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## G = C52⋊4M4(2)  order 400 = 24·52

### 3rd semidirect product of C52 and M4(2) acting via M4(2)/C2=C2×C4

Aliases: D10.3F5, Dic5.3F5, C524M4(2), C53(C4.F5), C10.5(C2×F5), C524C82C2, C525C82C2, (D5×C10).4C4, C2.5(D5⋊F5), (D5×Dic5).5C2, (C5×Dic5).2C4, C51(C22.F5), C526C4.5C22, (C5×C10).12(C2×C4), SmallGroup(400,128)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C52⋊4M4(2)
 Chief series C1 — C5 — C52 — C5×C10 — C52⋊6C4 — C52⋊4C8 — C52⋊4M4(2)
 Lower central C52 — C5×C10 — C52⋊4M4(2)
 Upper central C1 — C2

Generators and relations for C524M4(2)
G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, cac-1=a3, dad=a-1, cbc-1=b2, bd=db, dcd=c5 >

Character table of C524M4(2)

 class 1 2A 2B 4A 4B 4C 5A 5B 5C 5D 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 20A 20B size 1 1 10 10 25 25 4 4 8 8 50 50 50 50 4 4 8 8 20 20 20 20 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 -1 -1 1 1 1 1 1 1 -1 1 1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 -1 -1 -1 1 1 1 1 -i i -i i 1 1 1 1 1 1 -1 -1 linear of order 4 ρ6 1 1 -1 1 -1 -1 1 1 1 1 -i -i i i 1 1 1 1 -1 -1 1 1 linear of order 4 ρ7 1 1 1 -1 -1 -1 1 1 1 1 i -i i -i 1 1 1 1 1 1 -1 -1 linear of order 4 ρ8 1 1 -1 1 -1 -1 1 1 1 1 i i -i -i 1 1 1 1 -1 -1 1 1 linear of order 4 ρ9 2 -2 0 0 2i -2i 2 2 2 2 0 0 0 0 -2 -2 -2 -2 0 0 0 0 complex lifted from M4(2) ρ10 2 -2 0 0 -2i 2i 2 2 2 2 0 0 0 0 -2 -2 -2 -2 0 0 0 0 complex lifted from M4(2) ρ11 4 4 4 0 0 0 -1 4 -1 -1 0 0 0 0 -1 4 -1 -1 -1 -1 0 0 orthogonal lifted from F5 ρ12 4 4 0 -4 0 0 4 -1 -1 -1 0 0 0 0 4 -1 -1 -1 0 0 1 1 orthogonal lifted from C2×F5 ρ13 4 4 0 4 0 0 4 -1 -1 -1 0 0 0 0 4 -1 -1 -1 0 0 -1 -1 orthogonal lifted from F5 ρ14 4 4 -4 0 0 0 -1 4 -1 -1 0 0 0 0 -1 4 -1 -1 1 1 0 0 orthogonal lifted from C2×F5 ρ15 4 -4 0 0 0 0 -1 4 -1 -1 0 0 0 0 1 -4 1 1 -√5 √5 0 0 symplectic lifted from C22.F5, Schur index 2 ρ16 4 -4 0 0 0 0 -1 4 -1 -1 0 0 0 0 1 -4 1 1 √5 -√5 0 0 symplectic lifted from C22.F5, Schur index 2 ρ17 4 -4 0 0 0 0 4 -1 -1 -1 0 0 0 0 -4 1 1 1 0 0 √-5 -√-5 complex lifted from C4.F5 ρ18 4 -4 0 0 0 0 4 -1 -1 -1 0 0 0 0 -4 1 1 1 0 0 -√-5 √-5 complex lifted from C4.F5 ρ19 8 8 0 0 0 0 -2 -2 3 -2 0 0 0 0 -2 -2 -2 3 0 0 0 0 orthogonal lifted from D5⋊F5 ρ20 8 8 0 0 0 0 -2 -2 -2 3 0 0 0 0 -2 -2 3 -2 0 0 0 0 orthogonal lifted from D5⋊F5 ρ21 8 -8 0 0 0 0 -2 -2 -2 3 0 0 0 0 2 2 -3 2 0 0 0 0 symplectic faithful, Schur index 2 ρ22 8 -8 0 0 0 0 -2 -2 3 -2 0 0 0 0 2 2 2 -3 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C524M4(2)
On 80 points
Generators in S80
```(1 15 29 62 66)(2 63 16 67 30)(3 68 64 31 9)(4 32 69 10 57)(5 11 25 58 70)(6 59 12 71 26)(7 72 60 27 13)(8 28 65 14 61)(17 34 79 50 46)(18 51 35 47 80)(19 48 52 73 36)(20 74 41 37 53)(21 38 75 54 42)(22 55 39 43 76)(23 44 56 77 40)(24 78 45 33 49)
(1 66 62 29 15)(2 63 16 67 30)(3 9 31 64 68)(4 32 69 10 57)(5 70 58 25 11)(6 59 12 71 26)(7 13 27 60 72)(8 28 65 14 61)(17 34 79 50 46)(18 80 47 35 51)(19 48 52 73 36)(20 53 37 41 74)(21 38 75 54 42)(22 76 43 39 55)(23 44 56 77 40)(24 49 33 45 78)
(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)
