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## G = D4.D6order 96 = 25·3

### 4th non-split extension by D4 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — D4.D6
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×Q8 — D4.D6
 Lower central C3 — C6 — C12 — D4.D6
 Upper central C1 — C2 — C4 — SD16

Generators and relations for D4.D6
G = < a,b,c,d | a4=b2=1, c6=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=ab, dbd-1=a-1b, dcd-1=c5 >

Subgroups: 138 in 60 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, Dic3, Dic3, C12, C12, D6, C2×C6, M4(2), SD16, SD16, Q16, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C8.C22, C8⋊S3, Dic12, D4.S3, C3⋊Q16, C3×SD16, D42S3, S3×Q8, D4.D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C8.C22, S3×D4, D4.D6

Character table of D4.D6

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 6A 6B 8A 8B 12A 12B 24A 24B size 1 1 4 6 2 2 4 6 12 12 2 8 4 12 4 8 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 2 0 2 2 -2 0 -2 0 0 2 0 0 0 -2 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 0 -1 2 2 0 0 0 -1 1 -2 0 -1 -1 1 1 orthogonal lifted from D6 ρ11 2 2 0 -2 2 -2 0 2 0 0 2 0 0 0 -2 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 0 -1 2 2 0 0 0 -1 -1 2 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ13 2 2 2 0 -1 2 -2 0 0 0 -1 -1 -2 0 -1 1 1 1 orthogonal lifted from D6 ρ14 2 2 -2 0 -1 2 -2 0 0 0 -1 1 2 0 -1 1 -1 -1 orthogonal lifted from D6 ρ15 4 4 0 0 -2 -4 0 0 0 0 -2 0 0 0 2 0 0 0 orthogonal lifted from S3×D4 ρ16 4 -4 0 0 4 0 0 0 0 0 -4 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ17 4 -4 0 0 -2 0 0 0 0 0 2 0 0 0 0 0 √6 -√6 symplectic faithful, Schur index 2 ρ18 4 -4 0 0 -2 0 0 0 0 0 2 0 0 0 0 0 -√6 √6 symplectic faithful, Schur index 2

Smallest permutation representation of D4.D6
On 48 points
Generators in S48
```(1 40 7 46)(2 47 8 41)(3 42 9 48)(4 37 10 43)(5 44 11 38)(6 39 12 45)(13 30 19 36)(14 25 20 31)(15 32 21 26)(16 27 22 33)(17 34 23 28)(18 29 24 35)
(1 32)(2 16)(3 34)(4 18)(5 36)(6 20)(7 26)(8 22)(9 28)(10 24)(11 30)(12 14)(13 38)(15 40)(17 42)(19 44)(21 46)(23 48)(25 39)(27 41)(29 43)(31 45)(33 47)(35 37)
(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)
(1 6 7 12)(2 11 8 5)(3 4 9 10)(13 16 19 22)(14 21 20 15)(17 24 23 18)(25 32 31 26)(27 30 33 36)(28 35 34 29)(37 42 43 48)(38 47 44 41)(39 40 45 46)```

`G:=sub<Sym(48)| (1,40,7,46)(2,47,8,41)(3,42,9,48)(4,37,10,43)(5,44,11,38)(6,39,12,45)(13,30,19,36)(14,25,20,31)(15,32,21,26)(16,27,22,33)(17,34,23,28)(18,29,24,35), (1,32)(2,16)(3,34)(4,18)(5,36)(6,20)(7,26)(8,22)(9,28)(10,24)(11,30)(12,14)(13,38)(15,40)(17,42)(19,44)(21,46)(23,48)(25,39)(27,41)(29,43)(31,45)(33,47)(35,37), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,16,19,22)(14,21,20,15)(17,24,23,18)(25,32,31,26)(27,30,33,36)(28,35,34,29)(37,42,43,48)(38,47,44,41)(39,40,45,46)>;`

`G:=Group( (1,40,7,46)(2,47,8,41)(3,42,9,48)(4,37,10,43)(5,44,11,38)(6,39,12,45)(13,30,19,36)(14,25,20,31)(15,32,21,26)(16,27,22,33)(17,34,23,28)(18,29,24,35), (1,32)(2,16)(3,34)(4,18)(5,36)(6,20)(7,26)(8,22)(9,28)(10,24)(11,30)(12,14)(13,38)(15,40)(17,42)(19,44)(21,46)(23,48)(25,39)(27,41)(29,43)(31,45)(33,47)(35,37), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,16,19,22)(14,21,20,15)(17,24,23,18)(25,32,31,26)(27,30,33,36)(28,35,34,29)(37,42,43,48)(38,47,44,41)(39,40,45,46) );`

`G=PermutationGroup([[(1,40,7,46),(2,47,8,41),(3,42,9,48),(4,37,10,43),(5,44,11,38),(6,39,12,45),(13,30,19,36),(14,25,20,31),(15,32,21,26),(16,27,22,33),(17,34,23,28),(18,29,24,35)], [(1,32),(2,16),(3,34),(4,18),(5,36),(6,20),(7,26),(8,22),(9,28),(10,24),(11,30),(12,14),(13,38),(15,40),(17,42),(19,44),(21,46),(23,48),(25,39),(27,41),(29,43),(31,45),(33,47),(35,37)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,16,19,22),(14,21,20,15),(17,24,23,18),(25,32,31,26),(27,30,33,36),(28,35,34,29),(37,42,43,48),(38,47,44,41),(39,40,45,46)]])`

Matrix representation of D4.D6 in GL4(𝔽5) generated by

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

D4.D6 in GAP, Magma, Sage, TeX

`D_4.D_6`
`% in TeX`

`G:=Group("D4.D6");`
`// GroupNames label`

`G:=SmallGroup(96,122);`
`// by ID`

`G=gap.SmallGroup(96,122);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,362,116,297,159,69,2309]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=1,c^6=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^5>;`
`// generators/relations`

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