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## G = C4.12D28order 224 = 25·7

### 4th non-split extension by C4 of D28 acting via D28/D14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C4.12D28
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×Dic14 — C4.12D28
 Lower central C7 — C14 — C2×C14 — C4.12D28
 Upper central C1 — C2 — C2×C4 — M4(2)

Generators and relations for C4.12D28
G = < a,b,c | a28=1, b4=c2=a14, bab-1=cac-1=a-1, cbc-1=a21b3 >

Smallest permutation representation of C4.12D28
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 92 22 99 15 106 8 85)(2 91 23 98 16 105 9 112)(3 90 24 97 17 104 10 111)(4 89 25 96 18 103 11 110)(5 88 26 95 19 102 12 109)(6 87 27 94 20 101 13 108)(7 86 28 93 21 100 14 107)(29 66 36 59 43 80 50 73)(30 65 37 58 44 79 51 72)(31 64 38 57 45 78 52 71)(32 63 39 84 46 77 53 70)(33 62 40 83 47 76 54 69)(34 61 41 82 48 75 55 68)(35 60 42 81 49 74 56 67)
(1 61 15 75)(2 60 16 74)(3 59 17 73)(4 58 18 72)(5 57 19 71)(6 84 20 70)(7 83 21 69)(8 82 22 68)(9 81 23 67)(10 80 24 66)(11 79 25 65)(12 78 26 64)(13 77 27 63)(14 76 28 62)(29 104 43 90)(30 103 44 89)(31 102 45 88)(32 101 46 87)(33 100 47 86)(34 99 48 85)(35 98 49 112)(36 97 50 111)(37 96 51 110)(38 95 52 109)(39 94 53 108)(40 93 54 107)(41 92 55 106)(42 91 56 105)```

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

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

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

C4.12D28 is a maximal subgroup of
M4(2).19D14  D28.2D4  D7×C4.10D4  D28.7D4  D4.9D28  D4.10D28  C8.20D28  C8.24D28  M4(2).31D14  D4.3D28  D4.5D28  M4(2).13D14  D28.38D4  M4(2).16D14  D28.40D4
C4.12D28 is a maximal quotient of
C42.2D14  (C2×Dic7)⋊C8  C28.47D8  C28.2D8  M4(2)⋊Dic7

41 conjugacy classes

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

41 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + - - image C1 C2 C2 C2 C4 D4 D7 D14 D28 C7⋊D4 C4×D7 C4.10D4 C4.12D28 kernel C4.12D28 C4.Dic7 C7×M4(2) C2×Dic14 C2×Dic7 C28 M4(2) C2×C4 C4 C4 C22 C7 C1 # reps 1 1 1 1 4 2 3 3 6 6 6 1 6

Matrix representation of C4.12D28 in GL4(𝔽113) generated by

 67 81 0 0 112 100 0 0 0 0 67 81 0 0 112 100
,
 0 0 80 40 0 0 1 33 33 81 0 0 27 80 0 0
,
 80 40 0 0 1 33 0 0 0 0 80 40 0 0 1 33
`G:=sub<GL(4,GF(113))| [67,112,0,0,81,100,0,0,0,0,67,112,0,0,81,100],[0,0,33,27,0,0,81,80,80,1,0,0,40,33,0,0],[80,1,0,0,40,33,0,0,0,0,80,1,0,0,40,33] >;`

C4.12D28 in GAP, Magma, Sage, TeX

`C_4._{12}D_{28}`
`% in TeX`

`G:=Group("C4.12D28");`
`// GroupNames label`

`G:=SmallGroup(224,30);`
`// by ID`

`G=gap.SmallGroup(224,30);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,121,31,362,86,297,6917]);`
`// Polycyclic`

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

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