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## G = A4×C16order 192 = 26·3

### Direct product of C16 and A4

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

 Derived series C1 — C22 — A4×C16
 Chief series C1 — C22 — C23 — C22×C4 — C22×C8 — C8×A4 — A4×C16
 Lower central C22 — A4×C16
 Upper central C1 — 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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