C2^2xC2^2:A4 - GroupNames

G = C₂²×C₂²⋊A₄ order 192 = 2⁶·3

Direct product of C₂² and C₂²⋊A₄

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

Aliases: C₂²×C₂²⋊A₄, C₂² ≀C₃, C₂⁶⋊₂C₃, C₂⁵⋊₄C₆, C₂⁴⋊₇A₄, C₂³⋊₃(C₂×A₄), C₂²⋊(C₂²×A₄), C₂⁴⋊₁₀(C₂×C₆), SmallGroup(192,1540)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₂²×C₂²⋊A₄

Chief series

C₁ — C₂² — C₂⁴ — C₂²⋊A₄ — C₂×C₂²⋊A₄ — C₂²×C₂²⋊A₄

Lower central

C₂⁴ — C₂²×C₂²⋊A₄

Upper central

C₁ — C₂²

Generators and relations for C₂²×C₂²⋊A₄
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, gcg^-1=cd=dc, ce=ec, cf=fc, de=ed, df=fd, gdg^-1=c, geg^-1=ef=fe, gfg^-1=e >

Subgroups: 3010 in 1000 conjugacy classes, 40 normal (6 characteristic)
C₁, C₂, C₂, C₃, C₂², C₂², C₂², C₆, C₂³, C₂³, A₄, C₂×C₆, C₂⁴, C₂⁴, C₂⁴, C₂×A₄, C₂⁵, C₂⁵, C₂²×A₄, C₂²⋊A₄, C₂⁶, C₂×C₂²⋊A₄, C₂²×C₂²⋊A₄
Quotients: C₁, C₂, C₃, C₂², C₆, A₄, C₂×C₆, C₂×A₄, C₂²×A₄, C₂²⋊A₄, C₂×C₂²⋊A₄, C₂²×C₂²⋊A₄

Permutation representations of C₂²×C₂²⋊A₄
►On 12 points - transitive group 12T90

Generators in S₁₂

(1 10)(2 11)(3 12)(4 8)(5 9)(6 7)

(1 4)(2 5)(3 6)(7 12)(8 10)(9 11)

(1 4)(2 11)(3 7)(5 9)(6 12)(8 10)

(1 8)(2 5)(3 12)(4 10)(6 7)(9 11)

(2 9)(3 7)(5 11)(6 12)

(1 8)(3 7)(4 10)(6 12)

(1 2 3)(4 5 6)(7 8 9)(10 11 12)

G:=sub<Sym(12)| (1,10)(2,11)(3,12)(4,8)(5,9)(6,7), (1,4)(2,5)(3,6)(7,12)(8,10)(9,11), (1,4)(2,11)(3,7)(5,9)(6,12)(8,10), (1,8)(2,5)(3,12)(4,10)(6,7)(9,11), (2,9)(3,7)(5,11)(6,12), (1,8)(3,7)(4,10)(6,12), (1,2,3)(4,5,6)(7,8,9)(10,11,12)>;

G:=Group( (1,10)(2,11)(3,12)(4,8)(5,9)(6,7), (1,4)(2,5)(3,6)(7,12)(8,10)(9,11), (1,4)(2,11)(3,7)(5,9)(6,12)(8,10), (1,8)(2,5)(3,12)(4,10)(6,7)(9,11), (2,9)(3,7)(5,11)(6,12), (1,8)(3,7)(4,10)(6,12), (1,2,3)(4,5,6)(7,8,9)(10,11,12) );

G=PermutationGroup([[(1,10),(2,11),(3,12),(4,8),(5,9),(6,7)], [(1,4),(2,5),(3,6),(7,12),(8,10),(9,11)], [(1,4),(2,11),(3,7),(5,9),(6,12),(8,10)], [(1,8),(2,5),(3,12),(4,10),(6,7),(9,11)], [(2,9),(3,7),(5,11),(6,12)], [(1,8),(3,7),(4,10),(6,12)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12)]])

G:=TransitiveGroup(12,90);

