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## G = C56.Q8order 448 = 26·7

### 5th non-split extension by C56 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — C56.Q8
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C56 — C28.C8 — C56.Q8
 Lower central C7 — C14 — C28 — C56 — C56.Q8
 Upper central C1 — C2 — C2×C4 — C2×C8 — C4.Q8

Generators and relations for C56.Q8
G = < a,b,c | a56=b4=1, c2=a49b2, bab-1=a43, cac-1=a13, cbc-1=a7b-1 >

Smallest permutation representation of C56.Q8
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)
(2 44)(3 31)(4 18)(6 48)(7 35)(8 22)(10 52)(11 39)(12 26)(14 56)(15 43)(16 30)(19 47)(20 34)(23 51)(24 38)(27 55)(28 42)(32 46)(36 50)(40 54)(57 64 85 92)(58 107 86 79)(59 94 87 66)(60 81 88 109)(61 68 89 96)(62 111 90 83)(63 98 91 70)(65 72 93 100)(67 102 95 74)(69 76 97 104)(71 106 99 78)(73 80 101 108)(75 110 103 82)(77 84 105 112)
(1 106 50 71 43 92 36 57 29 78 22 99 15 64 8 85)(2 63 51 84 44 105 37 70 30 91 23 112 16 77 9 98)(3 76 52 97 45 62 38 83 31 104 24 69 17 90 10 111)(4 89 53 110 46 75 39 96 32 61 25 82 18 103 11 68)(5 102 54 67 47 88 40 109 33 74 26 95 19 60 12 81)(6 59 55 80 48 101 41 66 34 87 27 108 20 73 13 94)(7 72 56 93 49 58 42 79 35 100 28 65 21 86 14 107)```

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

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

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

58 conjugacy classes

 class 1 2A 2B 4A 4B 4C 4D 7A 7B 7C 8A 8B 8C 8D 8E 14A ··· 14I 16A 16B 16C 16D 28A ··· 28F 28G ··· 28R 56A ··· 56L order 1 2 2 4 4 4 4 7 7 7 8 8 8 8 8 14 ··· 14 16 16 16 16 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 2 2 2 8 8 2 2 2 2 2 4 56 56 2 ··· 2 28 28 28 28 4 ··· 4 8 ··· 8 4 ··· 4

58 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + - + + + - + - image C1 C2 C2 C2 C4 Q8 D4 D7 SD16 SD16 D14 Dic14 C4×D7 C7⋊D4 C8.Q8 Q8⋊D7 D4.D7 C56.Q8 kernel C56.Q8 C28.C8 C56.C4 C7×C4.Q8 C7⋊C16 C56 C2×C28 C4.Q8 C28 C2×C14 C2×C8 C8 C8 C2×C4 C7 C4 C22 C1 # reps 1 1 1 1 4 1 1 3 2 2 3 6 6 6 2 3 3 12

Matrix representation of C56.Q8 in GL4(𝔽113) generated by

 0 59 34 107 110 69 69 6 0 0 77 49 0 0 109 0
,
 1 90 83 37 0 112 29 53 0 0 0 4 0 0 28 0
,
 89 100 3 17 28 0 0 24 8 21 79 30 26 80 5 58
`G:=sub<GL(4,GF(113))| [0,110,0,0,59,69,0,0,34,69,77,109,107,6,49,0],[1,0,0,0,90,112,0,0,83,29,0,28,37,53,4,0],[89,28,8,26,100,0,21,80,3,0,79,5,17,24,30,58] >;`

C56.Q8 in GAP, Magma, Sage, TeX

`C_{56}.Q_8`
`% in TeX`

`G:=Group("C56.Q8");`
`// GroupNames label`

`G:=SmallGroup(448,44);`
`// by ID`

`G=gap.SmallGroup(448,44);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,365,36,758,346,80,851,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^56=b^4=1,c^2=a^49*b^2,b*a*b^-1=a^43,c*a*c^-1=a^13,c*b*c^-1=a^7*b^-1>;`
`// generators/relations`

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