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G = C56.D4order 448 = 26·7

48th non-split extension by C56 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C56.D4
 Chief series C1 — C7 — C14 — C28 — C56 — C2×C56 — C28.C8 — C56.D4
 Lower central C7 — C14 — C2×C14 — C56.D4
 Upper central C1 — C4 — C2×C8 — C2×M4(2)

Generators and relations for C56.D4
G = < a,b,c | a56=1, b4=a42, c2=a49, bab-1=a13, cac-1=a41, cbc-1=a7b3 >

Subgroups: 132 in 58 conjugacy classes, 31 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, C14, C14, C16, C2×C8, M4(2), C22×C4, C28, C28, C2×C14, C2×C14, M5(2), C2×M4(2), C56, C56, C2×C28, C2×C28, C22×C14, C23.C8, C7⋊C16, C2×C56, C7×M4(2), C22×C28, C28.C8, C14×M4(2), C56.D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D7, C22⋊C4, C2×C8, M4(2), Dic7, D14, C22⋊C8, C7⋊C8, C2×Dic7, C7⋊D4, C23.C8, C2×C7⋊C8, C4.Dic7, C23.D7, C28.55D4, C56.D4

Smallest permutation representation of C56.D4
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 99 50 64 43 85 36 106 29 71 22 92 15 57 8 78)(2 112 51 77 44 98 37 63 30 84 23 105 16 70 9 91)(3 69 52 90 45 111 38 76 31 97 24 62 17 83 10 104)(4 82 53 103 46 68 39 89 32 110 25 75 18 96 11 61)(5 95 54 60 47 81 40 102 33 67 26 88 19 109 12 74)(6 108 55 73 48 94 41 59 34 80 27 101 20 66 13 87)(7 65 56 86 49 107 42 72 35 93 28 58 21 79 14 100)
(1 71 50 64 43 57 36 106 29 99 22 92 15 85 8 78)(2 112 51 105 44 98 37 91 30 84 23 77 16 70 9 63)(3 97 52 90 45 83 38 76 31 69 24 62 17 111 10 104)(4 82 53 75 46 68 39 61 32 110 25 103 18 96 11 89)(5 67 54 60 47 109 40 102 33 95 26 88 19 81 12 74)(6 108 55 101 48 94 41 87 34 80 27 73 20 66 13 59)(7 93 56 86 49 79 42 72 35 65 28 58 21 107 14 100)```

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 7A 7B 7C 8A 8B 8C 8D 8E 8F 14A ··· 14I 14J ··· 14O 16A ··· 16H 28A ··· 28L 28M ··· 28R 56A ··· 56X order 1 2 2 2 4 4 4 4 7 7 7 8 8 8 8 8 8 14 ··· 14 14 ··· 14 16 ··· 16 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 2 4 1 1 2 4 2 2 2 2 2 2 2 4 4 2 ··· 2 4 ··· 4 28 ··· 28 2 ··· 2 4 ··· 4 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + - + - image C1 C2 C2 C4 C4 C8 C8 D4 D7 M4(2) Dic7 D14 Dic7 C7⋊D4 C7⋊C8 C7⋊C8 C4.Dic7 C23.C8 C56.D4 kernel C56.D4 C28.C8 C14×M4(2) C2×C56 C22×C28 C2×C28 C22×C14 C56 C2×M4(2) C28 C2×C8 C2×C8 C22×C4 C8 C2×C4 C23 C4 C7 C1 # reps 1 2 1 2 2 4 4 2 3 2 3 3 3 12 6 6 12 2 12

Matrix representation of C56.D4 in GL4(𝔽113) generated by

 0 30 0 0 111 0 0 0 0 0 0 49 0 0 57 0
,
 0 0 1 0 0 0 0 112 0 1 0 0 98 0 0 0
,
 0 0 1 0 0 0 0 1 0 1 0 0 15 0 0 0
`G:=sub<GL(4,GF(113))| [0,111,0,0,30,0,0,0,0,0,0,57,0,0,49,0],[0,0,0,98,0,0,1,0,1,0,0,0,0,112,0,0],[0,0,0,15,0,0,1,0,1,0,0,0,0,1,0,0] >;`

C56.D4 in GAP, Magma, Sage, TeX

`C_{56}.D_4`
`% in TeX`

`G:=Group("C56.D4");`
`// GroupNames label`

`G:=SmallGroup(448,110);`
`// by ID`

`G=gap.SmallGroup(448,110);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,387,100,1123,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^56=1,b^4=a^42,c^2=a^49,b*a*b^-1=a^13,c*a*c^-1=a^41,c*b*c^-1=a^7*b^3>;`
`// generators/relations`

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