►On 24 points - transitive group 24T457

Generators in S₂₄

(1 12)(2 10)(3 11)(4 16)(5 17)(6 18)(7 21)(8 19)(9 20)(13 22)(14 23)(15 24)

(1 19)(2 20)(3 21)(4 14)(5 15)(6 13)(7 11)(8 12)(9 10)(16 23)(17 24)(18 22)

(1 19)(3 21)(4 14)(6 13)(7 11)(8 12)(16 23)(18 22)

(1 19)(2 20)(4 14)(5 15)(8 12)(9 10)(16 23)(17 24)

(1 23)(2 9)(3 6)(4 8)(5 24)(7 22)(10 20)(11 18)(12 14)(13 21)(15 17)(16 19)

(1 4)(2 24)(3 7)(5 9)(6 22)(8 23)(10 15)(11 21)(12 16)(13 18)(14 19)(17 20)

(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)

G:=sub<Sym(24)| (1,12)(2,10)(3,11)(4,16)(5,17)(6,18)(7,21)(8,19)(9,20)(13,22)(14,23)(15,24), (1,19)(2,20)(3,21)(4,14)(5,15)(6,13)(7,11)(8,12)(9,10)(16,23)(17,24)(18,22), (1,19)(3,21)(4,14)(6,13)(7,11)(8,12)(16,23)(18,22), (1,19)(2,20)(4,14)(5,15)(8,12)(9,10)(16,23)(17,24), (1,23)(2,9)(3,6)(4,8)(5,24)(7,22)(10,20)(11,18)(12,14)(13,21)(15,17)(16,19), (1,4)(2,24)(3,7)(5,9)(6,22)(8,23)(10,15)(11,21)(12,16)(13,18)(14,19)(17,20), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)>;

G:=Group( (1,12)(2,10)(3,11)(4,16)(5,17)(6,18)(7,21)(8,19)(9,20)(13,22)(14,23)(15,24), (1,19)(2,20)(3,21)(4,14)(5,15)(6,13)(7,11)(8,12)(9,10)(16,23)(17,24)(18,22), (1,19)(3,21)(4,14)(6,13)(7,11)(8,12)(16,23)(18,22), (1,19)(2,20)(4,14)(5,15)(8,12)(9,10)(16,23)(17,24), (1,23)(2,9)(3,6)(4,8)(5,24)(7,22)(10,20)(11,18)(12,14)(13,21)(15,17)(16,19), (1,4)(2,24)(3,7)(5,9)(6,22)(8,23)(10,15)(11,21)(12,16)(13,18)(14,19)(17,20), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24) );

G=PermutationGroup([[(1,12),(2,10),(3,11),(4,16),(5,17),(6,18),(7,21),(8,19),(9,20),(13,22),(14,23),(15,24)], [(1,19),(2,20),(3,21),(4,14),(5,15),(6,13),(7,11),(8,12),(9,10),(16,23),(17,24),(18,22)], [(1,19),(3,21),(4,14),(6,13),(7,11),(8,12),(16,23),(18,22)], [(1,19),(2,20),(4,14),(5,15),(8,12),(9,10),(16,23),(17,24)], [(1,23),(2,9),(3,6),(4,8),(5,24),(7,22),(10,20),(11,18),(12,14),(13,21),(15,17),(16,19)], [(1,4),(2,24),(3,7),(5,9),(6,22),(8,23),(10,15),(11,21),(12,16),(13,18),(14,19),(17,20)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)]])

G:=TransitiveGroup(24,457);

►On 24 points - transitive group 24T458

Generators in S₂₄

(1 15)(2 13)(3 14)(4 20)(5 21)(6 19)(7 24)(8 22)(9 23)(10 18)(11 16)(12 17)

(1 4)(2 5)(3 6)(7 17)(8 18)(9 16)(10 22)(11 23)(12 24)(13 21)(14 19)(15 20)

(1 15)(2 11)(3 17)(4 20)(5 23)(6 7)(8 22)(9 21)(10 18)(12 14)(13 16)(19 24)

