metabelian, supersoluble, monomial
Aliases: C3⋊2D48, D24⋊1S3, C32⋊3D16, C12.9D12, C6.12D24, C24.48D6, C8.4S32, C3⋊C16⋊1S3, (C3×C6).7D8, (C3×D24)⋊6C2, C32⋊5D8⋊4C2, C3⋊1(C3⋊D16), C6.1(D4⋊S3), (C3×C12).22D4, (C3×C24).7C22, C4.1(C3⋊D12), C2.4(C3⋊D24), C12.66(C3⋊D4), (C3×C3⋊C16)⋊1C2, SmallGroup(288,194)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊D48
G = < a,b,c | a3=b48=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 498 in 67 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C32, C12, C12, D6, C2×C6, C16, D8, C3×S3, C3⋊S3, C3×C6, C24, C24, D12, C3×D4, D16, C3×C12, S3×C6, C2×C3⋊S3, C3⋊C16, C48, D24, D24, C3×D8, C3×C24, C3×D12, C12⋊S3, D48, C3⋊D16, C3×C3⋊C16, C3×D24, C32⋊5D8, C3⋊D48
Quotients: C1, C2, C22, S3, D4, D6, D8, D12, C3⋊D4, D16, S32, D24, D4⋊S3, C3⋊D12, D48, C3⋊D16, C3⋊D24, C3⋊D48
►On 48 points
(1 33 17)(2 18 34)(3 35 19)(4 20 36)(5 37 21)(6 22 38)(7 39 23)(8 24 40)(9 41 25)(10 26 42)(11 43 27)(12 28 44)(13 45 29)(14 30 46)(15 47 31)(16 32 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)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)
G:=sub<Sym(48)| (1,33,17)(2,18,34)(3,35,19)(4,20,36)(5,37,21)(6,22,38)(7,39,23)(8,24,40)(9,41,25)(10,26,42)(11,43,27)(12,28,44)(13,45,29)(14,30,46)(15,47,31)(16,32,48), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)>;
G:=Group( (1,33,17)(2,18,34)(3,35,19)(4,20,36)(5,37,21)(6,22,38)(7,39,23)(8,24,40)(9,41,25)(10,26,42)(11,43,27)(12,28,44)(13,45,29)(14,30,46)(15,47,31)(16,32,48), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33) );
G=PermutationGroup([[(1,33,17),(2,18,34),(3,35,19),(4,20,36),(5,37,21),(6,22,38),(7,39,23),(8,24,40),(9,41,25),(10,26,42),(11,43,27),(12,28,44),(13,45,29),(14,30,46),(15,47,31),(16,32,48)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,40),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(31,33)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4 | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 24E | ··· | 24J | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 24 | 72 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 24 | 24 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D8 | D12 | C3⋊D4 | D16 | D24 | D48 | S32 | D4⋊S3 | C3⋊D12 | C3⋊D16 | C3⋊D24 | C3⋊D48 |
| kernel | C3⋊D48 | C3×C3⋊C16 | C3×D24 | C32⋊5D8 | C3⋊C16 | D24 | C3×C12 | C24 | C3×C6 | C12 | C12 | C32 | C6 | C3 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of C3⋊D48 ►in GL4(𝔽97) generated by
| 0 | 96 | 0 | 0 |
| 1 | 96 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 96 | 0 | 0 |
| 96 | 0 | 0 | 0 |
| 0 | 0 | 32 | 84 |
| 0 | 0 | 13 | 19 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 95 | 18 |
| 0 | 0 | 16 | 2 |
G:=sub<GL(4,GF(97))| [0,1,0,0,96,96,0,0,0,0,1,0,0,0,0,1],[0,96,0,0,96,0,0,0,0,0,32,13,0,0,84,19],[0,1,0,0,1,0,0,0,0,0,95,16,0,0,18,2] >;
C3⋊D48 in GAP, Magma, Sage, TeX
C_3\rtimes D_{48} % in TeX
G:=Group("C3:D48"); // GroupNames label
G:=SmallGroup(288,194);
// by ID
G=gap.SmallGroup(288,194);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,92,254,142,675,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^3=b^48=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations