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## G = D4.5D4order 64 = 26

### 5th non-split extension by D4 of D4 acting via D4/C4=C2

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

Aliases: D4.5D4, C8.20D4, Q8.5D4, M4(2).5C22, (C2×Q16)⋊8C2, C8○D4.1C2, C4.60(C2×D4), C8.C22.C2, C8.C47C2, C4.10D44C2, (C2×C4).10C23, (C2×C8).19C22, C2.25(C4⋊D4), C4○D4.11C22, C22.8(C4○D4), (C2×Q8).16C22, SmallGroup(64,154)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — D4.5D4
 Chief series C1 — C2 — C4 — C2×C4 — C4○D4 — C8○D4 — D4.5D4
 Lower central C1 — C2 — C2×C4 — D4.5D4
 Upper central C1 — C2 — C2×C4 — D4.5D4
 Jennings C1 — C2 — C2 — C2×C4 — D4.5D4

Generators and relations for D4.5D4
G = < a,b,c,d | a4=b2=1, c4=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c3 >

Character table of D4.5D4

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

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

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

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

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

D4.5D4 is a maximal subgroup of
M4(2).10C23  C8.S4
D4p.D4: D8.3D4  D8.12D4  D8.13D4  D8○SD16  D8○Q16  D12.7D4  C24.18D4  C24.29D4 ...
M4(2).D2p: D4.4D8  D4.5D8  M4(2).38D4  Q8.10D12  M4(2).16D6  D4.5D20  M4(2).16D10  D4.5D28 ...
D4.5D4 is a maximal quotient of
M4(2).D2p: M4(2).49D4  M4(2).33D4  M4(2).6D4  M4(2).11D4  M4(2).13D4  D12.7D4  C24.18D4  Q8.10D12 ...
(C2×C8).D2p: C8.28D8  Q81Q16  C810SD16  C87Q16  D4.3SD16  Q8.3SD16  C8.D8  C8.SD16 ...

Matrix representation of D4.5D4 in GL4(𝔽7) generated by

 0 6 5 1 3 0 5 6 3 3 6 1 1 6 3 1
,
 1 2 5 3 4 0 2 2 6 6 0 2 6 1 4 6
,
 5 0 2 6 6 5 5 6 6 1 1 2 2 2 6 2
,
 4 2 4 5 6 3 3 5 6 1 4 1 2 2 5 3
`G:=sub<GL(4,GF(7))| [0,3,3,1,6,0,3,6,5,5,6,3,1,6,1,1],[1,4,6,6,2,0,6,1,5,2,0,4,3,2,2,6],[5,6,6,2,0,5,1,2,2,5,1,6,6,6,2,2],[4,6,6,2,2,3,1,2,4,3,4,5,5,5,1,3] >;`

D4.5D4 in GAP, Magma, Sage, TeX

`D_4._5D_4`
`% in TeX`

`G:=Group("D4.5D4");`
`// GroupNames label`

`G:=SmallGroup(64,154);`
`// by ID`

`G=gap.SmallGroup(64,154);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,247,362,963,117,1444,376,88]);`
`// Polycyclic`

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

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