A4xC4^2 - GroupNames

G = A₄×C₄² order 192 = 2⁶·3

Direct product of C₄² and A₄

direct product, metabelian, soluble, monomial, A-group

Aliases: A₄×C₄², C₂²⋊(C₄×C₁₂), (C₂²×C₄)⋊₆C₁₂, (C₂³×C₄).₄C₆, (C₂²×C₄²)⋊₂C₃, C₂⁴.₂₃(C₂×C₆), C₂³.₁₄(C₂×C₁₂), C₂².₁₀(C₂²×A₄), (C₂²×A₄).₁₃C₂², C₂.₁(C₂×C₄×A₄), (C₂×C₄×A₄).₁₀C₂, (C₂×C₄).₁₆(C₂×A₄), (C₂×A₄).₁₀(C₂×C₄), SmallGroup(192,993)

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

G = A4×C42 order 192 = 26·3

Direct product of C42 and A4

G = A₄×C₄² order 192 = 2⁶·3

Direct product of C₄² and A₄