direct product, metabelian, supersoluble, monomial
Aliases: D8×C3⋊S3, C24⋊5D6, C3⋊4(S3×D8), (C3×D4)⋊3D6, (C3×D8)⋊2S3, C32⋊5D8⋊8C2, C32⋊12(C2×D8), (C3×C24)⋊9C22, C6.118(S3×D4), (C32×D8)⋊5C2, C32⋊7D8⋊5C2, C3⋊Dic3.48D4, C12⋊S3⋊7C22, C12.87(C22×S3), (C3×C12).91C23, (D4×C32)⋊7C22, C32⋊4C8⋊20C22, C8⋊4(C2×C3⋊S3), (C8×C3⋊S3)⋊3C2, (D4×C3⋊S3)⋊3C2, D4⋊1(C2×C3⋊S3), C2.15(D4×C3⋊S3), (C2×C3⋊S3).72D4, C4.1(C22×C3⋊S3), (C3×C6).239(C2×D4), (C4×C3⋊S3).70C22, SmallGroup(288,767)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8×C3⋊S3
G = < a,b,c,d,e | a8=b2=c3=d3=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 1236 in 228 conjugacy classes, 57 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, C23, C32, Dic3, C12, D6, C2×C6, C2×C8, D8, D8, C2×D4, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C2×D8, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C8, D24, D4⋊S3, C3×D8, S3×D4, C32⋊4C8, C3×C24, C4×C3⋊S3, C12⋊S3, C32⋊7D4, D4×C32, C22×C3⋊S3, S3×D8, C8×C3⋊S3, C32⋊5D8, C32⋊7D8, C32×D8, D4×C3⋊S3, D8×C3⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊S3, C22×S3, C2×D8, C2×C3⋊S3, S3×D4, C22×C3⋊S3, S3×D8, D4×C3⋊S3, D8×C3⋊S3
►On 72 points
Generators in S72
(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)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)
(2 8)(3 7)(4 6)(10 16)(11 15)(12 14)(17 23)(18 22)(19 21)(25 29)(26 28)(30 32)(33 35)(36 40)(37 39)(41 47)(42 46)(43 45)(50 56)(51 55)(52 54)(58 64)(59 63)(60 62)(65 69)(66 68)(70 72)
(1 13 20)(2 14 21)(3 15 22)(4 16 23)(5 9 24)(6 10 17)(7 11 18)(8 12 19)(25 51 46)(26 52 47)(27 53 48)(28 54 41)(29 55 42)(30 56 43)(31 49 44)(32 50 45)(33 70 60)(34 71 61)(35 72 62)(36 65 63)(37 66 64)(38 67 57)(39 68 58)(40 69 59)
(1 27 61)(2 28 62)(3 29 63)(4 30 64)(5 31 57)(6 32 58)(7 25 59)(8 26 60)(9 49 38)(10 50 39)(11 51 40)(12 52 33)(13 53 34)(14 54 35)(15 55 36)(16 56 37)(17 45 68)(18 46 69)(19 47 70)(20 48 71)(21 41 72)(22 42 65)(23 43 66)(24 44 67)
(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 65)(56 66)
G:=sub<Sym(72)| (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72), (2,8)(3,7)(4,6)(10,16)(11,15)(12,14)(17,23)(18,22)(19,21)(25,29)(26,28)(30,32)(33,35)(36,40)(37,39)(41,47)(42,46)(43,45)(50,56)(51,55)(52,54)(58,64)(59,63)(60,62)(65,69)(66,68)(70,72), (1,13,20)(2,14,21)(3,15,22)(4,16,23)(5,9,24)(6,10,17)(7,11,18)(8,12,19)(25,51,46)(26,52,47)(27,53,48)(28,54,41)(29,55,42)(30,56,43)(31,49,44)(32,50,45)(33,70,60)(34,71,61)(35,72,62)(36,65,63)(37,66,64)(38,67,57)(39,68,58)(40,69,59), (1,27,61)(2,28,62)(3,29,63)(4,30,64)(5,31,57)(6,32,58)(7,25,59)(8,26,60)(9,49,38)(10,50,39)(11,51,40)(12,52,33)(13,53,34)(14,54,35)(15,55,36)(16,56,37)(17,45,68)(18,46,69)(19,47,70)(20,48,71)(21,41,72)(22,42,65)(23,43,66)(24,44,67), (9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,65)(56,66)>;
