A4xC2^2xC4

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

Aliases: A₄×C₂²×C₄, C₂⁴⋊₅C₁₂, C₂⁵.₃C₆, (C₂⁴×C₄)⋊₁C₃, (C₂³×C₄)⋊₄C₆, C₂³⋊₄(C₂×C₁₂), C₂²⋊(C₂²×C₁₂), C₂.₁(C₂³×A₄), (C₂³×A₄).₄C₂, C₂⁴.₂₆(C₂×C₆), C₂³.₃₀(C₂×A₄), (C₂×A₄).₁₁C₂³, C₂².₁₇(C₂²×A₄), C₂³.₂₈(C₂²×C₆), (C₂²×A₄).₁₆C₂², (C₂²×C₄)⋊₁₆(C₂×C₆), SmallGroup(192,1496)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — A₄×C₂²×C₄

Chief series

C₁ — C₂² — C₂³ — C₂×A₄ — C₂²×A₄ — C₂³×A₄ — A₄×C₂²×C₄

Lower central

C₂² — A₄×C₂²×C₄

Upper central

C₁ — C₂²×C₄

Subgroups: 816 in 317 conjugacy classes, 81 normal (12 characteristic)
C₁, C₂, C₂^[×6], C₂^[×8], C₃, C₄^[×4], C₄^[×4], C₂², C₂²^[×7], C₂²^[×49], C₆^[×7], C₂×C₄^[×6], C₂×C₄^[×38], C₂³^[×2], C₂³^[×6], C₂³^[×49], C₁₂^[×4], A₄, C₂×C₆^[×7], C₂²×C₄, C₂²×C₄^[×4], C₂²×C₄^[×45], C₂⁴^[×7], C₂⁴^[×8], C₂×C₁₂^[×6], C₂×A₄, C₂×A₄^[×6], C₂²×C₆, C₂³×C₄^[×6], C₂³×C₄^[×8], C₂⁵, C₄×A₄^[×4], C₂²×C₁₂, C₂²×A₄^[×7], C₂⁴×C₄, C₂×C₄×A₄^[×6], C₂³×A₄, A₄×C₂²×C₄

Quotients:
C₁, C₂^[×7], C₃, C₄^[×4], C₂²^[×7], C₆^[×7], C₂×C₄^[×6], C₂³, C₁₂^[×4], A₄, C₂×C₆^[×7], C₂²×C₄, C₂×C₁₂^[×6], C₂×A₄^[×7], C₂²×C₆, C₄×A₄^[×4], C₂²×C₁₂, C₂²×A₄^[×7], C₂×C₄×A₄^[×6], C₂³×A₄, A₄×C₂²×C₄

Generators and relations
G = < a,b,c,d,e,f | a²=b²=c⁴=d²=e²=f³=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf^-1=de=ed, fef^-1=d >

Smallest permutation representation
►On 48 points

Generators in S₄₈

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

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

(5 22)(6 23)(7 24)(8 21)(13 29)(14 30)(15 31)(16 32)(17 27)(18 28)(19 25)(20 26)(37 45)(38 46)(39 47)(40 48)

(1 11)(2 12)(3 9)(4 10)(5 22)(6 23)(7 24)(8 21)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)

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

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

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

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

Matrix representation ►G ⊆ GL₅(𝔽₁₃)

12	0	0	0	0
0	1	0	0	0
0	0	12	0	0
0	0	0	12	0
0	0	0	0	12

12	0	0	0	0
0	12	0	0	0
0	0	12	0	0
0	0	0	12	0
0	0	0	0	12

8	0	0	0	0
0	1	0	0	0
0	0	12	0	0
0	0	0	12	0
0	0	0	0	12

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	12	0
0	0	12	0	12

1	0	0	0	0
0	1	0	0	0
0	0	12	0	0
0	0	0	12	0
0	0	1	1	1

1	0	0	0	0
0	1	0	0	0
0	0	0	1	0
0	0	12	12	11
0	0	0	0	1

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[8,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,12,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,1,0,0,0,12,1,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,1,12,0,0,0,0,11,1] >;

64 conjugacy classes

class	1	2A	···	2G	2H	···	2O	3A	3B	4A	···	4H	4I	···	4P	6A	···	6N	12A	···	12P
order	1	2	···	2	2	···	2	3	3	4	···	4	4	···	4	6	···	6	12	···	12
size	1	1	···	1	3	···	3	4	4	1	···	1	3	···	3	4	···	4	4	···	4

64 irreducible representations

dim	1	1	1	1	1	1	1	1	3	3	3	3
type	+	+	+						+	+	+
image	C₁	C₂	C₂	C₃	C₄	C₆	C₆	C₁₂	A₄	C₂×A₄	C₂×A₄	C₄×A₄
kernel	A₄×C₂²×C₄	C₂×C₄×A₄	C₂³×A₄	C₂⁴×C₄	C₂²×A₄	C₂³×C₄	C₂⁵	C₂⁴	C₂²×C₄	C₂×C₄	C₂³	C₂²
# reps	1	6	1	2	8	12	2	16	1	6	1	8

In GAP, Magma, Sage, TeX

A_4\times C_2^2\times C_4

% in TeX

G:=Group("A4xC2^2xC4");

// GroupNames label

G:=SmallGroup(192,1496);

// by ID

G=gap.SmallGroup(192,1496);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,2,142,530,909]);

// Polycyclic

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

// generators/relations

G = A₄×C₂²×C₄ order 192 = 2⁶·3

Direct product of C₂²×C₄ and A₄

G = A4×C22×C4 order 192 = 26·3

Direct product of C22×C4 and A4

G = A₄×C₂²×C₄ order 192 = 2⁶·3

Direct product of C₂²×C₄ and A₄