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G = C2×C22⋊A4order 96 = 25·3

Direct product of C2 and C22⋊A4

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

Aliases: C2×C22⋊A4, C252C3, C233A4, C244C6, C22⋊(C2×A4), SmallGroup(96,229)

Series: Derived Chief Lower central Upper central

C1C24 — C2×C22⋊A4
C1C22C24C22⋊A4 — C2×C22⋊A4
C24 — C2×C22⋊A4
C1C2

Generators and relations for C2×C22⋊A4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, fcf-1=b, fdf-1=de=ed, fef-1=d >

Subgroups: 448 in 148 conjugacy classes, 16 normal (6 characteristic)
C1, C2, C2 [×10], C3, C22 [×5], C22 [×50], C6, C23 [×5], C23 [×50], A4 [×5], C24, C24 [×10], C2×A4 [×5], C25, C22⋊A4, C2×C22⋊A4
Quotients: C1, C2, C3, C6, A4 [×5], C2×A4 [×5], C22⋊A4, C2×C22⋊A4

Character table of C2×C22⋊A4

 class 12A2B2C2D2E2F2G2H2I2J2K3A3B6A6B
 size 11333333333316161616
ρ11111111111111111    trivial
ρ21-111111-1-1-1-1-111-1-1    linear of order 2
ρ3111111111111ζ32ζ3ζ32ζ3    linear of order 3
ρ41-111111-1-1-1-1-1ζ32ζ3ζ6ζ65    linear of order 6
ρ51-111111-1-1-1-1-1ζ3ζ32ζ65ζ6    linear of order 6
ρ6111111111111ζ3ζ32ζ3ζ32    linear of order 3
ρ733-13-1-1-1-1-13-1-10000    orthogonal lifted from A4
ρ83-3-1-13-1-1111-310000    orthogonal lifted from C2×A4
ρ93-3-13-1-1-111-3110000    orthogonal lifted from C2×A4
ρ1033-1-13-1-1-1-1-13-10000    orthogonal lifted from A4
ρ1133-1-1-13-1-1-1-1-130000    orthogonal lifted from A4
ρ123-3-1-1-1-13-311110000    orthogonal lifted from C2×A4
ρ13333-1-1-1-1-13-1-1-10000    orthogonal lifted from A4
ρ1433-1-1-1-133-1-1-1-10000    orthogonal lifted from A4
ρ153-3-1-1-13-11111-30000    orthogonal lifted from C2×A4
ρ163-33-1-1-1-11-31110000    orthogonal lifted from C2×A4

Permutation representations of C2×C22⋊A4
On 12 points - transitive group 12T56
Generators in S12
(1 10)(2 11)(3 12)(4 8)(5 9)(6 7)
(2 5)(3 6)(7 12)(9 11)
(1 4)(3 6)(7 12)(8 10)
(1 8)(2 5)(3 12)(4 10)(6 7)(9 11)
(1 10)(2 9)(3 6)(4 8)(5 11)(7 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), (2,5)(3,6)(7,12)(9,11), (1,4)(3,6)(7,12)(8,10), (1,8)(2,5)(3,12)(4,10)(6,7)(9,11), (1,10)(2,9)(3,6)(4,8)(5,11)(7,12), (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), (2,5)(3,6)(7,12)(9,11), (1,4)(3,6)(7,12)(8,10), (1,8)(2,5)(3,12)(4,10)(6,7)(9,11), (1,10)(2,9)(3,6)(4,8)(5,11)(7,12), (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)], [(2,5),(3,6),(7,12),(9,11)], [(1,4),(3,6),(7,12),(8,10)], [(1,8),(2,5),(3,12),(4,10),(6,7),(9,11)], [(1,10),(2,9),(3,6),(4,8),(5,11),(7,12)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12)])

G:=TransitiveGroup(12,56);