(1 18)(2 13)(3 12)(4 8)(5 21)(6 24)(7 19)(9 23)(10 15)(11 16)(14 17)(20 22)

(2 11)(3 12)(5 23)(6 24)(7 19)(9 21)(13 16)(14 17)

(1 10)(3 12)(4 22)(6 24)(7 19)(8 20)(14 17)(15 18)

(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)

G:=sub<Sym(24)| (1,15)(2,13)(3,14)(4,20)(5,21)(6,19)(7,24)(8,22)(9,23)(10,18)(11,16)(12,17), (1,4)(2,5)(3,6)(7,17)(8,18)(9,16)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20), (1,15)(2,11)(3,17)(4,20)(5,23)(6,7)(8,22)(9,21)(10,18)(12,14)(13,16)(19,24), (1,18)(2,13)(3,12)(4,8)(5,21)(6,24)(7,19)(9,23)(10,15)(11,16)(14,17)(20,22), (2,11)(3,12)(5,23)(6,24)(7,19)(9,21)(13,16)(14,17), (1,10)(3,12)(4,22)(6,24)(7,19)(8,20)(14,17)(15,18), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)>;

G:=Group( (1,15)(2,13)(3,14)(4,20)(5,21)(6,19)(7,24)(8,22)(9,23)(10,18)(11,16)(12,17), (1,4)(2,5)(3,6)(7,17)(8,18)(9,16)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20), (1,15)(2,11)(3,17)(4,20)(5,23)(6,7)(8,22)(9,21)(10,18)(12,14)(13,16)(19,24), (1,18)(2,13)(3,12)(4,8)(5,21)(6,24)(7,19)(9,23)(10,15)(11,16)(14,17)(20,22), (2,11)(3,12)(5,23)(6,24)(7,19)(9,21)(13,16)(14,17), (1,10)(3,12)(4,22)(6,24)(7,19)(8,20)(14,17)(15,18), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24) );

G=PermutationGroup([[(1,15),(2,13),(3,14),(4,20),(5,21),(6,19),(7,24),(8,22),(9,23),(10,18),(11,16),(12,17)], [(1,4),(2,5),(3,6),(7,17),(8,18),(9,16),(10,22),(11,23),(12,24),(13,21),(14,19),(15,20)], [(1,15),(2,11),(3,17),(4,20),(5,23),(6,7),(8,22),(9,21),(10,18),(12,14),(13,16),(19,24)], [(1,18),(2,13),(3,12),(4,8),(5,21),(6,24),(7,19),(9,23),(10,15),(11,16),(14,17),(20,22)], [(2,11),(3,12),(5,23),(6,24),(7,19),(9,21),(13,16),(14,17)], [(1,10),(3,12),(4,22),(6,24),(7,19),(8,20),(14,17),(15,18)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)]])

G:=TransitiveGroup(24,458);

►On 24 points - transitive group 24T459

Generators in S₂₄

(1 4)(2 5)(3 6)(7 22)(8 23)(9 24)(10 17)(11 18)(12 16)(13 21)(14 19)(15 20)

(1 12)(2 10)(3 11)(4 16)(5 17)(6 18)(7 21)(8 19)(9 20)(13 22)(14 23)(15 24)

(1 19)(3 21)(4 14)(6 13)(7 11)(8 12)(16 23)(18 22)

(1 19)(2 20)(4 14)(5 15)(8 12)(9 10)(16 23)(17 24)

(1 23)(2 9)(3 6)(4 8)(5 24)(7 22)(10 20)(11 18)(12 14)(13 21)(15 17)(16 19)

(1 4)(2 24)(3 7)(5 9)(6 22)(8 23)(10 15)(11 21)(12 16)(13 18)(14 19)(17 20)

(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)

