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## G = C42⋊9C4order 64 = 26

### 6th semidirect product of C42 and C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C42⋊9C4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C42⋊9C4
 Lower central C1 — C22 — C42⋊9C4
 Upper central C1 — C23 — C42⋊9C4
 Jennings C1 — C23 — C42⋊9C4

Generators and relations for C429C4
G = < a,b,c | a4=b4=c4=1, ab=ba, cac-1=a-1, cbc-1=b-1 >

Subgroups: 129 in 93 conjugacy classes, 65 normal (6 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C42, C4⋊C4, C22×C4, C2×C42, C2×C4⋊C4, C429C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, C41D4, C4⋊Q8, C429C4

Character table of C429C4

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 4Q 4R 4S 4T size 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 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 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 -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 -1 1 1 1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 1 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 ρ6 1 1 1 1 1 1 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 ρ7 1 1 1 1 1 1 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 ρ8 1 1 1 1 1 1 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 ρ9 1 -1 1 -1 1 -1 1 -1 -1 -1 1 1 -1 1 1 -1 1 -1 1 -1 -i -i i i -i i i -i linear of order 4 ρ10 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -i i -i i -i i -i i linear of order 4 ρ11 1 -1 1 -1 1 -1 1 -1 1 -1 -1 -1 1 -1 1 -1 1 1 -1 1 i -i -i i -i -i i i linear of order 4 ρ12 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 i i i i -i -i -i -i linear of order 4 ρ13 1 -1 1 -1 1 -1 1 -1 -1 -1 1 1 -1 1 1 -1 1 -1 1 -1 i i -i -i i -i -i i linear of order 4 ρ14 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 i -i i -i i -i i -i linear of order 4 ρ15 1 -1 1 -1 1 -1 1 -1 1 -1 -1 -1 1 -1 1 -1 1 1 -1 1 -i i i -i i i -i -i linear of order 4 ρ16 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 -i -i -i -i i i i i linear of order 4 ρ17 2 2 -2 -2 -2 -2 2 2 0 0 2 0 0 0 0 0 0 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 -2 2 2 -2 -2 -2 2 2 0 0 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 2 2 -2 -2 -2 2 -2 0 0 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 -2 -2 -2 2 2 -2 2 0 2 0 0 0 0 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 -2 -2 -2 2 2 -2 2 0 -2 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ22 2 2 -2 -2 -2 -2 2 2 0 0 -2 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 -2 2 2 -2 -2 -2 0 2 0 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ24 2 -2 -2 2 -2 2 2 -2 0 0 -2 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ25 2 2 2 -2 -2 2 -2 -2 2 0 0 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ26 2 -2 -2 2 -2 2 2 -2 0 0 2 0 0 0 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ27 2 2 -2 2 2 -2 -2 -2 0 -2 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ28 2 2 2 -2 -2 2 -2 -2 -2 0 0 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2

Smallest permutation representation of C429C4
Regular action on 64 points
Generators in S64
```(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)
(1 13 51 40)(2 14 52 37)(3 15 49 38)(4 16 50 39)(5 44 55 20)(6 41 56 17)(7 42 53 18)(8 43 54 19)(9 31 36 28)(10 32 33 25)(11 29 34 26)(12 30 35 27)(21 61 45 60)(22 62 46 57)(23 63 47 58)(24 64 48 59)
(1 63 29 56)(2 62 30 55)(3 61 31 54)(4 64 32 53)(5 52 57 27)(6 51 58 26)(7 50 59 25)(8 49 60 28)(9 19 38 45)(10 18 39 48)(11 17 40 47)(12 20 37 46)(13 23 34 41)(14 22 35 44)(15 21 36 43)(16 24 33 42)```

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

`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), (1,13,51,40)(2,14,52,37)(3,15,49,38)(4,16,50,39)(5,44,55,20)(6,41,56,17)(7,42,53,18)(8,43,54,19)(9,31,36,28)(10,32,33,25)(11,29,34,26)(12,30,35,27)(21,61,45,60)(22,62,46,57)(23,63,47,58)(24,64,48,59), (1,63,29,56)(2,62,30,55)(3,61,31,54)(4,64,32,53)(5,52,57,27)(6,51,58,26)(7,50,59,25)(8,49,60,28)(9,19,38,45)(10,18,39,48)(11,17,40,47)(12,20,37,46)(13,23,34,41)(14,22,35,44)(15,21,36,43)(16,24,33,42) );`

`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)], [(1,13,51,40),(2,14,52,37),(3,15,49,38),(4,16,50,39),(5,44,55,20),(6,41,56,17),(7,42,53,18),(8,43,54,19),(9,31,36,28),(10,32,33,25),(11,29,34,26),(12,30,35,27),(21,61,45,60),(22,62,46,57),(23,63,47,58),(24,64,48,59)], [(1,63,29,56),(2,62,30,55),(3,61,31,54),(4,64,32,53),(5,52,57,27),(6,51,58,26),(7,50,59,25),(8,49,60,28),(9,19,38,45),(10,18,39,48),(11,17,40,47),(12,20,37,46),(13,23,34,41),(14,22,35,44),(15,21,36,43),(16,24,33,42)]])`

Matrix representation of C429C4 in GL5(𝔽5)

 4 0 0 0 0 0 3 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 3
,
 4 0 0 0 0 0 2 0 0 0 0 0 3 0 0 0 0 0 4 0 0 0 0 0 4
,
 2 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 4 0

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

C429C4 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_9C_4`
`% in TeX`

`G:=Group("C4^2:9C4");`
`// GroupNames label`

`G:=SmallGroup(64,65);`
`// by ID`

`G=gap.SmallGroup(64,65);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,2,192,121,55,362,86]);`
`// Polycyclic`

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

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