Copied to
clipboard

## G = D12.38D4order 192 = 26·3

### 8th non-split extension by D12 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D12.38D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — Q8○D12 — D12.38D4
 Lower central C3 — C6 — C2×C12 — D12.38D4
 Upper central C1 — C2 — C2×C4 — C8⋊C22

Generators and relations for D12.38D4
G = < a,b,c,d | a12=b2=1, c4=d2=a6, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, bd=db, dcd-1=c3 >

Subgroups: 432 in 146 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, M4(2), M4(2), D8, SD16, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, C4.10D4, C4≀C2, C4.4D4, C8⋊C22, C8⋊C22, 2- 1+4, C4.Dic3, C4×Dic3, D4⋊S3, D4.S3, C6.D4, C3×M4(2), C3×D8, C3×SD16, C2×Dic6, C2×Dic6, C4○D12, C4○D12, D42S3, S3×Q8, C6×D4, C3×C4○D4, D4.8D4, C12.47D4, D12⋊C4, Q83Dic3, D126C22, C23.12D6, C3×C8⋊C22, Q8○D12, D12.38D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, S3×D4, C2×C3⋊D4, D4.8D4, C232D6, D12.38D4

Character table of D12.38D4

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 8A 8B 12A 12B 12C 24A 24B size 1 1 2 4 8 12 2 2 2 4 12 12 12 12 12 2 4 8 8 8 8 24 4 4 8 8 8 ρ1 1 1 1 1 1 1 1 1 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 -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 -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 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 -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 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 -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 1 -1 1 1 1 -1 -1 -1 linear of order 2 ρ9 2 2 2 2 -2 0 -1 2 2 2 0 0 0 0 0 -1 -1 -1 1 1 -2 0 -1 -1 -1 1 1 orthogonal lifted from D6 ρ10 2 2 2 -2 -2 0 -1 2 2 -2 0 0 0 0 0 -1 -1 1 1 1 2 0 -1 -1 1 -1 -1 orthogonal lifted from D6 ρ11 2 2 -2 0 0 2 2 2 -2 0 0 -2 0 0 0 2 -2 0 0 0 0 0 -2 2 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 0 0 -2 2 2 -2 0 0 2 0 0 0 2 -2 0 0 0 0 0 -2 2 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 2 0 -1 2 2 2 0 0 0 0 0 -1 -1 -1 -1 -1 2 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ14 2 2 -2 -2 0 0 2 -2 2 2 0 0 0 0 0 2 -2 -2 0 0 0 0 2 -2 2 0 0 orthogonal lifted from D4 ρ15 2 2 2 0 0 0 2 -2 -2 0 0 0 -2 2 0 2 2 0 0 0 0 0 -2 -2 0 0 0 orthogonal lifted from D4 ρ16 2 2 2 0 0 0 2 -2 -2 0 0 0 2 -2 0 2 2 0 0 0 0 0 -2 -2 0 0 0 orthogonal lifted from D4 ρ17 2 2 -2 2 0 0 2 -2 2 -2 0 0 0 0 0 2 -2 2 0 0 0 0 2 -2 -2 0 0 orthogonal lifted from D4 ρ18 2 2 2 -2 2 0 -1 2 2 -2 0 0 0 0 0 -1 -1 1 -1 -1 -2 0 -1 -1 1 1 1 orthogonal lifted from D6 ρ19 2 2 -2 2 0 0 -1 -2 2 -2 0 0 0 0 0 -1 1 -1 √-3 -√-3 0 0 -1 1 1 -√-3 √-3 complex lifted from C3⋊D4 ρ20 2 2 -2 2 0 0 -1 -2 2 -2 0 0 0 0 0 -1 1 -1 -√-3 √-3 0 0 -1 1 1 √-3 -√-3 complex lifted from C3⋊D4 ρ21 2 2 -2 -2 0 0 -1 -2 2 2 0 0 0 0 0 -1 1 1 √-3 -√-3 0 0 -1 1 -1 √-3 -√-3 complex lifted from C3⋊D4 ρ22 2 2 -2 -2 