G:=sub<Sym(24)| (1,4)(2,5)(3,6)(7,22)(8,23)(9,24)(10,17)(11,18)(12,16)(13,21)(14,19)(15,20), (1,12)(2,10)(3,11)(4,16)(5,17)(6,18)(7,21)(8,19)(9,20)(13,22)(14,23)(15,24), (1,19)(3,21)(4,14)(6,13)(7,11)(8,12)(16,23)(18,22), (1,19)(2,20)(4,14)(5,15)(8,12)(9,10)(16,23)(17,24), (1,23)(2,9)(3,6)(4,8)(5,24)(7,22)(10,20)(11,18)(12,14)(13,21)(15,17)(16,19), (1,4)(2,24)(3,7)(5,9)(6,22)(8,23)(10,15)(11,21)(12,16)(13,18)(14,19)(17,20), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)>;

G:=Group( (1,4)(2,5)(3,6)(7,22)(8,23)(9,24)(10,17)(11,18)(12,16)(13,21)(14,19)(15,20), (1,12)(2,10)(3,11)(4,16)(5,17)(6,18)(7,21)(8,19)(9,20)(13,22)(14,23)(15,24), (1,19)(3,21)(4,14)(6,13)(7,11)(8,12)(16,23)(18,22), (1,19)(2,20)(4,14)(5,15)(8,12)(9,10)(16,23)(17,24), (1,23)(2,9)(3,6)(4,8)(5,24)(7,22)(10,20)(11,18)(12,14)(13,21)(15,17)(16,19), (1,4)(2,24)(3,7)(5,9)(6,22)(8,23)(10,15)(11,21)(12,16)(13,18)(14,19)(17,20), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24) );

G=PermutationGroup([[(1,4),(2,5),(3,6),(7,22),(8,23),(9,24),(10,17),(11,18),(12,16),(13,21),(14,19),(15,20)], [(1,12),(2,10),(3,11),(4,16),(5,17),(6,18),(7,21),(8,19),(9,20),(13,22),(14,23),(15,24)], [(1,19),(3,21),(4,14),(6,13),(7,11),(8,12),(16,23),(18,22)], [(1,19),(2,20),(4,14),(5,15),(8,12),(9,10),(16,23),(17,24)], [(1,23),(2,9),(3,6),(4,8),(5,24),(7,22),(10,20),(11,18),(12,14),(13,21),(15,17),(16,19)], [(1,4),(2,24),(3,7),(5,9),(6,22),(8,23),(10,15),(11,21),(12,16),(13,18),(14,19),(17,20)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)]])

G:=TransitiveGroup(24,459);

Polynomial with Galois group C₂²×C₂²⋊A₄ over ℚ

action	f(x)	Disc(f)
12T90	x¹²-13x¹⁰+60x⁸-116x⁶+85x⁴-17x²+1	2¹²·7⁸·13²·113²

32 conjugacy classes

class	1	2A	2B	2C	2D	···	2W	3A	3B	6A	···	6F
order	1	2	2	2	2	···	2	3	3	6	···	6
size	1	1	1	1	3	···	3	16	16	16	···	16

32 irreducible representations

dim	1	1	1	1	3	3
type	+	+			+	+
image	C₁	C₂	C₃	C₆	A₄	C₂×A₄
kernel	C₂²×C₂²⋊A₄	C₂×C₂²⋊A₄	C₂⁶	C₂⁵	C₂⁴	C₂³
# reps	1	3	2	6	5	15

Matrix representation of C₂²×C₂²⋊A₄ ►in 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

0	1	0	0	0	0
0	0	1	0	0	0
1	0	0	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],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;

C₂²×C₂²⋊A₄ in GAP, Magma, Sage, TeX

C_2^2\times C_2^2\rtimes A_4

% in TeX

G:=Group("C2^2xC2^2:A4");

// GroupNames label

G:=SmallGroup(192,1540);

// by ID

G=gap.SmallGroup(192,1540);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,185,333,1027,1784]);

// 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,g*c*g^-1=c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,g*d*g^-1=c,g*e*g^-1=e*f=f*e,g*f*g^-1=e>;

// generators/relations

G = C22×C22⋊A4 order 192 = 26·3

Direct product of C22 and C22⋊A4

G = C₂²×C₂²⋊A₄ order 192 = 2⁶·3

Direct product of C₂² and C₂²⋊A₄