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## G = C23.D4order 64 = 26

### 2nd non-split extension by C23 of D4 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C23.2D4, (C2×C4).2D4, C22⋊C42C4, (C22×C4)⋊2C4, C4.D4.C2, C23⋊C4.1C2, C23.2(C2×C4), C2.7(C23⋊C4), (C2×D4).2C22, C22.D4.1C2, C22.10(C22⋊C4), SmallGroup(64,33)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C23.D4
G = < a,b,c,d,e | a2=b2=c2=1, d4=c, e2=a, ab=ba, ac=ca, dad-1=abc, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=ad3 >

Character table of C23.D4

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 8A 8B size 1 1 2 4 4 4 4 4 8 8 8 8 8 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 1 1 -1 1 -1 -1 -1 -i i 1 i -i linear of order 4 ρ6 1 1 1 -1 1 1 -1 1 i -i -1 i -i linear of order 4 ρ7 1 1 1 -1 1 -1 -1 -1 i -i 1 -i i linear of order 4 ρ8 1 1 1 -1 1 1 -1 1 -i i -1 -i i linear of order 4 ρ9 2 2 2 -2 -2 0 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 -4 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ12 4 -4 0 0 0 2i 0 -2i 0 0 0 0 0 complex faithful ρ13 4 -4 0 0 0 -2i 0 2i 0 0 0 0 0 complex faithful

Permutation representations of C23.D4
On 16 points - transitive group 16T140
Generators in S16
```(1 3)(2 8)(4 6)(5 7)(9 11)(10 12)(13 15)(14 16)
(2 6)(4 8)(9 13)(11 15)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 11 3 9)(2 16 8 14)(4 10 6 12)(5 15 7 13)```

`G:=sub<Sym(16)| (1,3)(2,8)(4,6)(5,7)(9,11)(10,12)(13,15)(14,16), (2,6)(4,8)(9,13)(11,15), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,11,3,9)(2,16,8,14)(4,10,6,12)(5,15,7,13)>;`

`G:=Group( (1,3)(2,8)(4,6)(5,7)(9,11)(10,12)(13,15)(14,16), (2,6)(4,8)(9,13)(11,15), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,11,3,9)(2,16,8,14)(4,10,6,12)(5,15,7,13) );`

`G=PermutationGroup([(1,3),(2,8),(4,6),(5,7),(9,11),(10,12),(13,15),(14,16)], [(2,6),(4,8),(9,13),(11,15)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,11,3,9),(2,16,8,14),(4,10,6,12),(5,15,7,13)])`

`G:=TransitiveGroup(16,140);`

On 16 points - transitive group 16T148
Generators in S16
```(2 10)(3 7)(4 16)(6 14)(8 12)(11 15)
(1 13)(2 10)(3 15)(4 12)(5 9)(6 14)(7 11)(8 16)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 16 10 4)(3 15 7 11)(6 12 14 8)(9 13)```

`G:=sub<Sym(16)| (2,10)(3,7)(4,16)(6,14)(8,12)(11,15), (1,13)(2,10)(3,15)(4,12)(5,9)(6,14)(7,11)(8,16), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,16,10,4)(3,15,7,11)(6,12,14,8)(9,13)>;`

`G:=Group( (2,10)(3,7)(4,16)(6,14)(8,12)(11,15), (1,13)(2,10)(3,15)(4,12)(5,9)(6,14)(7,11)(8,16), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,16,10,4)(3,15,7,11)(6,12,14,8)(9,13) );`

`G=PermutationGroup([(2,10),(3,7),(4,16),(6,14),(8,12),(11,15)], [(1,13),(2,10),(3,15),(4,12),(5,9),(6,14),(7,11),(8,16)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,16,10,4),(3,15,7,11),(6,12,14,8),(9,13)])`

`G:=TransitiveGroup(16,148);`

On 16 points - transitive group 16T153
Generators in S16
```(1 9)(2 10)(3 15)(4 16)(5 13)(6 14)(7 11)(8 12)
(1 5)(3 7)(9 13)(11 15)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 4 9 16)(2 11 10 7)(3 6 15 14)(5 8 13 12)```

`G:=sub<Sym(16)| (1,9)(2,10)(3,15)(4,16)(5,13)(6,14)(7,11)(8,12), (1,5)(3,7)(9,13)(11,15), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,4,9,16)(2,11,10,7)(3,6,15,14)(5,8,13,12)>;`

`G:=Group( (1,9)(2,10)(3,15)(4,16)(5,13)(6,14)(7,11)(8,12), (1,5)(3,7)(9,13)(11,15), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,4,9,16)(2,11,10,7)(3,6,15,14)(5,8,13,12) );`

`G=PermutationGroup([(1,9),(2,10),(3,15),(4,16),(5,13),(6,14),(7,11),(8,12)], [(1,5),(3,7),(9,13),(11,15)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,4,9,16),(2,11,10,7),(3,6,15,14),(5,8,13,12)])`

`G:=TransitiveGroup(16,153);`

On 16 points - transitive group 16T160
Generators in S16
```(2 6)(3 7)(10 14)(11 15)
(2 6)(4 8)(10 14)(12 16)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 16)(2 15 6 11)(3 14 7 10)(4 9)(5 12)(8 13)```

`G:=sub<Sym(16)| (2,6)(3,7)(10,14)(11,15), (2,6)(4,8)(10,14)(12,16), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,16)(2,15,6,11)(3,14,7,10)(4,9)(5,12)(8,13)>;`

`G:=Group( (2,6)(3,7)(10,14)(11,15), (2,6)(4,8)(10,14)(12,16), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,16)(2,15,6,11)(3,14,7,10)(4,9)(5,12)(8,13) );`

`G=PermutationGroup([(2,6),(3,7),(10,14),(11,15)], [(2,6),(4,8),(10,14),(12,16)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,16),(2,15,6,11),(3,14,7,10),(4,9),(5,12),(8,13)])`

`G:=TransitiveGroup(16,160);`

C23.D4 is a maximal subgroup of
C424D4  C426D4  C42.14D4  C22⋊C4⋊F5  (C22×C4)⋊F5
(C2×D4).D2p: C4○C2≀C4  C24.36D4  C23.(C2×D4)  C42.13D4  (C2×D4).D6  C23.4D12  (C22×C12)⋊C4  (C2×C20).D4 ...
C23.D4 is a maximal quotient of
C23.15M4(2)  C23.2M4(2)  C24.4D4  (C2×C4).Q16  C22⋊C4⋊F5  (C22×C4)⋊F5
C23.D4p: C23.4D8  C23.4D12  C23.4D20  C23.4D28 ...
(C2×C4).D4p: (C2×C4).D8  (C2×D4).D6  (C2×C20).D4  (C2×C28).D4 ...
(C2×D4).D2p: C24.5D4  (C22×C12)⋊C4  (C22×C20)⋊C4  (C22×C28)⋊C4 ...

Matrix representation of C23.D4 in GL4(𝔽5) generated by

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

C23.D4 in GAP, Magma, Sage, TeX

`C_2^3.D_4`
`% in TeX`

`G:=Group("C2^3.D4");`
`// GroupNames label`

`G:=SmallGroup(64,33);`
`// by ID`

`G=gap.SmallGroup(64,33);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,199,362,297,255,1444]);`
`// Polycyclic`

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

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