direct product, metabelian, supersoluble, monomial
Aliases: C3×S3×D8, D24⋊4C6, C24⋊18D6, C8⋊4(S3×C6), C3⋊2(C6×D8), (S3×C8)⋊1C6, C24⋊2(C2×C6), D4⋊S3⋊1C6, (C3×D8)⋊2C6, (S3×D4)⋊1C6, D4⋊1(S3×C6), (S3×C24)⋊5C2, (C3×D4)⋊12D6, D12⋊1(C2×C6), C6.27(C6×D4), C32⋊11(C2×D8), (C3×D24)⋊12C2, (C3×C24)⋊8C22, D6.12(C3×D4), (S3×C6).48D4, C6.187(S3×D4), (C32×D8)⋊3C2, C12.1(C22×C6), Dic3.3(C3×D4), (C3×D12)⋊10C22, (C3×C12).72C23, (C3×Dic3).30D4, (D4×C32)⋊5C22, (S3×C12).47C22, C12.152(C22×S3), C3⋊C8⋊5(C2×C6), (C3×S3×D4)⋊4C2, C4.1(S3×C2×C6), C2.15(C3×S3×D4), (C3×D4⋊S3)⋊9C2, (C3×D4)⋊1(C2×C6), (C3×C3⋊C8)⋊31C22, (C4×S3).7(C2×C6), (C3×C6).215(C2×D4), SmallGroup(288,681)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×S3×D8
G = < a,b,c,d,e | a3=b3=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 522 in 163 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, S3, C6, C6, C8, C8, C2×C4, D4, D4, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, D8, D8, C2×D4, C3×S3, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, C24, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, C2×D8, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, S3×C8, D24, D4⋊S3, C2×C24, C3×D8, C3×D8, S3×D4, C6×D4, C3×C3⋊C8, C3×C24, S3×C12, C3×D12, C3×C3⋊D4, D4×C32, S3×C2×C6, S3×D8, C6×D8, S3×C24, C3×D24, C3×D4⋊S3, C32×D8, C3×S3×D4, C3×S3×D8
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, D8, C2×D4, C3×S3, C3×D4, C22×S3, C22×C6, C2×D8, S3×C6, C3×D8, S3×D4, C6×D4, S3×C2×C6, S3×D8, C6×D8, C3×S3×D4, C3×S3×D8
►On 48 points
(1 36 27)(2 37 28)(3 38 29)(4 39 30)(5 40 31)(6 33 32)(7 34 25)(8 35 26)(9 21 45)(10 22 46)(11 23 47)(12 24 48)(13 17 41)(14 18 42)(15 19 43)(16 20 44)
(1 36 27)(2 37 28)(3 38 29)(4 39 30)(5 40 31)(6 33 32)(7 34 25)(8 35 26)(9 45 21)(10 46 22)(11 47 23)(12 48 24)(13 41 17)(14 42 18)(15 43 19)(16 44 20)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(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 8)(3 7)(4 6)(9 11)(12 16)(13 15)(17 19)(20 24)(21 23)(25 29)(26 28)(30 32)(33 39)(34 38)(35 37)(41 43)(44 48)(45 47)
G:=sub<Sym(48)| (1,36,27)(2,37,28)(3,38,29)(4,39,30)(5,40,31)(6,33,32)(7,34,25)(8,35,26)(9,21,45)(10,22,46)(11,23,47)(12,24,48)(13,17,41)(14,18,42)(15,19,43)(16,20,44), (1,36,27)(2,37,28)(3,38,29)(4,39,30)(5,40,31)(6,33,32)(7,34,25)(8,35,26)(9,45,21)(10,46,22)(11,47,23)(12,48,24)(13,41,17)(14,42,18)(15,43,19)(16,44,20), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (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), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,19)(20,24)(21,23)(25,29)(26,28)(30,32)(33,39)(34,38)(35,37)(41,43)(44,48)(45,47)>;
