Copied to
clipboard

## G = C102⋊C4order 400 = 24·52

### 3rd semidirect product of C102 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C102⋊C4
 Chief series C1 — C5 — C52 — C5⋊D5 — C2×C5⋊D5 — C2×C5⋊F5 — C102⋊C4
 Lower central C52 — C5×C10 — C102⋊C4
 Upper central C1 — C2 — C22

Generators and relations for C102⋊C4
G = < a,b,c | a10=b10=c4=1, ab=ba, cac-1=a3b5, cbc-1=b3 >

Subgroups: 1048 in 136 conjugacy classes, 32 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C23, D5, C10, C10, C22⋊C4, F5, D10, C2×C10, C52, C2×F5, C22×D5, C5⋊D5, C5⋊D5, C5×C10, C5×C10, C22⋊F5, C5⋊F5, C2×C5⋊D5, C2×C5⋊D5, C102, C2×C5⋊F5, C22×C5⋊D5, C102⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, F5, C2×F5, C22⋊F5, C5⋊F5, C2×C5⋊F5, C102⋊C4

Smallest permutation representation of C102⋊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)
(1 30 21 50 14 7 18 40 44 34)(2 26 22 46 15 8 19 36 45 35)(3 27 23 47 11 9 20 37 41 31)(4 28 24 48 12 10 16 38 42 32)(5 29 25 49 13 6 17 39 43 33)(51 100 78 62 88 56 95 73 67 83)(52 91 79 63 89 57 96 74 68 84)(53 92 80 64 90 58 97 75 69 85)(54 93 71 65 81 59 98 76 70 86)(55 94 72 66 82 60 99 77 61 87)
(1 75 48 65)(2 77 47 63)(3 79 46 61)(4 71 50 69)(5 73 49 67)(6 78 43 62)(7 80 42 70)(8 72 41 68)(9 74 45 66)(10 76 44 64)(11 52 35 55)(12 54 34 53)(13 56 33 51)(14 58 32 59)(15 60 31 57)(16 81 40 97)(17 83 39 95)(18 85 38 93)(19 87 37 91)(20 89 36 99)(21 92 28 86)(22 94 27 84)(23 96 26 82)(24 98 30 90)(25 100 29 88)```

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

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

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

34 conjugacy classes

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

34 irreducible representations

 dim 1 1 1 1 1 2 4 4 4 type + + + + + + + image C1 C2 C2 C4 C4 D4 F5 C2×F5 C22⋊F5 kernel C102⋊C4 C2×C5⋊F5 C22×C5⋊D5 C2×C5⋊D5 C102 C5⋊D5 C2×C10 C10 C5 # reps 1 2 1 2 2 2 6 6 12

Matrix representation of C102⋊C4 in GL8(𝔽41)

 35 35 0 0 0 0 0 0 6 40 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 1 35 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 6 6 0 0 0 0 0 0 35 1 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 1 35 0 0 0 0 0 0 0 0 40 7 0 0 0 0 0 0 34 7 0 0 0 0 0 0 1 40 35 35 0 0 0 0 0 1 6 40
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 35 1 0 0 0 0 0 0 0 0 0 0 19 32 6 40 0 0 0 0 19 0 0 40 0 0 0 0 7 0 0 19 0 0 0 0 34 27 32 22

`G:=sub<GL(8,GF(41))| [35,6,0,0,0,0,0,0,35,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,35,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[6,35,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,35,0,0,0,0,0,0,0,0,40,34,1,0,0,0,0,0,7,7,40,1,0,0,0,0,0,0,35,6,0,0,0,0,0,0,35,40],[0,0,40,35,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,19,19,7,34,0,0,0,0,32,0,0,27,0,0,0,0,6,0,0,32,0,0,0,0,40,40,19,22] >;`

C102⋊C4 in GAP, Magma, Sage, TeX

`C_{10}^2\rtimes C_4`
`% in TeX`

`G:=Group("C10^2:C4");`
`// GroupNames label`

`G:=SmallGroup(400,155);`
`// by ID`

`G=gap.SmallGroup(400,155);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,964,496,5765,2897]);`
`// Polycyclic`

`G:=Group<a,b,c|a^10=b^10=c^4=1,a*b=b*a,c*a*c^-1=a^3*b^5,c*b*c^-1=b^3>;`
`// generators/relations`

׿
×
𝔽