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G = C100⋊C4order 400 = 24·52

1st semidirect product of C100 and C4 acting faithfully

Aliases: C1001C4, D25.Q8, C20.2F5, D25.1D4, Dic253C4, D50.5C22, C4⋊(C25⋊C4), C25⋊(C4⋊C4), C5.(C4⋊F5), C50.4(C2×C4), C10.9(C2×F5), (C4×D25).4C2, (C2×C25⋊C4).C2, C2.5(C2×C25⋊C4), SmallGroup(400,31)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C50 — C100⋊C4
 Chief series C1 — C5 — C25 — D25 — D50 — C2×C25⋊C4 — C100⋊C4
 Lower central C25 — C50 — C100⋊C4
 Upper central C1 — C2 — C4

Generators and relations for C100⋊C4
G = < a,b | a100=b4=1, bab-1=a43 >

Smallest permutation representation of C100⋊C4
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)
(2 8 50 44)(3 15 99 87)(4 22 48 30)(5 29 97 73)(6 36 46 16)(7 43 95 59)(9 57 93 45)(10 64 42 88)(11 71 91 31)(12 78 40 74)(13 85 89 17)(14 92 38 60)(18 20 34 32)(19 27 83 75)(21 41 81 61)(23 55 79 47)(24 62 28 90)(25 69 77 33)(26 76)(35 39 67 63)(37 53 65 49)(52 58 100 94)(54 72 98 80)(56 86 96 66)(68 70 84 82)```

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 4A 4B ··· 4F 5 10 20A 20B 25A ··· 25E 50A ··· 50E 100A ··· 100J order 1 2 2 2 4 4 ··· 4 5 10 20 20 25 ··· 25 50 ··· 50 100 ··· 100 size 1 1 25 25 2 50 ··· 50 4 4 4 4 4 ··· 4 4 ··· 4 4 ··· 4

34 irreducible representations

 dim 1 1 1 1 1 2 2 4 4 4 4 4 4 type + + + + - + + + + image C1 C2 C2 C4 C4 D4 Q8 F5 C2×F5 C4⋊F5 C25⋊C4 C2×C25⋊C4 C100⋊C4 kernel C100⋊C4 C4×D25 C2×C25⋊C4 Dic25 C100 D25 D25 C20 C10 C5 C4 C2 C1 # reps 1 1 2 2 2 1 1 1 1 2 5 5 10

Matrix representation of C100⋊C4 in GL4(𝔽7) generated by

 5 0 5 2 2 2 1 2 1 4 3 6 0 3 2 2
,
 3 3 2 0 4 1 3 6 3 1 5 4 2 4 4 5
`G:=sub<GL(4,GF(7))| [5,2,1,0,0,2,4,3,5,1,3,2,2,2,6,2],[3,4,3,2,3,1,1,4,2,3,5,4,0,6,4,5] >;`

C100⋊C4 in GAP, Magma, Sage, TeX

`C_{100}\rtimes C_4`
`% in TeX`

`G:=Group("C100:C4");`
`// GroupNames label`

`G:=SmallGroup(400,31);`
`// by ID`

`G=gap.SmallGroup(400,31);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,55,3364,2896,178,5765,2897]);`
`// Polycyclic`

`G:=Group<a,b|a^100=b^4=1,b*a*b^-1=a^43>;`
`// generators/relations`

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