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## G = C23.48D4order 64 = 26

### 19th 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 — C23.48D4
 Chief series C1 — C2 — C4 — C2×C4 — C4⋊C4 — C2×C4⋊C4 — C23.48D4
 Lower central C1 — C2 — C2×C4 — C23.48D4
 Upper central C1 — C22 — C22×C4 — C23.48D4
 Jennings C1 — C2 — C2 — C2×C4 — C23.48D4

Generators and relations for C23.48D4
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=bcd3 >

Character table of C23.48D4

 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 -2 2 0 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 symplectic lifted from Q16, Schur index 2 ρ12 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 symplectic lifted from Q16, Schur index 2 ρ13 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 symplectic lifted from Q16, Schur index 2 ρ14 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 -√2 -√2 √2 √2 symplectic lifted from Q16, Schur index 2 ρ15 2 -2 2 -2 0 0 -2 2 0 0 0 2i -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ16 2 -2 2 -2 0 0 -2 2 0 0 0 -2i 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ17 2 -2 2 -2 0 0 2 -2 -2i 0 2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ18 2 -2 2 -2 0 0 2 -2 2i 0 -2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ19 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22

Smallest permutation representation of C23.48D4
On 32 points
Generators in S32
```(2 31)(4 25)(6 27)(8 29)(10 20)(12 22)(14 24)(16 18)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)
(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 18 5 22)(2 15 6 11)(3 24 7 20)(4 13 8 9)(10 32 14 28)(12 30 16 26)(17 27 21 31)(19 25 23 29)```

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

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

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

C23.48D4 is a maximal subgroup of
C24.115D4  C24.183D4  C24.118D4  (C2×D4).302D4  (C2×D4).303D4  C42.224D4  C42.225D4  C42.228D4  C42.235D4  C24.121D4  C233Q16  C24.126D4  C24.129D4  C4.162+ 1+4  C4.182+ 1+4  C42.284D4  C42.288D4  C42.291D4
C4⋊C4.D2p: C24.17D4  C4⋊C4.18D4  C4⋊C4.20D4  C24.18D4  C42.354C23  C42.358C23  C42.424C23  C42.425C23 ...
(C2×C2p).Q16: C42.282D4  C23.40D12  C23.35D20  C23.35D28 ...
C23.48D4 is a maximal quotient of
C24.157D4  C24.88D4
(C2×C2p).Q16: C2.D85C4  C2.(C4×Q16)  (C2×C4).19Q16  (C2×C8).1Q8  (C2×C8).60D4  (C2×C4).23Q16  C23.40D12  C4⋊C4.230D6 ...
C4⋊C4.D2p: C23.37D8  C24.160D4  C24.86D4  D6.Q16  D6.2Q16  D10.7Q16  D10.8Q16  D14.Q16 ...

Matrix representation of C23.48D4 in GL4(𝔽17) generated by

 1 0 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
,
 0 4 0 0 13 0 0 0 0 0 11 11 0 0 3 0
,
 0 1 0 0 1 0 0 0 0 0 10 10 0 0 12 7
`G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[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],[0,13,0,0,4,0,0,0,0,0,11,3,0,0,11,0],[0,1,0,0,1,0,0,0,0,0,10,12,0,0,10,7] >;`

C23.48D4 in GAP, Magma, Sage, TeX

`C_2^3._{48}D_4`
`% in TeX`

`G:=Group("C2^3.48D4");`
`// GroupNames label`

`G:=SmallGroup(64,165);`
`// by ID`

`G=gap.SmallGroup(64,165);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,362,194,1444,376,88]);`
`// 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*c*d^3>;`
`// generators/relations`

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