(1 79)(2 76)(3 73)(4 78)(5 75)(6 80)(7 77)(8 74)(9 36)(10 33)(11 38)(12 35)(13 40)(14 37)(15 34)(16 39)(17 29)(18 26)(19 31)(20 28)(21 25)(22 30)(23 27)(24 32)(41 61)(42 58)(43 63)(44 60)(45 57)(46 62)(47 59)(48 64)(49 69)(50 66)(51 71)(52 68)(53 65)(54 70)(55 67)(56 72)```

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

`G:=Group( (1,15,29,62,66)(2,63,16,67,30)(3,68,64,31,9)(4,32,69,10,57)(5,11,25,58,70)(6,59,12,71,26)(7,72,60,27,13)(8,28,65,14,61)(17,34,79,50,46)(18,51,35,47,80)(19,48,52,73,36)(20,74,41,37,53)(21,38,75,54,42)(22,55,39,43,76)(23,44,56,77,40)(24,78,45,33,49), (1,66,62,29,15)(2,63,16,67,30)(3,9,31,64,68)(4,32,69,10,57)(5,70,58,25,11)(6,59,12,71,26)(7,13,27,60,72)(8,28,65,14,61)(17,34,79,50,46)(18,80,47,35,51)(19,48,52,73,36)(20,53,37,41,74)(21,38,75,54,42)(22,76,43,39,55)(23,44,56,77,40)(24,49,33,45,78), (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), (1,79)(2,76)(3,73)(4,78)(5,75)(6,80)(7,77)(8,74)(9,36)(10,33)(11,38)(12,35)(13,40)(14,37)(15,34)(16,39)(17,29)(18,26)(19,31)(20,28)(21,25)(22,30)(23,27)(24,32)(41,61)(42,58)(43,63)(44,60)(45,57)(46,62)(47,59)(48,64)(49,69)(50,66)(51,71)(52,68)(53,65)(54,70)(55,67)(56,72) );`

`G=PermutationGroup([[(1,15,29,62,66),(2,63,16,67,30),(3,68,64,31,9),(4,32,69,10,57),(5,11,25,58,70),(6,59,12,71,26),(7,72,60,27,13),(8,28,65,14,61),(17,34,79,50,46),(18,51,35,47,80),(19,48,52,73,36),(20,74,41,37,53),(21,38,75,54,42),(22,55,39,43,76),(23,44,56,77,40),(24,78,45,33,49)], [(1,66,62,29,15),(2,63,16,67,30),(3,9,31,64,68),(4,32,69,10,57),(5,70,58,25,11),(6,59,12,71,26),(7,13,27,60,72),(8,28,65,14,61),(17,34,79,50,46),(18,80,47,35,51),(19,48,52,73,36),(20,53,37,41,74),(21,38,75,54,42),(22,76,43,39,55),(23,44,56,77,40),(24,49,33,45,78)], [(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)], [(1,79),(2,76),(3,73),(4,78),(5,75),(6,80),(7,77),(8,74),(9,36),(10,33),(11,38),(12,35),(13,40),(14,37),(15,34),(16,39),(17,29),(18,26),(19,31),(20,28),(21,25),(22,30),(23,27),(24,32),(41,61),(42,58),(43,63),(44,60),(45,57),(46,62),(47,59),(48,64),(49,69),(50,66),(51,71),(52,68),(53,65),(54,70),(55,67),(56,72)]])`

Matrix representation of C524M4(2) in GL10(𝔽41)

 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 40 40 40 0 0 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 40 1 0 0 0 0 0 0 0 1 40 0 0 0 0 0 0 0 1 0 40 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 14 0 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 40 40 40 40
,
 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 40 40 40 40 0 0 0 0 0 0 0 1 0 0

`G:=sub<GL(10,GF(41))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,40,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,40,40,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[14,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,1,40],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,40,0] >;`

C524M4(2) in GAP, Magma, Sage, TeX

`C_5^2\rtimes_4M_4(2)`
`% in TeX`

`G:=Group("C5^2:4M4(2)");`
`// GroupNames label`

`G:=SmallGroup(400,128);`
`// by ID`

`G=gap.SmallGroup(400,128);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,50,1444,970,496,8645,2897]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^5=c^8=d^2=1,a*b=b*a,c*a*c^-1=a^3,d*a*d=a^-1,c*b*c^-1=b^2,b*d=d*b,d*c*d=c^5>;`
`// generators/relations`

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