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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72), (2,8)(3,7)(4,6)(10,16)(11,15)(12,14)(17,23)(18,22)(19,21)(25,29)(26,28)(30,32)(33,35)(36,40)(37,39)(41,47)(42,46)(43,45)(50,56)(51,55)(52,54)(58,64)(59,63)(60,62)(65,69)(66,68)(70,72), (1,13,20)(2,14,21)(3,15,22)(4,16,23)(5,9,24)(6,10,17)(7,11,18)(8,12,19)(25,51,46)(26,52,47)(27,53,48)(28,54,41)(29,55,42)(30,56,43)(31,49,44)(32,50,45)(33,70,60)(34,71,61)(35,72,62)(36,65,63)(37,66,64)(38,67,57)(39,68,58)(40,69,59), (1,27,61)(2,28,62)(3,29,63)(4,30,64)(5,31,57)(6,32,58)(7,25,59)(8,26,60)(9,49,38)(10,50,39)(11,51,40)(12,52,33)(13,53,34)(14,54,35)(15,55,36)(16,56,37)(17,45,68)(18,46,69)(19,47,70)(20,48,71)(21,41,72)(22,42,65)(23,43,66)(24,44,67), (9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,65)(56,66) );
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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72)], [(2,8),(3,7),(4,6),(10,16),(11,15),(12,14),(17,23),(18,22),(19,21),(25,29),(26,28),(30,32),(33,35),(36,40),(37,39),(41,47),(42,46),(43,45),(50,56),(51,55),(52,54),(58,64),(59,63),(60,62),(65,69),(66,68),(70,72)], [(1,13,20),(2,14,21),(3,15,22),(4,16,23),(5,9,24),(6,10,17),(7,11,18),(8,12,19),(25,51,46),(26,52,47),(27,53,48),(28,54,41),(29,55,42),(30,56,43),(31,49,44),(32,50,45),(33,70,60),(34,71,61),(35,72,62),(36,65,63),(37,66,64),(38,67,57),(39,68,58),(40,69,59)], [(1,27,61),(2,28,62),(3,29,63),(4,30,64),(5,31,57),(6,32,58),(7,25,59),(8,26,60),(9,49,38),(10,50,39),(11,51,40),(12,52,33),(13,53,34),(14,54,35),(15,55,36),(16,56,37),(17,45,68),(18,46,69),(19,47,70),(20,48,71),(21,41,72),(22,42,65),(23,43,66),(24,44,67)], [(9,24),(10,17),(11,18),(12,19),(13,20),(14,21),(15,22),(16,23),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,57),(32,58),(33,47),(34,48),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(55,65),(56,66)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 6E | ··· | 6L | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 4 | 4 | 9 | 9 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 18 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | 2 | 18 | 18 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D8 | S3×D4 | S3×D8 |
| kernel | D8×C3⋊S3 | C8×C3⋊S3 | C32⋊5D8 | C32⋊7D8 | C32×D8 | D4×C3⋊S3 | C3×D8 | C3⋊Dic3 | C2×C3⋊S3 | C24 | C3×D4 | C3⋊S3 | C6 | C3 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 1 | 1 | 4 | 8 | 4 | 4 | 8 |
Matrix representation of D8×C3⋊S3 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 48 |
| 0 | 0 | 0 | 0 | 38 | 41 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 48 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 70 | 0 | 0 | 0 | 0 |
| 1 | 71 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 3 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,38,0,0,0,0,48,41],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,48,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,70,71,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,3,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
D8×C3⋊S3 in GAP, Magma, Sage, TeX
D_8\times C_3\rtimes S_3
% in TeX
G:=Group("D8xC3:S3"); // GroupNames label
G:=SmallGroup(288,767);
// by ID
G=gap.SmallGroup(288,767);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,135,346,185,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^8=b^2=c^3=d^3=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations