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## G = D56.C4order 448 = 26·7

### 4th non-split extension by D56 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — D56.C4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C56 — C2×D56 — D56.C4
 Lower central C7 — C14 — C28 — C56 — D56.C4
 Upper central C1 — C2 — C2×C4 — C2×C8 — C8.C4

Generators and relations for D56.C4
G = < a,b,c | a56=b2=1, c4=a28, bab=a-1, cac-1=a43, cbc-1=a7b >

Subgroups: 540 in 62 conjugacy classes, 25 normal (23 characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C8, C2×C4, D4, C23, D7, C14, C14, C16, C2×C8, M4(2), D8, C2×D4, C28, D14, C2×C14, C8.C4, M5(2), C2×D8, C56, C56, D28, C2×C28, C22×D7, M5(2)⋊C2, C7⋊C16, D56, D56, C2×C56, C7×M4(2), C2×D28, C28.C8, C7×C8.C4, C2×D56, D56.C4
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, D14, D4⋊C4, C4×D7, D28, C7⋊D4, M5(2)⋊C2, D14⋊C4, D4⋊D7, Q8⋊D7, C14.D8, D56.C4

Smallest permutation representation of D56.C4
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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(57 91)(58 90)(59 89)(60 88)(61 87)(62 86)(63 85)(64 84)(65 83)(66 82)(67 81)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(92 112)(93 111)(94 110)(95 109)(96 108)(97 107)(98 106)(99 105)(100 104)(101 103)
(1 71 43 85 29 99 15 57)(2 58 44 72 30 86 16 100)(3 101 45 59 31 73 17 87)(4 88 46 102 32 60 18 74)(5 75 47 89 33 103 19 61)(6 62 48 76 34 90 20 104)(7 105 49 63 35 77 21 91)(8 92 50 106 36 64 22 78)(9 79 51 93 37 107 23 65)(10 66 52 80 38 94 24 108)(11 109 53 67 39 81 25 95)(12 96 54 110 40 68 26 82)(13 83 55 97 41 111 27 69)(14 70 56 84 42 98 28 112)```

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

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

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

58 conjugacy classes

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

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 D4 D4 D7 D8 SD16 D14 C4×D7 D28 C7⋊D4 M5(2)⋊C2 D4⋊D7 Q8⋊D7 D56.C4 kernel D56.C4 C28.C8 C7×C8.C4 C2×D56 D56 C56 C2×C28 C8.C4 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 D56.C4 in GL4(𝔽113) generated by

 52 44 0 0 69 91 0 0 0 0 33 106 0 0 7 91
,
 80 7 0 0 6 33 0 0 0 0 36 17 0 0 90 77
,
 0 0 1 0 0 0 0 1 96 105 0 0 8 17 0 0
`G:=sub<GL(4,GF(113))| [52,69,0,0,44,91,0,0,0,0,33,7,0,0,106,91],[80,6,0,0,7,33,0,0,0,0,36,90,0,0,17,77],[0,0,96,8,0,0,105,17,1,0,0,0,0,1,0,0] >;`

D56.C4 in GAP, Magma, Sage, TeX

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

`G:=Group("D56.C4");`
`// GroupNames label`

`G:=SmallGroup(448,52);`
`// by ID`

`G=gap.SmallGroup(448,52);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,758,184,675,794,192,1684,851,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^56=b^2=1,c^4=a^28,b*a*b=a^-1,c*a*c^-1=a^43,c*b*c^-1=a^7*b>;`
`// generators/relations`

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