Copied to
clipboard

## G = C56.C4order 224 = 25·7

### 1st non-split extension by C56 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C56.C4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4.Dic7 — C56.C4
 Lower central C7 — C14 — C28 — C56.C4
 Upper central C1 — C4 — C2×C4 — C2×C8

Generators and relations for C56.C4
G = < a,b,c | a8=1, b14=a4, c2=a4b7, ab=ba, cac-1=a-1, cbc-1=b13 >

Smallest permutation representation of C56.C4
On 112 points
Generators in S112
```(1 56 22 49 15 42 8 35)(2 29 23 50 16 43 9 36)(3 30 24 51 17 44 10 37)(4 31 25 52 18 45 11 38)(5 32 26 53 19 46 12 39)(6 33 27 54 20 47 13 40)(7 34 28 55 21 48 14 41)(57 92 64 99 71 106 78 85)(58 93 65 100 72 107 79 86)(59 94 66 101 73 108 80 87)(60 95 67 102 74 109 81 88)(61 96 68 103 75 110 82 89)(62 97 69 104 76 111 83 90)(63 98 70 105 77 112 84 91)
(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 57 22 78 15 71 8 64)(2 70 23 63 16 84 9 77)(3 83 24 76 17 69 10 62)(4 68 25 61 18 82 11 75)(5 81 26 74 19 67 12 60)(6 66 27 59 20 80 13 73)(7 79 28 72 21 65 14 58)(29 98 50 91 43 112 36 105)(30 111 51 104 44 97 37 90)(31 96 52 89 45 110 38 103)(32 109 53 102 46 95 39 88)(33 94 54 87 47 108 40 101)(34 107 55 100 48 93 41 86)(35 92 56 85 49 106 42 99)```

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

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

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

C56.C4 is a maximal subgroup of
C56.Q8  C8.7Dic14  C112.C4  D56.1C4  C16⋊Dic7  C28.3D8  C28.4D8  D8.Dic7  Q16.Dic7  C28.58D8  D5611C4  D564C4  D7×C8.C4  M4(2).25D14  M4(2).Dic7  D4.3D28  D4.4D28  D4.5D28  C56.23D4  C56.44D4  C56.29D4  D85Dic7  D84Dic7
C56.C4 is a maximal quotient of
C562C8  C561C8  C28.10C42

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + - + - + + - image C1 C2 C2 C4 D4 Q8 D7 Dic7 D14 C8.C4 D28 Dic14 C56.C4 kernel C56.C4 C4.Dic7 C2×C56 C56 C28 C2×C14 C2×C8 C8 C2×C4 C7 C4 C22 C1 # reps 1 2 1 4 1 1 3 6 3 4 6 6 24

Matrix representation of C56.C4 in GL4(𝔽113) generated by

 18 0 0 0 0 44 0 0 0 0 1 0 0 0 0 1
,
 98 0 0 0 0 98 0 0 0 0 5 1 0 0 19 4
,
 0 1 0 0 98 0 0 0 0 0 85 33 0 0 79 28
`G:=sub<GL(4,GF(113))| [18,0,0,0,0,44,0,0,0,0,1,0,0,0,0,1],[98,0,0,0,0,98,0,0,0,0,5,19,0,0,1,4],[0,98,0,0,1,0,0,0,0,0,85,79,0,0,33,28] >;`

C56.C4 in GAP, Magma, Sage, TeX

`C_{56}.C_4`
`% in TeX`

`G:=Group("C56.C4");`
`// GroupNames label`

`G:=SmallGroup(224,25);`
`// by ID`

`G=gap.SmallGroup(224,25);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,121,55,86,579,69,6917]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=1,b^14=a^4,c^2=a^4*b^7,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^13>;`
`// generators/relations`

Export

׿
×
𝔽