G:=Group( (1,36,27)(2,37,28)(3,38,29)(4,39,30)(5,40,31)(6,33,32)(7,34,25)(8,35,26)(9,21,45)(10,22,46)(11,23,47)(12,24,48)(13,17,41)(14,18,42)(15,19,43)(16,20,44), (1,36,27)(2,37,28)(3,38,29)(4,39,30)(5,40,31)(6,33,32)(7,34,25)(8,35,26)(9,45,21)(10,46,22)(11,47,23)(12,48,24)(13,41,17)(14,42,18)(15,43,19)(16,44,20), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (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), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,19)(20,24)(21,23)(25,29)(26,28)(30,32)(33,39)(34,38)(35,37)(41,43)(44,48)(45,47) );
G=PermutationGroup([[(1,36,27),(2,37,28),(3,38,29),(4,39,30),(5,40,31),(6,33,32),(7,34,25),(8,35,26),(9,21,45),(10,22,46),(11,23,47),(12,24,48),(13,17,41),(14,18,42),(15,19,43),(16,20,44)], [(1,36,27),(2,37,28),(3,38,29),(4,39,30),(5,40,31),(6,33,32),(7,34,25),(8,35,26),(9,45,21),(10,46,22),(11,47,23),(12,48,24),(13,41,17),(14,42,18),(15,43,19),(16,44,20)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,17),(9,30),(10,31),(11,32),(12,25),(13,26),(14,27),(15,28),(16,29),(33,47),(34,48),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46)], [(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)], [(2,8),(3,7),(4,6),(9,11),(12,16),(13,15),(17,19),(20,24),(21,23),(25,29),(26,28),(30,32),(33,39),(34,38),(35,37),(41,43),(44,48),(45,47)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | ··· | 6S | 6T | 6U | 6V | 6W | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 24A | 24B | 24C | 24D | 24E | ··· | 24J | 24K | 24L | 24M | 24N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 3 | 3 | 4 | 4 | 12 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 2 | 2 | 6 | 6 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 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 | D8 | C3×S3 | C3×D4 | C3×D4 | S3×C6 | S3×C6 | C3×D8 | S3×D4 | S3×D8 | C3×S3×D4 | C3×S3×D8 |
| kernel | C3×S3×D8 | S3×C24 | C3×D24 | C3×D4⋊S3 | C32×D8 | C3×S3×D4 | S3×D8 | S3×C8 | D24 | D4⋊S3 | C3×D8 | S3×D4 | C3×D8 | C3×Dic3 | S3×C6 | C24 | C3×D4 | C3×S3 | D8 | Dic3 | D6 | C8 | D4 | S3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 |
Matrix representation of C3×S3×D8 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 2 | 3 | 3 | 1 |
| 6 | 1 | 1 | 4 |
| 2 | 2 | 0 | 6 |
| 4 | 3 | 2 | 2 |
| 4 | 2 | 5 | 3 |
| 6 | 2 | 1 | 4 |
| 0 | 6 | 5 | 4 |
| 5 | 0 | 3 | 3 |
| 3 | 3 | 0 | 2 |
| 1 | 3 | 1 | 1 |
| 2 | 2 | 6 | 3 |
| 3 | 4 | 2 | 3 |
| 2 | 0 | 6 | 1 |
| 6 | 4 | 6 | 6 |
| 2 | 4 | 1 | 2 |
| 6 | 4 | 3 | 0 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,6,2,4,3,1,2,3,3,1,0,2,1,4,6,2],[4,6,0,5,2,2,6,0,5,1,5,3,3,4,4,3],[3,1,2,3,3,3,2,4,0,1,6,2,2,1,3,3],[2,6,2,6,0,4,4,4,6,6,1,3,1,6,2,0] >;
C3×S3×D8 in GAP, Magma, Sage, TeX
C_3\times S_3\times D_8
% in TeX
G:=Group("C3xS3xD8"); // GroupNames label
G:=SmallGroup(288,681);
// by ID
G=gap.SmallGroup(288,681);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,303,1271,648,102,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^8=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,c*e=e*c,e*d*e=d^-1>;
// generators/relations