metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊10D6, Q16⋊9D6, SD16⋊10D6, D24⋊22C22, C12.15C24, C24.37C23, D12.10C23, Dic12⋊19C22, Dic6.10C23, C4○D8⋊3S3, (C2×C8)⋊12D6, C4○D24⋊7C2, C4○D4⋊11D6, Q8⋊3D6⋊7C2, D8⋊S3⋊7C2, C3⋊C8.6C23, (C2×C24)⋊5C22, (C4×S3).51D4, D6.53(C2×D4), Q16⋊S3⋊7C2, C4.222(S3×D4), D4.D6⋊7C2, (S3×D4)⋊7C22, (S3×Q8)⋊8C22, C22.5(S3×D4), D4⋊S3⋊13C22, Q8.13D6⋊2C2, C12.381(C2×D4), C4○D12⋊6C22, (C3×D8)⋊15C22, (C4×S3).8C23, C4.15(S3×C23), C8.15(C22×S3), (C3×D4).9C23, D4.9(C22×S3), C24⋊C2⋊16C22, C8⋊S3⋊15C22, (C3×Q8).9C23, C3⋊2(D8⋊C22), D4.S3⋊12C22, (C2×Dic3).80D4, Dic3.58(C2×D4), (C3×Q16)⋊13C22, (C22×S3).42D4, C3⋊Q16⋊11C22, C6.116(C22×D4), Q8.19(C22×S3), (C2×C12).532C23, Q8⋊2S3⋊12C22, (C3×SD16)⋊10C22, D4⋊2S3.5C22, Q8⋊3S3.5C22, C2.89(C2×S3×D4), (C3×C4○D8)⋊3C2, (S3×C4○D4)⋊2C2, (C2×C8⋊S3)⋊1C2, (C2×C3⋊C8)⋊16C22, (C2×C6).12(C2×D4), (C3×C4○D4)⋊2C22, (S3×C2×C4).159C22, (C2×C4).619(C22×S3), SmallGroup(192,1327)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 712 in 262 conjugacy classes, 99 normal (33 characteristic)
C1, C2, C2 [×7], C3, C4 [×2], C4 [×6], C22, C22 [×11], S3 [×4], C6, C6 [×3], C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4 [×2], D4 [×12], Q8 [×2], Q8 [×4], C23 [×3], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×7], C2×C6, C2×C6 [×2], C2×C8, C2×C8, M4(2) [×4], D8, D8 [×3], SD16 [×2], SD16 [×6], Q16, Q16 [×3], C22×C4 [×3], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×2], C4○D4 [×10], C3⋊C8 [×2], C24 [×2], Dic6 [×2], Dic6 [×2], C4×S3 [×4], C4×S3 [×6], D12 [×2], D12 [×2], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×6], C2×C12, C2×C12 [×2], C3×D4 [×2], C3×D4 [×2], C3×Q8 [×2], C22×S3, C22×S3 [×2], C2×M4(2), C4○D8, C4○D8 [×3], C8⋊C22 [×4], C8.C22 [×4], C2×C4○D4 [×2], C8⋊S3 [×4], C24⋊C2 [×2], D24, Dic12, C2×C3⋊C8, D4⋊S3 [×2], D4.S3 [×2], Q8⋊2S3 [×2], C3⋊Q16 [×2], C2×C24, C3×D8, C3×SD16 [×2], C3×Q16, S3×C2×C4, S3×C2×C4 [×2], C4○D12 [×2], C4○D12 [×2], S3×D4 [×2], S3×D4 [×2], D4⋊2S3 [×2], D4⋊2S3 [×2], S3×Q8 [×2], Q8⋊3S3 [×2], C3×C4○D4 [×2], D8⋊C22, C2×C8⋊S3, C4○D24, D8⋊S3 [×2], Q8⋊3D6 [×2], D4.D6 [×2], Q16⋊S3 [×2], Q8.13D6 [×2], C3×C4○D8, S3×C4○D4 [×2], SD16⋊D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C22×D4, S3×D4 [×2], S3×C23, D8⋊C22, C2×S3×D4, SD16⋊D6
Generators and relations
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 >
►On 48 points
(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)(10 12)(11 15)(14 16)(17 19)(18 22)(21 23)(26 28)(27 31)(30 32)(33 35)(34 38)(37 39)(42 44)(43 47)(46 48)
(1 37 45 21 9 30)(2 40 46 24 10 25)(3 35 47 19 11 28)(4 38 48 22 12 31)(5 33 41 17 13 26)(6 36 42 20 14 29)(7 39 43 23 15 32)(8 34 44 18 16 27)
(1 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 12)(10 11)(13 16)(14 15)(17 27)(18 26)(19 25)(20 32)(21 31)(22 30)(23 29)(24 28)(33 34)(35 40)(36 39)(37 38)
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)(10,12)(11,15)(14,16)(17,19)(18,22)(21,23)(26,28)(27,31)(30,32)(33,35)(34,38)(37,39)(42,44)(43,47)(46,48), (1,37,45,21,9,30)(2,40,46,24,10,25)(3,35,47,19,11,28)(4,38,48,22,12,31)(5,33,41,17,13,26)(6,36,42,20,14,29)(7,39,43,23,15,32)(8,34,44,18,16,27), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,12)(10,11)(13,16)(14,15)(17,27)(18,26)(19,25)(20,32)(21,31)(22,30)(23,29)(24,28)(33,34)(35,40)(36,39)(37,38)>;
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)(10,12)(11,15)(14,16)(17,19)(18,22)(21,23)(26,28)(27,31)(30,32)(33,35)(34,38)(37,39)(42,44)(43,47)(46,48), (1,37,45,21,9,30)(2,40,46,24,10,25)(3,35,47,19,11,28)(4,38,48,22,12,31)(5,33,41,17,13,26)(6,36,42,20,14,29)(7,39,43,23,15,32)(8,34,44,18,16,27), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,12)(10,11)(13,16)(14,15)(17,27)(18,26)(19,25)(20,32)(21,31)(22,30)(23,29)(24,28)(33,34)(35,40)(36,39)(37,38) );
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),(10,12),(11,15),(14,16),(17,19),(18,22),(21,23),(26,28),(27,31),(30,32),(33,35),(34,38),(37,39),(42,44),(43,47),(46,48)], [(1,37,45,21,9,30),(2,40,46,24,10,25),(3,35,47,19,11,28),(4,38,48,22,12,31),(5,33,41,17,13,26),(6,36,42,20,14,29),(7,39,43,23,15,32),(8,34,44,18,16,27)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,42),(8,41),(9,12),(10,11),(13,16),(14,15),(17,27),(18,26),(19,25),(20,32),(21,31),(22,30),(23,29),(24,28),(33,34),(35,40),(36,39),(37,38)])
Matrix representation ►G ⊆ 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] >;
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 |
In GAP, Magma, Sage, TeX
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