direct product, metabelian, soluble, monomial, A-group
Aliases: C3×C32⋊2C16, C33⋊3C16, C32⋊3C48, (C3×C6).3C24, (C3×C12).7C12, (C32×C6).3C8, (C32×C12).2C4, C32⋊4C8.4C6, C12.10(C32⋊C4), C6.4(C32⋊2C8), C4.2(C3×C32⋊C4), C2.(C3×C32⋊2C8), (C3×C32⋊4C8).1C2, SmallGroup(432,412)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C3×C32⋊2C16 |
Generators and relations for C3×C32⋊2C16
G = < a,b,c,d | a3=b3=c3=d16=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >
Smallest permutation representation of C3×C32⋊2C16
►On 48 points
Generators in S48
(1 43 27)(2 44 28)(3 45 29)(4 46 30)(5 47 31)(6 48 32)(7 33 17)(8 34 18)(9 35 19)(10 36 20)(11 37 21)(12 38 22)(13 39 23)(14 40 24)(15 41 25)(16 42 26)
(2 44 28)(4 30 46)(6 48 32)(8 18 34)(10 36 20)(12 22 38)(14 40 24)(16 26 42)
(1 43 27)(2 44 28)(3 29 45)(4 30 46)(5 47 31)(6 48 32)(7 17 33)(8 18 34)(9 35 19)(10 36 20)(11 21 37)(12 22 38)(13 39 23)(14 40 24)(15 25 41)(16 26 42)
(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)
G:=sub<Sym(48)| (1,43,27)(2,44,28)(3,45,29)(4,46,30)(5,47,31)(6,48,32)(7,33,17)(8,34,18)(9,35,19)(10,36,20)(11,37,21)(12,38,22)(13,39,23)(14,40,24)(15,41,25)(16,42,26), (2,44,28)(4,30,46)(6,48,32)(8,18,34)(10,36,20)(12,22,38)(14,40,24)(16,26,42), (1,43,27)(2,44,28)(3,29,45)(4,30,46)(5,47,31)(6,48,32)(7,17,33)(8,18,34)(9,35,19)(10,36,20)(11,21,37)(12,22,38)(13,39,23)(14,40,24)(15,25,41)(16,26,42), (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)>;
G:=Group( (1,43,27)(2,44,28)(3,45,29)(4,46,30)(5,47,31)(6,48,32)(7,33,17)(8,34,18)(9,35,19)(10,36,20)(11,37,21)(12,38,22)(13,39,23)(14,40,24)(15,41,25)(16,42,26), (2,44,28)(4,30,46)(6,48,32)(8,18,34)(10,36,20)(12,22,38)(14,40,24)(16,26,42), (1,43,27)(2,44,28)(3,29,45)(4,30,46)(5,47,31)(6,48,32)(7,17,33)(8,18,34)(9,35,19)(10,36,20)(11,21,37)(12,22,38)(13,39,23)(14,40,24)(15,25,41)(16,26,42), (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) );
G=PermutationGroup([[(1,43,27),(2,44,28),(3,45,29),(4,46,30),(5,47,31),(6,48,32),(7,33,17),(8,34,18),(9,35,19),(10,36,20),(11,37,21),(12,38,22),(13,39,23),(14,40,24),(15,41,25),(16,42,26)], [(2,44,28),(4,30,46),(6,48,32),(8,18,34),(10,36,20),(12,22,38),(14,40,24),(16,26,42)], [(1,43,27),(2,44,28),(3,29,45),(4,30,46),(5,47,31),(6,48,32),(7,17,33),(8,18,34),(9,35,19),(10,36,20),(11,21,37),(12,22,38),(13,39,23),(14,40,24),(15,25,41),(16,26,42)], [(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)]])
72 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3H | 4A | 4B | 6A | 6B | 6C | ··· | 6H | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12P | 16A | ··· | 16H | 24A | ··· | 24H | 48A | ··· | 48P |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 9 | 9 | 9 | 9 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 9 | ··· | 9 | 9 | ··· | 9 | 9 | ··· | 9 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | ||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C16 | C24 | C48 | C32⋊C4 | C32⋊2C8 | C3×C32⋊C4 | C32⋊2C16 | C3×C32⋊2C8 | C3×C32⋊2C16 |
| kernel | C3×C32⋊2C16 | C3×C32⋊4C8 | C32⋊2C16 | C32×C12 | C32⋊4C8 | C32×C6 | C3×C12 | C33 | C3×C6 | C32 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C3×C32⋊2C16 ►in GL4(𝔽97) generated by
| 61 | 0 | 0 | 0 |
| 0 | 61 | 0 | 0 |
| 0 | 0 | 61 | 0 |
| 0 | 0 | 0 | 61 |
| 1 | 0 | 0 | 12 |
| 0 | 1 | 0 | 18 |
| 0 | 0 | 35 | 73 |
| 0 | 0 | 0 | 61 |
| 61 | 0 | 0 | 0 |
| 0 | 35 | 0 | 66 |
| 0 | 0 | 35 | 73 |
| 0 | 0 | 0 | 61 |
| 27 | 0 | 1 | 0 |
| 89 | 0 | 0 | 0 |
| 47 | 1 | 0 | 0 |
| 38 | 0 | 0 | 70 |
G:=sub<GL(4,GF(97))| [61,0,0,0,0,61,0,0,0,0,61,0,0,0,0,61],[1,0,0,0,0,1,0,0,0,0,35,0,12,18,73,61],[61,0,0,0,0,35,0,0,0,0,35,0,0,66,73,61],[27,89,47,38,0,0,1,0,1,0,0,0,0,0,0,70] >;
C3×C32⋊2C16 in GAP, Magma, Sage, TeX
C_3\times C_3^2\rtimes_2C_{16} % in TeX
G:=Group("C3xC3^2:2C16"); // GroupNames label
G:=SmallGroup(432,412);
// by ID
G=gap.SmallGroup(432,412);
# by ID
G:=PCGroup([7,-2,-3,-2,-2,-2,-3,3,42,58,80,14117,691,18822,2372]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^16=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations
Export