Copied to
clipboard

## G = SD16⋊D6order 192 = 26·3

### 3rd semidirect product of SD16 and D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — SD16⋊D6
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — S3×C4○D4 — SD16⋊D6
 Lower central C3 — C6 — C12 — SD16⋊D6
 Upper central C1 — C4 — C2×C4 — C4○D8

Generators and relations for SD16⋊D6
G = < a,b,c,d | a8=b2=c6=d2=1, bab=cac-1=a3, dad=a-1, cbc-1=a2b, dbd=a6b, dcd=c-1 >

Subgroups: 712 in 262 conjugacy classes, 99 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), D8, D8, SD16, SD16, Q16, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×S3, C2×M4(2), C4○D8, C4○D8, C8⋊C22, C8.C22, C2×C4○D4, C8⋊S3, C24⋊C2, D24, Dic12, C2×C3⋊C8, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C2×C24, C3×D8, C3×SD16, C3×Q16, S3×C2×C4, S3×C2×C4, C4○D12, C4○D12, S3×D4, S3×D4, D42S3, D42S3, S3×Q8, Q83S3, C3×C4○D4, D8⋊C22, C2×C8⋊S3, C4○D24, D8⋊S3, Q83D6, D4.D6, Q16⋊S3, Q8.13D6, C3×C4○D8, S3×C4○D4, SD16⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, S3×D4, S3×C23, D8⋊C22, C2×S3×D4, SD16⋊D6

Smallest permutation representation of SD16⋊D6
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)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 19)(18 22)(21 23)(26 28)(27 31)(30 32)(33 37)(34 40)(36 38)(42 44)(43 47)(46 48)
(1 15 45 21 39 30)(2 10 46 24 40 25)(3 13 47 19 33 28)(4 16 48 22 34 31)(5 11 41 17 35 26)(6 14 42 20 36 29)(7 9 43 23 37 32)(8 12 44 18 38 27)
(1 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 14)(10 13)(11 12)(15 16)(17 27)(18 26)(19 25)(20 32)(21 31)(22 30)(23 29)(24 28)(33 40)(34 39)(35 38)(36 37)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C 6D 8A 8B 8C 8D 12A 12B 12C 12D 12E 24A 24B 24C 24D order 1 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 6 6 6 6 8 8 8 8 12 12 12 12 12 24 24 24 24 size 1 1 2 4 4 6 6 12 12 2 1 1 2 4 4 6 6 12 12 2 4 8 8 4 4 12 12 2 2 4 8 8 4 4 4 4

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 D6 D6 S3×D4 S3×D4 D8⋊C22 SD16⋊D6 kernel SD16⋊D6 C2×C8⋊S3 C4○D24 D8⋊S3 Q8⋊3D6 D4.D6 Q16⋊S3 Q8.13D6 C3×C4○D8 S3×C4○D4 C4○D8 C4×S3 C2×Dic3 C22×S3 C2×C8 D8 SD16 Q16 C4○D4 C4 C22 C3 C1 # reps 1 1 1 2 2 2 2 2 1 2 1 2 1 1 1 1 2 1 2 1 1 2 4

Matrix representation of SD16⋊D6 in GL4(𝔽5) generated by

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

SD16⋊D6 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes D_6
% in TeX

G:=Group("SD16:D6");
// GroupNames label

G:=SmallGroup(192,1327);
// by ID

G=gap.SmallGroup(192,1327);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,1123,570,185,438,235,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=c*a*c^-1=a^3,d*a*d=a^-1,c*b*c^-1=a^2*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