direct product, metabelian, supersoluble, monomial
Aliases: C3×D8⋊3S3, C24.56D6, Dic12⋊4C6, (S3×C8)⋊2C6, (C3×D8)⋊3C6, D8⋊3(C3×S3), (C3×D8)⋊7S3, C8.8(S3×C6), (S3×C24)⋊6C2, C24.6(C2×C6), D4.S3⋊2C6, D6.1(C3×D4), D4.1(S3×C6), C6.29(C6×D4), D4⋊2S3⋊2C6, (C3×D4).24D6, (S3×C6).25D4, C6.189(S3×D4), (C32×D8)⋊4C2, C32⋊18(C4○D8), C12.3(C22×C6), Dic6.1(C2×C6), (C3×Dic12)⋊12C2, (C3×C24).26C22, (C3×C12).74C23, Dic3.12(C3×D4), (C3×Dic3).49D4, (S3×C12).48C22, C12.154(C22×S3), (C3×Dic6).24C22, (D4×C32).11C22, C4.3(S3×C2×C6), C3⋊2(C3×C4○D8), C3⋊C8.5(C2×C6), C2.17(C3×S3×D4), (C4×S3).8(C2×C6), (C3×D4).1(C2×C6), (C3×D4⋊2S3)⋊5C2, (C3×D4.S3)⋊13C2, (C3×C6).217(C2×D4), (C3×C3⋊C8).39C22, SmallGroup(288,683)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D8⋊3S3
G = < a,b,c,d,e | a3=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b4c, ede=d-1 >
Subgroups: 330 in 135 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C32, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, C24, Dic6, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C4○D8, C3×Dic3, C3×Dic3, C3×C12, S3×C6, C62, S3×C8, Dic12, D4.S3, C2×C24, C3×D8, C3×D8, C3×SD16, C3×Q16, D4⋊2S3, C3×C4○D4, C3×C3⋊C8, C3×C24, C3×Dic6, S3×C12, C6×Dic3, C3×C3⋊D4, D4×C32, D8⋊3S3, C3×C4○D8, S3×C24, C3×Dic12, C3×D4.S3, C32×D8, C3×D4⋊2S3, C3×D8⋊3S3
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3×D4, C22×S3, C22×C6, C4○D8, S3×C6, S3×D4, C6×D4, S3×C2×C6, D8⋊3S3, C3×C4○D8, C3×S3×D4, C3×D8⋊3S3
►On 48 points
(1 25 40)(2 26 33)(3 27 34)(4 28 35)(5 29 36)(6 30 37)(7 31 38)(8 32 39)(9 42 17)(10 43 18)(11 44 19)(12 45 20)(13 46 21)(14 47 22)(15 48 23)(16 41 24)
(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 8)(2 7)(3 6)(4 5)(9 12)(10 11)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)(25 32)(26 31)(27 30)(28 29)(33 38)(34 37)(35 36)(39 40)(41 46)(42 45)(43 44)(47 48)
(1 40 25)(2 33 26)(3 34 27)(4 35 28)(5 36 29)(6 37 30)(7 38 31)(8 39 32)(9 42 17)(10 43 18)(11 44 19)(12 45 20)(13 46 21)(14 47 22)(15 48 23)(16 41 24)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 36)(18 37)(19 38)(20 39)(21 40)(22 33)(23 34)(24 35)(25 46)(26 47)(27 48)(28 41)(29 42)(30 43)(31 44)(32 45)
G:=sub<Sym(48)| (1,25,40)(2,26,33)(3,27,34)(4,28,35)(5,29,36)(6,30,37)(7,31,38)(8,32,39)(9,42,17)(10,43,18)(11,44,19)(12,45,20)(13,46,21)(14,47,22)(15,48,23)(16,41,24), (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), (1,8)(2,7)(3,6)(4,5)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,32)(26,31)(27,30)(28,29)(33,38)(34,37)(35,36)(39,40)(41,46)(42,45)(43,44)(47,48), (1,40,25)(2,33,26)(3,34,27)(4,35,28)(5,36,29)(6,37,30)(7,38,31)(8,39,32)(9,42,17)(10,43,18)(11,44,19)(12,45,20)(13,46,21)(14,47,22)(15,48,23)(16,41,24), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,36)(18,37)(19,38)(20,39)(21,40)(22,33)(23,34)(24,35)(25,46)(26,47)(27,48)(28,41)(29,42)(30,43)(31,44)(32,45)>;
