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## G = C52.56D4order 416 = 25·13

### 13rd non-split extension by C52 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52.56D4
 Chief series C1 — C13 — C26 — C52 — C2×C52 — C52.4C4 — C52.56D4
 Lower central C13 — C26 — C52 — C52.56D4
 Upper central C1 — C4 — C2×C4 — C4○D4

Generators and relations for C52.56D4
G = < a,b,c | a52=b4=1, c2=a13, bab-1=cac-1=a25, cbc-1=a13b-1 >

Smallest permutation representation of C52.56D4
On 104 points
Generators in S104
```(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)
(1 14 27 40)(2 39 28 13)(3 12 29 38)(4 37 30 11)(5 10 31 36)(6 35 32 9)(7 8 33 34)(15 52 41 26)(16 25 42 51)(17 50 43 24)(18 23 44 49)(19 48 45 22)(20 21 46 47)(53 89)(54 62)(55 87)(56 60)(57 85)(59 83)(61 81)(63 79)(64 104)(65 77)(66 102)(67 75)(68 100)(69 73)(70 98)(72 96)(74 94)(76 92)(78 90)(80 88)(82 86)(91 103)(93 101)(95 99)
(1 58 14 71 27 84 40 97)(2 83 15 96 28 57 41 70)(3 56 16 69 29 82 42 95)(4 81 17 94 30 55 43 68)(5 54 18 67 31 80 44 93)(6 79 19 92 32 53 45 66)(7 104 20 65 33 78 46 91)(8 77 21 90 34 103 47 64)(9 102 22 63 35 76 48 89)(10 75 23 88 36 101 49 62)(11 100 24 61 37 74 50 87)(12 73 25 86 38 99 51 60)(13 98 26 59 39 72 52 85)```

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 13A ··· 13F 26A ··· 26F 26G ··· 26X 52A ··· 52L 52M ··· 52AD order 1 2 2 2 4 4 4 4 4 4 4 4 8 8 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 size 1 1 2 4 1 1 2 4 26 26 26 26 52 52 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + - - image C1 C2 C2 C2 C4 C4 D4 D4 D13 C4≀C2 D26 Dic13 Dic13 C13⋊D4 C13⋊D4 C52.56D4 kernel C52.56D4 C52.4C4 C4×Dic13 C13×C4○D4 D4×C13 Q8×C13 C52 C2×C26 C4○D4 C13 C2×C4 D4 Q8 C4 C22 C1 # reps 1 1 1 1 2 2 1 1 6 4 6 6 6 12 12 12

Matrix representation of C52.56D4 in GL4(𝔽313) generated by

 192 78 0 0 2 97 0 0 0 0 25 0 0 0 0 25
,
 65 222 0 0 184 248 0 0 0 0 25 0 0 0 80 1
,
 99 215 0 0 215 214 0 0 0 0 68 230 0 0 161 245
`G:=sub<GL(4,GF(313))| [192,2,0,0,78,97,0,0,0,0,25,0,0,0,0,25],[65,184,0,0,222,248,0,0,0,0,25,80,0,0,0,1],[99,215,0,0,215,214,0,0,0,0,68,161,0,0,230,245] >;`

C52.56D4 in GAP, Magma, Sage, TeX

`C_{52}._{56}D_4`
`% in TeX`

`G:=Group("C52.56D4");`
`// GroupNames label`

`G:=SmallGroup(416,44);`
`// by ID`

`G=gap.SmallGroup(416,44);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,86,579,297,69,13829]);`
`// Polycyclic`

`G:=Group<a,b,c|a^52=b^4=1,c^2=a^13,b*a*b^-1=c*a*c^-1=a^25,c*b*c^-1=a^13*b^-1>;`
`// generators/relations`

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