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## G = C23.47D4order 64 = 26

### 18th non-split extension by C23 of D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C23.47D4
 Chief series C1 — C2 — C4 — C2×C4 — C4⋊C4 — C2×C4⋊C4 — C23.47D4
 Lower central C1 — C2 — C2×C4 — C23.47D4
 Upper central C1 — C22 — C22×C4 — C23.47D4
 Jennings C1 — C2 — C2 — C2×C4 — C23.47D4

Generators and relations for C23.47D4
G = < a,b,c,d,e | a2=b2=c2=1, d4=e2=c, dad-1=eae-1=ab=ba, ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd3 >

Character table of C23.47D4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D size 1 1 1 1 2 2 2 2 4 4 4 4 4 8 8 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 2 2 2 -2 -2 -2 -2 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 -2 -2 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 0 -2 2 0 0 0 2i -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ12 2 -2 2 -2 0 0 -2 2 0 0 0 -2i 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ13 2 -2 2 -2 0 0 2 -2 -2i 0 2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ14 2 -2 2 -2 0 0 2 -2 2i 0 -2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ15 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 √-2 -√-2 √-2 -√-2 complex lifted from SD16 ρ16 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 complex lifted from SD16 ρ17 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 complex lifted from SD16 ρ18 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 -√-2 √-2 -√-2 √-2 complex lifted from SD16 ρ19 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C23.47D4
On 32 points
Generators in S32
```(2 30)(4 32)(6 26)(8 28)(9 22)(11 24)(13 18)(15 20)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
(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)
(1 22 5 18)(2 12 6 16)(3 20 7 24)(4 10 8 14)(9 25 13 29)(11 31 15 27)(17 26 21 30)(19 32 23 28)```

`G:=sub<Sym(32)| (2,30)(4,32)(6,26)(8,28)(9,22)(11,24)(13,18)(15,20), (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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), (1,22,5,18)(2,12,6,16)(3,20,7,24)(4,10,8,14)(9,25,13,29)(11,31,15,27)(17,26,21,30)(19,32,23,28)>;`

`G:=Group( (2,30)(4,32)(6,26)(8,28)(9,22)(11,24)(13,18)(15,20), (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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), (1,22,5,18)(2,12,6,16)(3,20,7,24)(4,10,8,14)(9,25,13,29)(11,31,15,27)(17,26,21,30)(19,32,23,28) );`

`G=PermutationGroup([(2,30),(4,32),(6,26),(8,28),(9,22),(11,24),(13,18),(15,20)], [(1,29),(2,30),(3,31),(4,32),(5,25),(6,26),(7,27),(8,28),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)], [(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)], [(1,22,5,18),(2,12,6,16),(3,20,7,24),(4,10,8,14),(9,25,13,29),(11,31,15,27),(17,26,21,30),(19,32,23,28)])`

C23.47D4 is a maximal subgroup of
C24.115D4  C24.183D4  C24.118D4  (C2×D4).302D4  (C2×D4).304D4  C42.226D4  C42.231D4  C42.232D4  C234SD16  C24.123D4  C24.127D4  C24.128D4  C4.162+ 1+4  C4.192+ 1+4  C42.284D4  C42.288D4  C42.290D4
C4⋊C4.D2p: C24.14D4  C4⋊C4.12D4  (C2×C4).SD16  C24.15D4  C42.354C23  C42.359C23  C42.424C23  C42.426C23 ...
C2p.(C2×SD16): C42.222D4  C42.281D4  C23.39D12  C23.34D20  C23.34D28 ...
C23.47D4 is a maximal quotient of
C4.Q810C4  (C2×C4).19Q16  C24.89D4  (C2×C8).170D4  (C2×C4).28D8
C23.D4p: C23.36D8  C23.39D12  C23.34D20  C23.34D28 ...
C4⋊C4.D2p: C24.159D4  C24.160D4  C4.68(C4×D4)  C24.85D4  C2.(C83Q8)  D6.1SD16  D6.2SD16  C4⋊C4.231D6 ...

Matrix representation of C23.47D4 in GL4(𝔽17) generated by

 1 0 0 0 16 16 0 0 0 0 1 0 0 0 0 1
,
 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 16 0 0 0 0 16
,
 4 8 0 0 13 13 0 0 0 0 12 5 0 0 12 12
,
 16 15 0 0 0 1 0 0 0 0 16 10 0 0 10 1
`G:=sub<GL(4,GF(17))| [1,16,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[4,13,0,0,8,13,0,0,0,0,12,12,0,0,5,12],[16,0,0,0,15,1,0,0,0,0,16,10,0,0,10,1] >;`

C23.47D4 in GAP, Magma, Sage, TeX

`C_2^3._{47}D_4`
`% in TeX`

`G:=Group("C2^3.47D4");`
`// GroupNames label`

`G:=SmallGroup(64,164);`
`// by ID`

`G=gap.SmallGroup(64,164);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,362,50,1444,376,88]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^4=e^2=c,d*a*d^-1=e*a*e^-1=a*b=b*a,a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^3>;`
`// generators/relations`

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