direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C16, C22⋊C48, C23.2C24, (C22×C16)⋊C3, C8.6(C2×A4), C4.4(C4×A4), C2.1(C8×A4), (C2×A4).2C8, (C8×A4).4C2, (C4×A4).5C4, (C22×C8).2C6, (C22×C4).8C12, SmallGroup(192,203)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C16 |
Generators and relations for A4×C16
G = < a,b,c,d | a16=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of A4×C16
►On 48 points
Generators in S48
(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)
(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(1 30 41)(2 31 42)(3 32 43)(4 17 44)(5 18 45)(6 19 46)(7 20 47)(8 21 48)(9 22 33)(10 23 34)(11 24 35)(12 25 36)(13 26 37)(14 27 38)(15 28 39)(16 29 40)
G:=sub<Sym(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), (17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (1,30,41)(2,31,42)(3,32,43)(4,17,44)(5,18,45)(6,19,46)(7,20,47)(8,21,48)(9,22,33)(10,23,34)(11,24,35)(12,25,36)(13,26,37)(14,27,38)(15,28,39)(16,29,40)>;
G:=Group( (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), (17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (1,30,41)(2,31,42)(3,32,43)(4,17,44)(5,18,45)(6,19,46)(7,20,47)(8,21,48)(9,22,33)(10,23,34)(11,24,35)(12,25,36)(13,26,37)(14,27,38)(15,28,39)(16,29,40) );
G=PermutationGroup([[(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)], [(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(1,30,41),(2,31,42),(3,32,43),(4,17,44),(5,18,45),(6,19,46),(7,20,47),(8,21,48),(9,22,33),(10,23,34),(11,24,35),(12,25,36),(13,26,37),(14,27,38),(15,28,39),(16,29,40)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 16A | ··· | 16H | 16I | ··· | 16P | 24A | ··· | 24H | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 16 | ··· | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 3 | 3 | 4 | 4 | 1 | 1 | 3 | 3 | 4 | 4 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | |||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C16 | C24 | C48 | A4 | C2×A4 | C4×A4 | C8×A4 | A4×C16 |
| kernel | A4×C16 | C8×A4 | C22×C16 | C4×A4 | C22×C8 | C2×A4 | C22×C4 | A4 | C23 | C22 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 1 | 1 | 2 | 4 | 8 |
Matrix representation of A4×C16 ►in GL3(𝔽97) generated by
| 85 | 0 | 0 |
| 0 | 85 | 0 |
| 0 | 0 | 85 |
| 1 | 35 | 61 |
| 0 | 96 | 0 |
| 0 | 0 | 96 |
| 96 | 0 | 36 |
| 0 | 96 | 0 |
| 0 | 0 | 1 |
| 61 | 0 | 0 |
| 0 | 0 | 1 |
| 95 | 62 | 36 |
G:=sub<GL(3,GF(97))| [85,0,0,0,85,0,0,0,85],[1,0,0,35,96,0,61,0,96],[96,0,0,0,96,0,36,0,1],[61,0,95,0,0,62,0,1,36] >;
A4×C16 in GAP, Magma, Sage, TeX
A_4\times C_{16} % in TeX
G:=Group("A4xC16"); // GroupNames label
G:=SmallGroup(192,203);
// by ID
G=gap.SmallGroup(192,203);
# by ID
G:=PCGroup([7,-2,-3,-2,-2,-2,-2,2,42,58,80,2028,3541]);
// Polycyclic
G:=Group<a,b,c,d|a^16=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
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