metabelian, soluble, monomial, A-group
Aliases: C33⋊4C16, C32⋊4(C3⋊C16), C3⋊(C32⋊2C16), C6.(C32⋊2C8), C2.(C33⋊4C8), (C32×C6).4C8, C12.2(C32⋊C4), (C32×C12).3C4, C32⋊4C8.4S3, C4.2(C33⋊C4), (C3×C12).12Dic3, (C3×C6).6(C3⋊C8), (C3×C32⋊4C8).6C2, SmallGroup(432,413)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — C33⋊4C16 |
Generators and relations for C33⋊4C16
G = < a,b,c,d | a3=b3=c3=d16=1, ab=ba, ac=ca, dad-1=ab-1, bc=cb, dbd-1=a-1b-1, dcd-1=c-1 >
Smallest permutation representation of C33⋊4C16
►On 48 points
Generators in S48
(1 18 46)(2 19 47)(3 48 20)(4 33 21)(5 22 34)(6 23 35)(7 36 24)(8 37 25)(9 26 38)(10 27 39)(11 40 28)(12 41 29)(13 30 42)(14 31 43)(15 44 32)(16 45 17)
(2 47 19)(4 21 33)(6 35 23)(8 25 37)(10 39 27)(12 29 41)(14 43 31)(16 17 45)
(1 46 18)(2 19 47)(3 48 20)(4 21 33)(5 34 22)(6 23 35)(7 36 24)(8 25 37)(9 38 26)(10 27 39)(11 40 28)(12 29 41)(13 42 30)(14 31 43)(15 44 32)(16 17 45)
(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)| (1,18,46)(2,19,47)(3,48,20)(4,33,21)(5,22,34)(6,23,35)(7,36,24)(8,37,25)(9,26,38)(10,27,39)(11,40,28)(12,41,29)(13,30,42)(14,31,43)(15,44,32)(16,45,17), (2,47,19)(4,21,33)(6,35,23)(8,25,37)(10,39,27)(12,29,41)(14,43,31)(16,17,45), (1,46,18)(2,19,47)(3,48,20)(4,21,33)(5,34,22)(6,23,35)(7,36,24)(8,25,37)(9,38,26)(10,27,39)(11,40,28)(12,29,41)(13,42,30)(14,31,43)(15,44,32)(16,17,45), (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( (1,18,46)(2,19,47)(3,48,20)(4,33,21)(5,22,34)(6,23,35)(7,36,24)(8,37,25)(9,26,38)(10,27,39)(11,40,28)(12,41,29)(13,30,42)(14,31,43)(15,44,32)(16,45,17), (2,47,19)(4,21,33)(6,35,23)(8,25,37)(10,39,27)(12,29,41)(14,43,31)(16,17,45), (1,46,18)(2,19,47)(3,48,20)(4,21,33)(5,34,22)(6,23,35)(7,36,24)(8,25,37)(9,38,26)(10,27,39)(11,40,28)(12,29,41)(13,42,30)(14,31,43)(15,44,32)(16,17,45), (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([[(1,18,46),(2,19,47),(3,48,20),(4,33,21),(5,22,34),(6,23,35),(7,36,24),(8,37,25),(9,26,38),(10,27,39),(11,40,28),(12,41,29),(13,30,42),(14,31,43),(15,44,32),(16,45,17)], [(2,47,19),(4,21,33),(6,35,23),(8,25,37),(10,39,27),(12,29,41),(14,43,31),(16,17,45)], [(1,46,18),(2,19,47),(3,48,20),(4,21,33),(5,34,22),(6,23,35),(7,36,24),(8,25,37),(9,38,26),(10,27,39),(11,40,28),(12,29,41),(13,42,30),(14,31,43),(15,44,32),(16,17,45)], [(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)]])
48 conjugacy classes
| class | 1 | 2 | 3A | 3B | ··· | 3G | 4A | 4B | 6A | 6B | ··· | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 12C | ··· | 12N | 16A | ··· | 16H | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 4 | ··· | 4 | 1 | 1 | 2 | 4 | ··· | 4 | 9 | 9 | 9 | 9 | 2 | 2 | 4 | ··· | 4 | 27 | ··· | 27 | 18 | 18 | 18 | 18 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | - | |||||||||
| image | C1 | C2 | C4 | C8 | C16 | S3 | Dic3 | C3⋊C8 | C3⋊C16 | C32⋊C4 | C32⋊2C8 | C33⋊C4 | C32⋊2C16 | C33⋊4C8 | C33⋊4C16 |
| kernel | C33⋊4C16 | C3×C32⋊4C8 | C32×C12 | C32×C6 | C33 | C32⋊4C8 | C3×C12 | C3×C6 | C32 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C33⋊4C16 ►in GL4(𝔽97) generated by
| 35 | 0 | 0 | 0 |
| 0 | 61 | 0 | 0 |
| 0 | 0 | 61 | 0 |
| 0 | 0 | 0 | 35 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 35 | 0 |
| 0 | 0 | 0 | 61 |
| 61 | 0 | 0 | 0 |
| 0 | 61 | 0 | 0 |
| 0 | 0 | 35 | 0 |
| 0 | 0 | 0 | 35 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 22 | 0 | 0 | 0 |
G:=sub<GL(4,GF(97))| [35,0,0,0,0,61,0,0,0,0,61,0,0,0,0,35],[1,0,0,0,0,1,0,0,0,0,35,0,0,0,0,61],[61,0,0,0,0,61,0,0,0,0,35,0,0,0,0,35],[0,0,0,22,0,0,1,0,1,0,0,0,0,1,0,0] >;
C33⋊4C16 in GAP, Magma, Sage, TeX
C_3^3\rtimes_4C_{16} % in TeX
G:=Group("C3^3:4C16"); // GroupNames label
G:=SmallGroup(432,413);
// by ID
G=gap.SmallGroup(432,413);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,14,36,58,2804,571,2693,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^16=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^-1,b*c=c*b,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^-1>;
// generators/relations
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