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G = C28.D4order 224 = 25·7

8th non-split extension by C28 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C28.D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4.Dic7 — C28.D4
 Lower central C7 — C14 — C2×C14 — C28.D4
 Upper central C1 — C2 — C2×C4 — C2×D4

Generators and relations for C28.D4
G = < a,b,c | a28=1, b4=a14, c2=a21, bab-1=a-1, cac-1=a13, cbc-1=a7b3 >

Smallest permutation representation of C28.D4
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 34 22 41 15 48 8 55)(2 33 23 40 16 47 9 54)(3 32 24 39 17 46 10 53)(4 31 25 38 18 45 11 52)(5 30 26 37 19 44 12 51)(6 29 27 36 20 43 13 50)(7 56 28 35 21 42 14 49)
(1 48 22 41 15 34 8 55)(2 33 23 54 16 47 9 40)(3 46 24 39 17 32 10 53)(4 31 25 52 18 45 11 38)(5 44 26 37 19 30 12 51)(6 29 27 50 20 43 13 36)(7 42 28 35 21 56 14 49)```

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

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

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

C28.D4 is a maximal subgroup of
C7⋊C2≀C4  (C2×C28).D4  C24⋊Dic7  (C22×C28)⋊C4  D7×C4.D4  M4(2).19D14  C425D14  D285D4  C56.23D4  C56.44D4  M4(2).D14  M4(2).13D14  (D4×C14).16C4  2+ 1+4⋊D7  2+ 1+4.D7
C28.D4 is a maximal quotient of
C24.Dic7  C28.(C4⋊C4)  C42.7D14  C28.9D8  C28.5Q16

41 conjugacy classes

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

41 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + - + image C1 C2 C2 C4 D4 D7 D14 Dic7 C7⋊D4 C4.D4 C28.D4 kernel C28.D4 C4.Dic7 D4×C14 C22×C14 C28 C2×D4 C2×C4 C23 C4 C7 C1 # reps 1 2 1 4 2 3 3 6 12 1 6

Matrix representation of C28.D4 in GL4(𝔽113) generated by

 7 14 0 0 106 106 0 0 14 38 0 16 89 75 97 0
,
 70 0 111 0 43 0 1 112 21 1 43 0 92 0 70 0
,
 70 0 111 0 0 0 1 1 21 1 43 0 93 0 70 0
`G:=sub<GL(4,GF(113))| [7,106,14,89,14,106,38,75,0,0,0,97,0,0,16,0],[70,43,21,92,0,0,1,0,111,1,43,70,0,112,0,0],[70,0,21,93,0,0,1,0,111,1,43,70,0,1,0,0] >;`

C28.D4 in GAP, Magma, Sage, TeX

`C_{28}.D_4`
`% in TeX`

`G:=Group("C28.D4");`
`// GroupNames label`

`G:=SmallGroup(224,39);`
`// by ID`

`G=gap.SmallGroup(224,39);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,121,188,86,579,6917]);`
`// Polycyclic`

`G:=Group<a,b,c|a^28=1,b^4=a^14,c^2=a^21,b*a*b^-1=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^7*b^3>;`
`// generators/relations`

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