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## G = C52⋊7D4order 200 = 23·52

### 2nd semidirect product of C52 and D4 acting via D4/C22=C2

Aliases: C527D4, C1023C2, C10.16D10, (C2×C10)⋊2D5, C22⋊(C5⋊D5), C53(C5⋊D4), C526C43C2, (C5×C10).15C22, (C2×C5⋊D5)⋊3C2, C2.5(C2×C5⋊D5), SmallGroup(200,36)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C52⋊7D4
 Chief series C1 — C5 — C52 — C5×C10 — C2×C5⋊D5 — C52⋊7D4
 Lower central C52 — C5×C10 — C52⋊7D4
 Upper central C1 — C2 — C22

Generators and relations for C527D4
G = < a,b,c,d | a5=b5=c4=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 320 in 64 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2, C4, C22, C22, C5, D4, D5, C10, C10, Dic5, D10, C2×C10, C52, C5⋊D4, C5⋊D5, C5×C10, C5×C10, C526C4, C2×C5⋊D5, C102, C527D4
Quotients: C1, C2, C22, D4, D5, D10, C5⋊D4, C5⋊D5, C2×C5⋊D5, C527D4

Smallest permutation representation of C527D4
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 44 38 33 28)(2 45 39 34 29)(3 41 40 35 30)(4 42 36 31 26)(5 43 37 32 27)(6 100 22 16 11)(7 96 23 17 12)(8 97 24 18 13)(9 98 25 19 14)(10 99 21 20 15)(46 67 61 56 51)(47 68 62 57 52)(48 69 63 58 53)(49 70 64 59 54)(50 66 65 60 55)(71 92 86 81 76)(72 93 87 82 77)(73 94 88 83 78)(74 95 89 84 79)(75 91 90 85 80)
(1 73 48 98)(2 72 49 97)(3 71 50 96)(4 75 46 100)(5 74 47 99)(6 42 80 67)(7 41 76 66)(8 45 77 70)(9 44 78 69)(10 43 79 68)(11 36 85 61)(12 40 81 65)(13 39 82 64)(14 38 83 63)(15 37 84 62)(16 31 90 56)(17 35 86 60)(18 34 87 59)(19 33 88 58)(20 32 89 57)(21 27 95 52)(22 26 91 51)(23 30 92 55)(24 29 93 54)(25 28 94 53)
(2 5)(3 4)(6 92)(7 91)(8 95)(9 94)(10 93)(11 86)(12 90)(13 89)(14 88)(15 87)(16 81)(17 85)(18 84)(19 83)(20 82)(21 77)(22 76)(23 80)(24 79)(25 78)(26 41)(27 45)(28 44)(29 43)(30 42)(31 40)(32 39)(33 38)(34 37)(35 36)(46 50)(47 49)(51 66)(52 70)(53 69)(54 68)(55 67)(56 65)(57 64)(58 63)(59 62)(60 61)(71 100)(72 99)(73 98)(74 97)(75 96)```

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

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

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

C527D4 is a maximal subgroup of   Dic5.D10  D5×C5⋊D4  C20.50D10  D4×C5⋊D5  C20.D10
C527D4 is a maximal quotient of   C102.22C22  C10.11D20  C527D8  C528SD16  C5210SD16  C527Q16  C10211C4

53 conjugacy classes

 class 1 2A 2B 2C 4 5A ··· 5L 10A ··· 10AJ order 1 2 2 2 4 5 ··· 5 10 ··· 10 size 1 1 2 50 50 2 ··· 2 2 ··· 2

53 irreducible representations

 dim 1 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 D4 D5 D10 C5⋊D4 kernel C52⋊7D4 C52⋊6C4 C2×C5⋊D5 C102 C52 C2×C10 C10 C5 # reps 1 1 1 1 1 12 12 24

Matrix representation of C527D4 in GL4(𝔽41) generated by

 40 34 0 0 7 7 0 0 0 0 0 1 0 0 40 34
,
 0 1 0 0 40 34 0 0 0 0 0 1 0 0 40 34
,
 17 40 0 0 3 24 0 0 0 0 38 3 0 0 24 3
,
 1 0 0 0 34 40 0 0 0 0 7 7 0 0 40 34
`G:=sub<GL(4,GF(41))| [40,7,0,0,34,7,0,0,0,0,0,40,0,0,1,34],[0,40,0,0,1,34,0,0,0,0,0,40,0,0,1,34],[17,3,0,0,40,24,0,0,0,0,38,24,0,0,3,3],[1,34,0,0,0,40,0,0,0,0,7,40,0,0,7,34] >;`

C527D4 in GAP, Magma, Sage, TeX

`C_5^2\rtimes_7D_4`
`% in TeX`

`G:=Group("C5^2:7D4");`
`// GroupNames label`

`G:=SmallGroup(200,36);`
`// by ID`

`G=gap.SmallGroup(200,36);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-5,-5,61,643,4004]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^5=c^4=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;`
`// generators/relations`

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