0 0 -1 -2 2 2 0 0 0 0 0 -1 1 1 -√-3 √-3 0 0 -1 1 -1 -√-3 √-3 complex lifted from C3⋊D4 ρ23 4 4 4 0 0 0 -2 -4 -4 0 0 0 0 0 0 -2 -2 0 0 0 0 0 2 2 0 0 0 orthogonal lifted from S3×D4 ρ24 4 4 -4 0 0 0 -2 4 -4 0 0 0 0 0 0 -2 2 0 0 0 0 0 2 -2 0 0 0 orthogonal lifted from S3×D4 ρ25 4 -4 0 0 0 0 4 0 0 0 -2i 0 0 0 2i -4 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.8D4 ρ26 4 -4 0 0 0 0 4 0 0 0 2i 0 0 0 -2i -4 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.8D4 ρ27 8 -8 0 0 0 0 -4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of D12.38D4
On 48 points
Generators in S48
```(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 39)(2 38)(3 37)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)
(1 43 10 46 7 37 4 40)(2 38 11 41 8 44 5 47)(3 45 12 48 9 39 6 42)(13 28 22 31 19 34 16 25)(14 35 23 26 20 29 17 32)(15 30 24 33 21 36 18 27)
(1 28 7 34)(2 29 8 35)(3 30 9 36)(4 31 10 25)(5 32 11 26)(6 33 12 27)(13 43 19 37)(14 44 20 38)(15 45 21 39)(16 46 22 40)(17 47 23 41)(18 48 24 42)```

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

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

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

Matrix representation of D12.38D4 in GL8(𝔽73)

 0 1 0 0 0 0 0 0 72 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 0 27 58 0 0 0 0 0 0 0 46 0 0 0 0 0 0 0 0 27 1 0 0 0 0 0 0 0 46
,
 67 67 58 0 0 0 0 0 3 6 0 15 0 0 0 0 6 0 6 67 0 0 0 0 0 67 3 67 0 0 0 0 0 0 0 0 14 52 58 29 0 0 0 0 65 59 19 72 0 0 0 0 3 14 1 46 0 0 0 0 57 45 19 72
,
 9 12 15 15 0 0 0 0 3 6 0 15 0 0 0 0 0 61 67 61 0 0 0 0 70 0 70 64 0 0 0 0 0 0 0 0 14 70 9 32 0 0 0 0 65 59 47 59 0 0 0 0 3 33 1 0 0 0 0 0 57 45 19 72
,
 70 12 3 3 0 0 0 0 64 6 0 3 0 0 0 0 0 61 67 61 0 0 0 0 9 0 9 3 0 0 0 0 0 0 0 0 1 0 25 11 0 0 0 0 0 1 0 72 0 0 0 0 70 40 72 0 0 0 0 0 0 2 0 72

`G:=sub<GL(8,GF(73))| [0,72,1,0,0,0,0,0,1,1,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,58,46,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,1,46],[67,3,6,0,0,0,0,0,67,6,0,67,0,0,0,0,58,0,6,3,0,0,0,0,0,15,67,67,0,0,0,0,0,0,0,0,14,65,3,57,0,0,0,0,52,59,14,45,0,0,0,0,58,19,1,19,0,0,0,0,29,72,46,72],[9,3,0,70,0,0,0,0,12,6,61,0,0,0,0,0,15,0,67,70,0,0,0,0,15,15,61,64,0,0,0,0,0,0,0,0,14,65,3,57,0,0,0,0,70,59,33,45,0,0,0,0,9,47,1,19,0,0,0,0,32,59,0,72],[70,64,0,9,0,0,0,0,12,6,61,0,0,0,0,0,3,0,67,9,0,0,0,0,3,3,61,3,0,0,0,0,0,0,0,0,1,0,70,0,0,0,0,0,0,1,40,2,0,0,0,0,25,0,72,0,0,0,0,0,11,72,0,72] >;`

D12.38D4 in GAP, Magma, Sage, TeX

`D_{12}._{38}D_4`
`% in TeX`

`G:=Group("D12.38D4");`
`// GroupNames label`

`G:=SmallGroup(192,760);`
`// by ID`

`G=gap.SmallGroup(192,760);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,1123,297,136,851,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=1,c^4=d^2=a^6,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^3>;`
`// generators/relations`

Export

׿
×
𝔽