direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C16⋊C4, C48⋊4C4, C16⋊2C12, C42.1C12, C12.35C42, M5(2).2C6, C12.33M4(2), (C4×C12).4C4, (C2×C24).5C4, (C2×C8).2C12, C8⋊C4.4C6, C4.11(C4×C12), C8.19(C2×C12), C24.87(C2×C4), C6.9(C8⋊C4), C4.6(C3×M4(2)), (C3×M5(2)).6C2, (C2×C6).16M4(2), (C2×C24).308C22, C22.4(C3×M4(2)), C2.3(C3×C8⋊C4), (C2×C8).45(C2×C6), (C3×C8⋊C4).9C2, (C2×C4).66(C2×C12), (C2×C12).327(C2×C4), SmallGroup(192,153)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C16⋊C4
G = < a,b,c | a3=b16=c4=1, ab=ba, ac=ca, cbc-1=b13 >
Smallest permutation representation of C3×C16⋊C4
►On 48 points
(1 40 28)(2 41 29)(3 42 30)(4 43 31)(5 44 32)(6 45 17)(7 46 18)(8 47 19)(9 48 20)(10 33 21)(11 34 22)(12 35 23)(13 36 24)(14 37 25)(15 38 26)(16 39 27)
(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 6 10 14)(3 11)(4 16 12 8)(7 15)(17 21 25 29)(18 26)(19 31 27 23)(22 30)(33 37 41 45)(34 42)(35 47 43 39)(38 46)
G:=sub<Sym(48)| (1,40,28)(2,41,29)(3,42,30)(4,43,31)(5,44,32)(6,45,17)(7,46,18)(8,47,19)(9,48,20)(10,33,21)(11,34,22)(12,35,23)(13,36,24)(14,37,25)(15,38,26)(16,39,27), (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), (2,6,10,14)(3,11)(4,16,12,8)(7,15)(17,21,25,29)(18,26)(19,31,27,23)(22,30)(33,37,41,45)(34,42)(35,47,43,39)(38,46)>;
G:=Group( (1,40,28)(2,41,29)(3,42,30)(4,43,31)(5,44,32)(6,45,17)(7,46,18)(8,47,19)(9,48,20)(10,33,21)(11,34,22)(12,35,23)(13,36,24)(14,37,25)(15,38,26)(16,39,27), (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), (2,6,10,14)(3,11)(4,16,12,8)(7,15)(17,21,25,29)(18,26)(19,31,27,23)(22,30)(33,37,41,45)(34,42)(35,47,43,39)(38,46) );
G=PermutationGroup([[(1,40,28),(2,41,29),(3,42,30),(4,43,31),(5,44,32),(6,45,17),(7,46,18),(8,47,19),(9,48,20),(10,33,21),(11,34,22),(12,35,23),(13,36,24),(14,37,25),(15,38,26),(16,39,27)], [(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)], [(2,6,10,14),(3,11),(4,16,12,8),(7,15),(17,21,25,29),(18,26),(19,31,27,23),(22,30),(33,37,41,45),(34,42),(35,47,43,39),(38,46)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 16A | ··· | 16H | 24A | ··· | 24H | 24I | 24J | 24K | 24L | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C12 | C12 | C12 | M4(2) | M4(2) | C3×M4(2) | C3×M4(2) | C16⋊C4 | C3×C16⋊C4 |
| kernel | C3×C16⋊C4 | C3×C8⋊C4 | C3×M5(2) | C16⋊C4 | C48 | C4×C12 | C2×C24 | C8⋊C4 | M5(2) | C16 | C42 | C2×C8 | C12 | C2×C6 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 16 | 4 | 4 | 2 | 2 | 4 | 4 | 2 | 4 |
Matrix representation of C3×C16⋊C4 ►in GL4(𝔽97) generated by
| 61 | 0 | 0 | 0 |
| 0 | 61 | 0 | 0 |
| 0 | 0 | 61 | 0 |
| 0 | 0 | 0 | 61 |
| 96 | 0 | 1 | 59 |
| 0 | 0 | 0 | 49 |
| 75 | 22 | 0 | 86 |
| 2 | 0 | 0 | 1 |
| 1 | 0 | 0 | 38 |
| 0 | 96 | 0 | 0 |
| 0 | 0 | 75 | 1 |
| 0 | 0 | 0 | 22 |
G:=sub<GL(4,GF(97))| [61,0,0,0,0,61,0,0,0,0,61,0,0,0,0,61],[96,0,75,2,0,0,22,0,1,0,0,0,59,49,86,1],[1,0,0,0,0,96,0,0,0,0,75,0,38,0,1,22] >;
C3×C16⋊C4 in GAP, Magma, Sage, TeX
C_3\times C_{16}\rtimes C_4 % in TeX
G:=Group("C3xC16:C4"); // GroupNames label
G:=SmallGroup(192,153);
// by ID
G=gap.SmallGroup(192,153);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,84,701,176,1522,136,4204,124]);
// Polycyclic
G:=Group<a,b,c|a^3=b^16=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^13>;
// generators/relations
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