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## G = D25.D4order 400 = 24·52

### The non-split extension by D25 of D4 acting via D4/C22=C2

Aliases: D502C4, D25.2D4, D50.6C22, (C2×C50)⋊1C4, C25⋊(C22⋊C4), C22⋊(C25⋊C4), C50.7(C2×C4), (C2×C10).3F5, C5.(C22⋊F5), C10.12(C2×F5), (C22×D25).2C2, (C2×C25⋊C4)⋊C2, C2.7(C2×C25⋊C4), SmallGroup(400,34)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C50 — D25.D4
 Chief series C1 — C5 — C25 — D25 — D50 — C2×C25⋊C4 — D25.D4
 Lower central C25 — C50 — D25.D4
 Upper central C1 — C2 — C22

Generators and relations for D25.D4
G = < a,b,c,d | a25=b2=c4=1, d2=a-1b, bab=a-1, cac-1=dad-1=a18, cbc-1=dbd-1=a17b, dcd-1=a-1bc-1 >

2C2
25C2
25C2
50C2
25C22
25C22
50C4
50C22
50C22
50C4
2C10
5D5
5D5
10D5
25C23
25C2×C4
25C2×C4
5D10
5D10
10D10
10D10
10F5
10F5
2D25
2C50
25C22⋊C4
2D50
2D50

Smallest permutation representation of D25.D4
On 100 points
Generators in S100
```(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)
(1 38)(2 37)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 30)(10 29)(11 28)(12 27)(13 26)(14 50)(15 49)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(25 39)(51 80)(52 79)(53 78)(54 77)(55 76)(56 100)(57 99)(58 98)(59 97)(60 96)(61 95)(62 94)(63 93)(64 92)(65 91)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)
(2 8 25 19)(3 15 24 12)(4 22 23 5)(6 11 21 16)(7 18 20 9)(10 14 17 13)(26 48 27 30)(28 37 50 41)(29 44 49 34)(31 33 47 45)(32 40 46 38)(35 36 43 42)(51 97 72 94)(52 79 71 87)(53 86 70 80)(54 93 69 98)(55 100 68 91)(56 82 67 84)(57 89 66 77)(58 96 65 95)(59 78 64 88)(60 85 63 81)(61 92 62 99)(73 76 75 90)(74 83)
(1 74 39 83)(2 56 38 76)(3 63 37 94)(4 70 36 87)(5 52 35 80)(6 59 34 98)(7 66 33 91)(8 73 32 84)(9 55 31 77)(10 62 30 95)(11 69 29 88)(12 51 28 81)(13 58 27 99)(14 65 26 92)(15 72 50 85)(16 54 49 78)(17 61 48 96)(18 68 47 89)(19 75 46 82)(20 57 45 100)(21 64 44 93)(22 71 43 86)(23 53 42 79)(24 60 41 97)(25 67 40 90)```

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5 10A 10B 10C 25A ··· 25E 50A ··· 50O order 1 2 2 2 2 2 4 4 4 4 5 10 10 10 25 ··· 25 50 ··· 50 size 1 1 2 25 25 50 50 50 50 50 4 4 4 4 4 ··· 4 4 ··· 4

34 irreducible representations

 dim 1 1 1 1 1 2 4 4 4 4 4 4 type + + + + + + + + + + image C1 C2 C2 C4 C4 D4 F5 C2×F5 C22⋊F5 C25⋊C4 C2×C25⋊C4 D25.D4 kernel D25.D4 C2×C25⋊C4 C22×D25 D50 C2×C50 D25 C2×C10 C10 C5 C22 C2 C1 # reps 1 2 1 2 2 2 1 1 2 5 5 10

Matrix representation of D25.D4 in GL6(𝔽101)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 27 93 55 23 0 0 78 4 70 32 0 0 69 46 73 38 0 0 63 31 8 35
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 63 35 31 0 0 55 27 23 93 0 0 73 69 38 46 0 0 97 66 74 28
,
 91 0 0 0 0 0 55 10 0 0 0 0 0 0 78 46 8 74 0 0 63 28 55 32 0 0 27 4 73 35 0 0 69 31 97 23
,
 46 81 0 0 0 0 30 55 0 0 0 0 0 0 78 46 8 74 0 0 63 28 55 32 0 0 27 4 73 35 0 0 69 31 97 23

`G:=sub<GL(6,GF(101))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,78,69,63,0,0,93,4,46,31,0,0,55,70,73,8,0,0,23,32,38,35],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,55,73,97,0,0,63,27,69,66,0,0,35,23,38,74,0,0,31,93,46,28],[91,55,0,0,0,0,0,10,0,0,0,0,0,0,78,63,27,69,0,0,46,28,4,31,0,0,8,55,73,97,0,0,74,32,35,23],[46,30,0,0,0,0,81,55,0,0,0,0,0,0,78,63,27,69,0,0,46,28,4,31,0,0,8,55,73,97,0,0,74,32,35,23] >;`

D25.D4 in GAP, Magma, Sage, TeX

`D_{25}.D_4`
`% in TeX`

`G:=Group("D25.D4");`
`// GroupNames label`

`G:=SmallGroup(400,34);`
`// by ID`

`G=gap.SmallGroup(400,34);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,3364,2896,178,5765,2897]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^25=b^2=c^4=1,d^2=a^-1*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^18,c*b*c^-1=d*b*d^-1=a^17*b,d*c*d^-1=a^-1*b*c^-1>;`
`// generators/relations`

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