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## G = A4×C42order 192 = 26·3

### Direct product of C42 and A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C42
 Chief series C1 — C22 — C23 — C24 — C22×A4 — C2×C4×A4 — A4×C42
 Lower central C22 — A4×C42
 Upper central C1 — C42

Generators and relations for A4×C42
G = < a,b,c,d,e | a4=b4=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Subgroups: 324 in 133 conjugacy classes, 45 normal (9 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C2×C4, C2×C4, C23, C23, C12, A4, C2×C6, C42, C42, C22×C4, C22×C4, C24, C2×C12, C2×A4, C2×C42, C23×C4, C4×C12, C4×A4, C22×A4, C22×C42, C2×C4×A4, A4×C42
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, A4, C2×C6, C42, C2×C12, C2×A4, C4×C12, C4×A4, C22×A4, C2×C4×A4, A4×C42

Smallest permutation representation of A4×C42
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)
(1 12 31 21)(2 9 32 22)(3 10 29 23)(4 11 30 24)(5 48 18 40)(6 45 19 37)(7 46 20 38)(8 47 17 39)(13 36 28 44)(14 33 25 41)(15 34 26 42)(16 35 27 43)
(5 7)(6 8)(13 15)(14 16)(17 19)(18 20)(25 27)(26 28)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)(29 31)(30 32)(37 39)(38 40)(45 47)(46 48)
(1 43 17)(2 44 18)(3 41 19)(4 42 20)(5 32 36)(6 29 33)(7 30 34)(8 31 35)(9 13 40)(10 14 37)(11 15 38)(12 16 39)(21 27 47)(22 28 48)(23 25 45)(24 26 46)

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), (1,12,31,21)(2,9,32,22)(3,10,29,23)(4,11,30,24)(5,48,18,40)(6,45,19,37)(7,46,20,38)(8,47,17,39)(13,36,28,44)(14,33,25,41)(15,34,26,42)(16,35,27,43), (5,7)(6,8)(13,15)(14,16)(17,19)(18,20)(25,27)(26,28)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(17,19)(18,20)(21,23)(22,24)(29,31)(30,32)(37,39)(38,40)(45,47)(46,48), (1,43,17)(2,44,18)(3,41,19)(4,42,20)(5,32,36)(6,29,33)(7,30,34)(8,31,35)(9,13,40)(10,14,37)(11,15,38)(12,16,39)(21,27,47)(22,28,48)(23,25,45)(24,26,46)>;

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), (1,12,31,21)(2,9,32,22)(3,10,29,23)(4,11,30,24)(5,48,18,40)(6,45,19,37)(7,46,20,38)(8,47,17,39)(13,36,28,44)(14,33,25,41)(15,34,26,42)(16,35,27,43), (5,7)(6,8)(13,15)(14,16)(17,19)(18,20)(25,27)(26,28)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(17,19)(18,20)(21,23)(22,24)(29,31)(30,32)(37,39)(38,40)(45,47)(46,48), (1,43,17)(2,44,18)(3,41,19)(4,42,20)(5,32,36)(6,29,33)(7,30,34)(8,31,35)(9,13,40)(10,14,37)(11,15,38)(12,16,39)(21,27,47)(22,28,48)(23,25,45)(24,26,46) );

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)], [(1,12,31,21),(2,9,32,22),(3,10,29,23),(4,11,30,24),(5,48,18,40),(6,45,19,37),(7,46,20,38),(8,47,17,39),(13,36,28,44),(14,33,25,41),(15,34,26,42),(16,35,27,43)], [(5,7),(6,8),(13,15),(14,16),(17,19),(18,20),(25,27),(26,28),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(17,19),(18,20),(21,23),(22,24),(29,31),(30,32),(37,39),(38,40),(45,47),(46,48)], [(1,43,17),(2,44,18),(3,41,19),(4,42,20),(5,32,36),(6,29,33),(7,30,34),(8,31,35),(9,13,40),(10,14,37),(11,15,38),(12,16,39),(21,27,47),(22,28,48),(23,25,45),(24,26,46)]])

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A ··· 4L 4M ··· 4X 6A ··· 6F 12A ··· 12X order 1 2 2 2 2 2 2 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 1 1 3 3 3 3 4 4 1 ··· 1 3 ··· 3 4 ··· 4 4 ··· 4

64 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 type + + + + image C1 C2 C3 C4 C6 C12 A4 C2×A4 C4×A4 kernel A4×C42 C2×C4×A4 C22×C42 C4×A4 C23×C4 C22×C4 C42 C2×C4 C4 # reps 1 3 2 12 6 24 1 3 12

Matrix representation of A4×C42 in GL4(𝔽13) generated by

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 12 0 0 0 0 5 0 0 0 0 5 0 0 0 0 5
,
 1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 12 0 0 0 0 12 0 0 0 0 1
,
 3 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0
G:=sub<GL(4,GF(13))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[12,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,1],[3,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;

A4×C42 in GAP, Magma, Sage, TeX

A_4\times C_4^2
% in TeX

G:=Group("A4xC4^2");
// GroupNames label

G:=SmallGroup(192,993);
// by ID

G=gap.SmallGroup(192,993);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,2,168,92,1027,1784]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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