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## G = C28.53D4order 224 = 25·7

### 10th non-split extension by C28 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C28.53D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C7⋊C8 — C28.53D4
 Lower central C7 — C14 — C28 — C28.53D4
 Upper central C1 — C4 — C2×C4 — M4(2)

Generators and relations for C28.53D4
G = < a,b,c | a28=1, b4=a14, c2=a21, bab-1=cac-1=a13, cbc-1=a14b3 >

Smallest permutation representation of C28.53D4
On 112 points
Generators in S112
```(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)(57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 67 8 74 15 81 22 60)(2 80 9 59 16 66 23 73)(3 65 10 72 17 79 24 58)(4 78 11 57 18 64 25 71)(5 63 12 70 19 77 26 84)(6 76 13 83 20 62 27 69)(7 61 14 68 21 75 28 82)(29 99 50 92 43 85 36 106)(30 112 51 105 44 98 37 91)(31 97 52 90 45 111 38 104)(32 110 53 103 46 96 39 89)(33 95 54 88 47 109 40 102)(34 108 55 101 48 94 41 87)(35 93 56 86 49 107 42 100)
(1 109 22 102 15 95 8 88)(2 94 23 87 16 108 9 101)(3 107 24 100 17 93 10 86)(4 92 25 85 18 106 11 99)(5 105 26 98 19 91 12 112)(6 90 27 111 20 104 13 97)(7 103 28 96 21 89 14 110)(29 78 50 71 43 64 36 57)(30 63 51 84 44 77 37 70)(31 76 52 69 45 62 38 83)(32 61 53 82 46 75 39 68)(33 74 54 67 47 60 40 81)(34 59 55 80 48 73 41 66)(35 72 56 65 49 58 42 79)```

`G:=sub<Sym(112)| (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)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,67,8,74,15,81,22,60)(2,80,9,59,16,66,23,73)(3,65,10,72,17,79,24,58)(4,78,11,57,18,64,25,71)(5,63,12,70,19,77,26,84)(6,76,13,83,20,62,27,69)(7,61,14,68,21,75,28,82)(29,99,50,92,43,85,36,106)(30,112,51,105,44,98,37,91)(31,97,52,90,45,111,38,104)(32,110,53,103,46,96,39,89)(33,95,54,88,47,109,40,102)(34,108,55,101,48,94,41,87)(35,93,56,86,49,107,42,100), (1,109,22,102,15,95,8,88)(2,94,23,87,16,108,9,101)(3,107,24,100,17,93,10,86)(4,92,25,85,18,106,11,99)(5,105,26,98,19,91,12,112)(6,90,27,111,20,104,13,97)(7,103,28,96,21,89,14,110)(29,78,50,71,43,64,36,57)(30,63,51,84,44,77,37,70)(31,76,52,69,45,62,38,83)(32,61,53,82,46,75,39,68)(33,74,54,67,47,60,40,81)(34,59,55,80,48,73,41,66)(35,72,56,65,49,58,42,79)>;`

`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)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,67,8,74,15,81,22,60)(2,80,9,59,16,66,23,73)(3,65,10,72,17,79,24,58)(4,78,11,57,18,64,25,71)(5,63,12,70,19,77,26,84)(6,76,13,83,20,62,27,69)(7,61,14,68,21,75,28,82)(29,99,50,92,43,85,36,106)(30,112,51,105,44,98,37,91)(31,97,52,90,45,111,38,104)(32,110,53,103,46,96,39,89)(33,95,54,88,47,109,40,102)(34,108,55,101,48,94,41,87)(35,93,56,86,49,107,42,100), (1,109,22,102,15,95,8,88)(2,94,23,87,16,108,9,101)(3,107,24,100,17,93,10,86)(4,92,25,85,18,106,11,99)(5,105,26,98,19,91,12,112)(6,90,27,111,20,104,13,97)(7,103,28,96,21,89,14,110)(29,78,50,71,43,64,36,57)(30,63,51,84,44,77,37,70)(31,76,52,69,45,62,38,83)(32,61,53,82,46,75,39,68)(33,74,54,67,47,60,40,81)(34,59,55,80,48,73,41,66)(35,72,56,65,49,58,42,79) );`

`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),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,67,8,74,15,81,22,60),(2,80,9,59,16,66,23,73),(3,65,10,72,17,79,24,58),(4,78,11,57,18,64,25,71),(5,63,12,70,19,77,26,84),(6,76,13,83,20,62,27,69),(7,61,14,68,21,75,28,82),(29,99,50,92,43,85,36,106),(30,112,51,105,44,98,37,91),(31,97,52,90,45,111,38,104),(32,110,53,103,46,96,39,89),(33,95,54,88,47,109,40,102),(34,108,55,101,48,94,41,87),(35,93,56,86,49,107,42,100)], [(1,109,22,102,15,95,8,88),(2,94,23,87,16,108,9,101),(3,107,24,100,17,93,10,86),(4,92,25,85,18,106,11,99),(5,105,26,98,19,91,12,112),(6,90,27,111,20,104,13,97),(7,103,28,96,21,89,14,110),(29,78,50,71,43,64,36,57),(30,63,51,84,44,77,37,70),(31,76,52,69,45,62,38,83),(32,61,53,82,46,75,39,68),(33,74,54,67,47,60,40,81),(34,59,55,80,48,73,41,66),(35,72,56,65,49,58,42,79)]])`

C28.53D4 is a maximal subgroup of
D28.2D4  D28.3D4  D28.6D4  D28.7D4  M4(2).22D14  C42.196D14  D7×C8.C4  M4(2).25D14  C23.Dic14  C56.93D4  C56.50D4  M4(2).D14  M4(2).13D14  M4(2).15D14  M4(2).16D14
C28.53D4 is a maximal quotient of
C28.53D8  C28.39SD16  C28.4C42

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + - + + - image C1 C2 C2 C2 C4 D4 Q8 D7 D14 C8.C4 C4×D7 C7⋊D4 Dic14 C28.53D4 kernel C28.53D4 C2×C7⋊C8 C4.Dic7 C7×M4(2) C7⋊C8 C28 C2×C14 M4(2) C2×C4 C7 C4 C4 C22 C1 # reps 1 1 1 1 4 1 1 3 3 4 6 6 6 6

Matrix representation of C28.53D4 in GL4(𝔽113) generated by

 10 104 0 0 99 24 0 0 0 0 98 0 0 0 0 98
,
 61 110 0 0 73 52 0 0 0 0 69 0 0 0 47 95
,
 82 4 0 0 42 31 0 0 0 0 39 36 0 0 108 74
`G:=sub<GL(4,GF(113))| [10,99,0,0,104,24,0,0,0,0,98,0,0,0,0,98],[61,73,0,0,110,52,0,0,0,0,69,47,0,0,0,95],[82,42,0,0,4,31,0,0,0,0,39,108,0,0,36,74] >;`

C28.53D4 in GAP, Magma, Sage, TeX

`C_{28}._{53}D_4`
`% in TeX`

`G:=Group("C28.53D4");`
`// GroupNames label`

`G:=SmallGroup(224,28);`
`// by ID`

`G=gap.SmallGroup(224,28);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,48,121,31,86,297,69,6917]);`
`// Polycyclic`

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

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