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## G = C28.55D4order 224 = 25·7

### 12nd non-split extension by C28 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C28.55D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C7⋊C8 — C28.55D4
 Lower central C7 — C14 — C28.55D4
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C28.55D4
G = < a,b,c | a28=1, b4=a14, c2=a21, bab-1=cac-1=a13, cbc-1=a7b3 >

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

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 7A 7B 7C 8A ··· 8H 14A ··· 14U 28A ··· 28X order 1 2 2 2 2 2 4 4 4 4 4 4 7 7 7 8 ··· 8 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 1 1 1 1 2 2 2 2 2 14 ··· 14 2 ··· 2 2 ··· 2

68 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + - + - image C1 C2 C2 C4 C4 C8 D4 D7 M4(2) Dic7 D14 Dic7 C7⋊D4 C7⋊C8 C4.Dic7 kernel C28.55D4 C2×C7⋊C8 C22×C28 C2×C28 C22×C14 C2×C14 C28 C22×C4 C14 C2×C4 C2×C4 C23 C4 C22 C2 # reps 1 2 1 2 2 8 2 3 2 3 3 3 12 12 12

Matrix representation of C28.55D4 in GL4(𝔽113) generated by

 15 0 0 0 0 15 0 0 0 0 32 83 0 0 0 60
,
 0 1 0 0 98 0 0 0 0 0 2 103 0 0 81 111
,
 0 1 0 0 15 0 0 0 0 0 2 103 0 0 81 111
`G:=sub<GL(4,GF(113))| [15,0,0,0,0,15,0,0,0,0,32,0,0,0,83,60],[0,98,0,0,1,0,0,0,0,0,2,81,0,0,103,111],[0,15,0,0,1,0,0,0,0,0,2,81,0,0,103,111] >;`

C28.55D4 in GAP, Magma, Sage, TeX

`C_{28}._{55}D_4`
`% in TeX`

`G:=Group("C28.55D4");`
`// GroupNames label`

`G:=SmallGroup(224,36);`
`// by ID`

`G=gap.SmallGroup(224,36);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,121,86,6917]);`
`// 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^7*b^3>;`
`// generators/relations`

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