A4xC2^4

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

Aliases: A₄×C₂⁴, C₂⁶⋊₁C₃, C₂⁵⋊₃C₆, C₂⁴⋊₉(C₂×C₆), C₂²⋊(C₂³×C₆), C₂³⋊₃(C₂²×C₆), SmallGroup(192,1539)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — A₄×C₂⁴

Chief series

C₁ — C₂² — A₄ — C₂×A₄ — C₂²×A₄ — C₂³×A₄ — A₄×C₂⁴

Lower central

C₂² — A₄×C₂⁴

Upper central

C₁ — C₂⁴

Subgroups: 3160 in 1165 conjugacy classes, 201 normal (6 characteristic)
C₁, C₂^[×15], C₂^[×16], C₃, C₂², C₂²^[×35], C₂²^[×205], C₆^[×15], C₂³^[×30], C₂³^[×455], A₄, C₂×C₆^[×35], C₂⁴, C₂⁴^[×35], C₂⁴^[×205], C₂×A₄^[×15], C₂²×C₆^[×15], C₂⁵^[×15], C₂⁵^[×16], C₂²×A₄^[×35], C₂³×C₆, C₂⁶, C₂³×A₄^[×15], A₄×C₂⁴

Quotients:
C₁, C₂^[×15], C₃, C₂²^[×35], C₆^[×15], C₂³^[×15], A₄, C₂×C₆^[×35], C₂⁴, C₂×A₄^[×15], C₂²×C₆^[×15], C₂²×A₄^[×35], C₂³×C₆, C₂³×A₄^[×15], A₄×C₂⁴

Generators and relations
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=f²=g³=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg^-1=ef=fe, gfg^-1=e >

Smallest permutation representation
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

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

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

Matrix representation ►G ⊆ GL₆(ℤ)

1	0	0	0	0	0
0	-1	0	0	0	0
0	0	-1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	-1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	-1	0	0
0	0	0	0	-1	0
0	0	0	0	0	-1

-1	0	0	0	0	0
0	1	0	0	0	0
0	0	-1	0	0	0
0	0	0	-1	0	0
0	0	0	0	-1	0
0	0	0	0	0	-1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	-1	0	0
0	0	0	0	1	0
0	0	0	0	0	-1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	-1	0
0	0	0	0	0	-1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;

64 conjugacy classes

class	1	2A	···	2O	2P	···	2AE	3A	3B	6A	···	6AD
order	1	2	···	2	2	···	2	3	3	6	···	6
size	1	1	···	1	3	···	3	4	4	4	···	4

64 irreducible representations

dim	1	1	1	1	3	3
type	+	+			+	+
image	C₁	C₂	C₃	C₆	A₄	C₂×A₄
kernel	A₄×C₂⁴	C₂³×A₄	C₂⁶	C₂⁵	C₂⁴	C₂³
# reps	1	15	2	30	1	15

In GAP, Magma, Sage, TeX

A_4\times C_2^4

% in TeX

G:=Group("A4xC2^4");

// GroupNames label

G:=SmallGroup(192,1539);

// by ID

G=gap.SmallGroup(192,1539);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,285,475]);

// Polycyclic

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

// generators/relations

G = A₄×C₂⁴ order 192 = 2⁶·3

Direct product of C₂⁴ and A₄

G = A4×C24 order 192 = 26·3

Direct product of C24 and A4

G = A₄×C₂⁴ order 192 = 2⁶·3

Direct product of C₂⁴ and A₄