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4○D28 — D4⋊6D14 — D28⋊5D4
 Lower central C7 — C14 — C2×C28 — D28⋊5D4
 Upper central C1 — C2 — C2×C4 — C4⋊1D4

Generators and relations for D285D4
G = < a,b,c,d | a28=b2=c4=d2=1, bab=a-1, ac=ca, dad=a15, cbc-1=a21b, dbd=a14b, dcd=c-1 >

Subgroups: 940 in 168 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4.D4, C4≀C2, C41D4, C8⋊C22, 2+ 1+4, C7⋊C8, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C22×C14, D44D4, C4.Dic7, D4⋊D7, D4.D7, C4×C28, C4○D28, D4×D7, D42D7, C2×C7⋊D4, D4×C14, D4×C14, Dic14⋊C4, C28.D4, D4.D14, C7×C41D4, D46D14, D285D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C7⋊D4, C22×D7, D44D4, D4×D7, C2×C7⋊D4, C23⋊D14, D285D4

Smallest permutation representation of D285D4
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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 56)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)
(29 50 43 36)(30 51 44 37)(31 52 45 38)(32 53 46 39)(33 54 47 40)(34 55 48 41)(35 56 49 42)
(1 8)(2 23)(3 10)(4 25)(5 12)(6 27)(7 14)(9 16)(11 18)(13 20)(15 22)(17 24)(19 26)(21 28)(29 50)(30 37)(31 52)(32 39)(33 54)(34 41)(35 56)(36 43)(38 45)(40 47)(42 49)(44 51)(46 53)(48 55)```

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

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

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

58 conjugacy classes

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

58 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D4 D7 D14 D14 C7⋊D4 D4⋊4D4 D4×D7 D28⋊5D4 kernel D28⋊5D4 Dic14⋊C4 C28.D4 D4.D14 C7×C4⋊1D4 D4⋊6D14 Dic14 D28 C22×C14 C4⋊1D4 C42 C2×D4 C23 C7 C4 C1 # reps 1 2 1 2 1 1 2 2 2 3 3 6 12 2 6 12

Matrix representation of D285D4 in GL4(𝔽113) generated by

 0 16 0 0 97 0 0 0 0 0 0 106 0 0 7 0
,
 0 0 7 0 0 0 0 106 97 0 0 0 0 16 0 0
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 112 0
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(113))| [0,97,0,0,16,0,0,0,0,0,0,7,0,0,106,0],[0,0,97,0,0,0,0,16,7,0,0,0,0,106,0,0],[1,0,0,0,0,1,0,0,0,0,0,112,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;`

D285D4 in GAP, Magma, Sage, TeX

`D_{28}\rtimes_5D_4`
`% in TeX`

`G:=Group("D28:5D4");`
`// GroupNames label`

`G:=SmallGroup(448,611);`
`// by ID`

`G=gap.SmallGroup(448,611);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,1123,570,297,136,1684,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^28=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^15,c*b*c^-1=a^21*b,d*b*d=a^14*b,d*c*d=c^-1>;`
`// generators/relations`

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