Copied to
clipboard

## G = C42⋊8C4order 64 = 26

### 5th 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⋊8C4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C42⋊8C4
 Lower central C1 — C22 — C42⋊8C4
 Upper central C1 — C23 — C42⋊8C4
 Jennings C1 — C23 — C42⋊8C4

Generators and relations for C428C4
G = < a,b,c | a4=b4=c4=1, ab=ba, cac-1=ab2, cbc-1=a2b >

Subgroups: 113 in 77 conjugacy classes, 49 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C42, C4⋊C4, C22×C4, C22×C4, C2.C42, C2×C42, C2×C4⋊C4, C428C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C42⋊C2, C4.4D4, C42.C2, C428C4

Character table of C428C4

 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 0 0 -2 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 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 ρ20 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 ρ21 2 2 -2 2 2 -2 -2 -2 0 2i 0 0 0 0 -2i -2i 2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 -2 2 2 -2 -2 -2 0 -2i 0 0 0 0 2i 2i -2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 2 2 2 -2 -2 2 -2 -2 2i 0 0 -2i -2i 2i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ24 2 -2 -2 -2 2 2 -2 2 0 2i 0 0 0 0 2i -2i -2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ25 2 -2 2 2 -2 -2 -2 2 2i 0 0 2i -2i -2i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ26 2 -2 2 2 -2 -2 -2 2 -2i 0 0 -2i 2i 2i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ27 2 -2 -2 -2 2 2 -2 2 0 -2i 0 0 0 0 -2i 2i 2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ28 2 2 2 -2 -2 2 -2 -2 -2i 0 0 2i 2i -2i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4

Smallest permutation representation of C428C4
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 27 14 10)(2 28 15 11)(3 25 16 12)(4 26 13 9)(5 37 22 30)(6 38 23 31)(7 39 24 32)(8 40 21 29)(17 50 61 33)(18 51 62 34)(19 52 63 35)(20 49 64 36)(41 46 58 55)(42 47 59 56)(43 48 60 53)(44 45 57 54)
(1 50 6 42)(2 34 7 60)(3 52 8 44)(4 36 5 58)(9 62 30 48)(10 19 31 54)(11 64 32 46)(12 17 29 56)(13 49 22 41)(14 33 23 59)(15 51 24 43)(16 35 21 57)(18 37 53 26)(20 39 55 28)(25 61 40 47)(27 63 38 45)```

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

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

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

Matrix representation of C428C4 in GL5(𝔽5)

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

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

C428C4 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_8C_4`
`% in TeX`

`G:=Group("C4^2:8C4");`
`// GroupNames label`

`G:=SmallGroup(64,63);`
`// by ID`

`G=gap.SmallGroup(64,63);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,2,192,121,103,362,50]);`
`// Polycyclic`

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

Export

׿
×
𝔽