G:=Group( (1,25,40)(2,26,33)(3,27,34)(4,28,35)(5,29,36)(6,30,37)(7,31,38)(8,32,39)(9,42,17)(10,43,18)(11,44,19)(12,45,20)(13,46,21)(14,47,22)(15,48,23)(16,41,24), (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), (1,8)(2,7)(3,6)(4,5)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,32)(26,31)(27,30)(28,29)(33,38)(34,37)(35,36)(39,40)(41,46)(42,45)(43,44)(47,48), (1,40,25)(2,33,26)(3,34,27)(4,35,28)(5,36,29)(6,37,30)(7,38,31)(8,39,32)(9,42,17)(10,43,18)(11,44,19)(12,45,20)(13,46,21)(14,47,22)(15,48,23)(16,41,24), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,36)(18,37)(19,38)(20,39)(21,40)(22,33)(23,34)(24,35)(25,46)(26,47)(27,48)(28,41)(29,42)(30,43)(31,44)(32,45) );
G=PermutationGroup([[(1,25,40),(2,26,33),(3,27,34),(4,28,35),(5,29,36),(6,30,37),(7,31,38),(8,32,39),(9,42,17),(10,43,18),(11,44,19),(12,45,20),(13,46,21),(14,47,22),(15,48,23),(16,41,24)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11),(13,16),(14,15),(17,20),(18,19),(21,24),(22,23),(25,32),(26,31),(27,30),(28,29),(33,38),(34,37),(35,36),(39,40),(41,46),(42,45),(43,44),(47,48)], [(1,40,25),(2,33,26),(3,34,27),(4,35,28),(5,36,29),(6,37,30),(7,38,31),(8,39,32),(9,42,17),(10,43,18),(11,44,19),(12,45,20),(13,46,21),(14,47,22),(15,48,23),(16,41,24)], [(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12),(17,36),(18,37),(19,38),(20,39),(21,40),(22,33),(23,34),(24,35),(25,46),(26,47),(27,48),(28,41),(29,42),(30,43),(31,44),(32,45)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | ··· | 6Q | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | 12M | 24A | 24B | 24C | 24D | 24E | ··· | 24J | 24K | 24L | 24M | 24N |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 4 | 4 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 12 | 12 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | ··· | 8 | 2 | 2 | 6 | 6 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | D4 | D6 | D6 | C3×S3 | C3×D4 | C3×D4 | C4○D8 | S3×C6 | S3×C6 | C3×C4○D8 | S3×D4 | D8⋊3S3 | C3×S3×D4 | C3×D8⋊3S3 |
| kernel | C3×D8⋊3S3 | S3×C24 | C3×Dic12 | C3×D4.S3 | C32×D8 | C3×D4⋊2S3 | D8⋊3S3 | S3×C8 | Dic12 | D4.S3 | C3×D8 | D4⋊2S3 | C3×D8 | C3×Dic3 | S3×C6 | C24 | C3×D4 | D8 | Dic3 | D6 | C32 | C8 | D4 | C3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 8 | 1 | 2 | 2 | 4 |
Matrix representation of C3×D8⋊3S3 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 4 | 5 | 3 | 6 |
| 4 | 2 | 4 | 4 |
| 3 | 3 | 0 | 1 |
| 2 | 5 | 6 | 2 |
| 4 | 2 | 1 | 4 |
| 6 | 0 | 3 | 2 |
| 0 | 0 | 1 | 0 |
| 2 | 5 | 6 | 2 |
| 2 | 5 | 1 | 1 |
| 3 | 3 | 3 | 6 |
| 4 | 3 | 4 | 6 |
| 2 | 2 | 6 | 3 |
| 0 | 3 | 0 | 1 |
| 1 | 5 | 0 | 6 |
| 0 | 5 | 6 | 6 |
| 5 | 6 | 0 | 3 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,4,3,2,5,2,3,5,3,4,0,6,6,4,1,2],[4,6,0,2,2,0,0,5,1,3,1,6,4,2,0,2],[2,3,4,2,5,3,3,2,1,3,4,6,1,6,6,3],[0,1,0,5,3,5,5,6,0,0,6,0,1,6,6,3] >;
C3×D8⋊3S3 in GAP, Magma, Sage, TeX
C_3\times D_8\rtimes_3S_3
% in TeX
G:=Group("C3xD8:3S3"); // GroupNames label
G:=SmallGroup(288,683);
// by ID
G=gap.SmallGroup(288,683);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,1094,303,1271,648,102,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^8=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^4*c,e*d*e=d^-1>;
// generators/relations