On 24 points - transitive group 24T176
Generators in S24
(1 8)(2 9)(3 7)(4 23)(5 24)(6 22)(10 20)(11 21)(12 19)(13 18)(14 16)(15 17)
(2 9)(3 7)(5 24)(6 22)(10 20)(11 21)(13 18)(15 17)
(1 8)(3 7)(4 23)(6 22)(11 21)(12 19)(13 18)(14 16)
(1 4)(2 20)(3 13)(5 15)(6 21)(7 18)(8 23)(9 10)(11 22)(12 16)(14 19)(17 24)
(1 14)(2 5)(3 21)(4 19)(6 13)(7 11)(8 16)(9 24)(10 17)(12 23)(15 20)(18 22)
(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)| (1,8)(2,9)(3,7)(4,23)(5,24)(6,22)(10,20)(11,21)(12,19)(13,18)(14,16)(15,17), (2,9)(3,7)(5,24)(6,22)(10,20)(11,21)(13,18)(15,17), (1,8)(3,7)(4,23)(6,22)(11,21)(12,19)(13,18)(14,16), (1,4)(2,20)(3,13)(5,15)(6,21)(7,18)(8,23)(9,10)(11,22)(12,16)(14,19)(17,24), (1,14)(2,5)(3,21)(4,19)(6,13)(7,11)(8,16)(9,24)(10,17)(12,23)(15,20)(18,22), (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( (1,8)(2,9)(3,7)(4,23)(5,24)(6,22)(10,20)(11,21)(12,19)(13,18)(14,16)(15,17), (2,9)(3,7)(5,24)(6,22)(10,20)(11,21)(13,18)(15,17), (1,8)(3,7)(4,23)(6,22)(11,21)(12,19)(13,18)(14,16), (1,4)(2,20)(3,13)(5,15)(6,21)(7,18)(8,23)(9,10)(11,22)(12,16)(14,19)(17,24), (1,14)(2,5)(3,21)(4,19)(6,13)(7,11)(8,16)(9,24)(10,17)(12,23)(15,20)(18,22), (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([(1,8),(2,9),(3,7),(4,23),(5,24),(6,22),(10,20),(11,21),(12,19),(13,18),(14,16),(15,17)], [(2,9),(3,7),(5,24),(6,22),(10,20),(11,21),(13,18),(15,17)], [(1,8),(3,7),(4,23),(6,22),(11,21),(12,19),(13,18),(14,16)], [(1,4),(2,20),(3,13),(5,15),(6,21),(7,18),(8,23),(9,10),(11,22),(12,16),(14,19),(17,24)], [(1,14),(2,5),(3,21),(4,19),(6,13),(7,11),(8,16),(9,24),(10,17),(12,23),(15,20),(18,22)], [(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(24,176);

On 24 points - transitive group 24T177
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 12)(2 20)(3 7)(4 16)(5 15)(6 22)(8 19)(9 10)(11 21)(13 18)(14 23)(17 24)
(1 8)(2 10)(3 21)(4 23)(5 17)(6 13)(7 11)(9 20)(12 19)(14 16)(15 24)(18 22)
(1 16)(2 20)(3 22)(4 12)(5 15)(6 7)(8 14)(9 10)(11 13)(17 24)(18 21)(19 23)
(1 23)(2 17)(3 21)(4 8)(5 10)(6 13)(7 11)(9 15)(12 14)(16 19)(18 22)(20 24)
(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), (1,12)(2,20)(3,7)(4,16)(5,15)(6,22)(8,19)(9,10)(11,21)(13,18)(14,23)(17,24), (1,8)(2,10)(3,21)(4,23)(5,17)(6,13)(7,11)(9,20)(12,19)(14,16)(15,24)(18,22), (1,16)(2,20)(3,22)(4,12)(5,15)(6,7)(8,14)(9,10)(11,13)(17,24)(18,21)(19,23), (1,23)(2,17)(3,21)(4,8)(5,10)(6,13)(7,11)(9,15)(12,14)(16,19)(18,22)(20,24), (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), (1,12)(2,20)(3,7)(4,16)(5,15)(6,22)(8,19)(9,10)(11,21)(13,18)(14,23)(17,24), (1,8)(2,10)(3,21)(4,23)(5,17)(6,13)(7,11)(9,20)(12,19)(14,16)(15,24)(18,22), (1,16)(2,20)(3,22)(4,12)(5,15)(6,7)(8,14)(9,10)(11,13)(17,24)(18,21)(19,23), (1,23)(2,17)(3,21)(4,8)(5,10)(6,13)(7,11)(9,15)(12,14)(16,19)(18,22)(20,24), (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)], [(1,12),(2,20),(3,7),(4,16),(5,15),(6,22),(8,19),(9,10),(11,21),(13,18),(14,23),(17,24)], [(1,8),(2,10),(3,21),(4,23),(5,17),(6,13),(7,11),(9,20),(12,19),(14,16),(15,24),(18,22)], [(1,16),(2,20),(3,22),(4,12),(5,15),(6,7),(8,14),(9,10),(11,13),(17,24),(18,21),(19,23)], [(1,23),(2,17),(3,21),(4,8),(5,10),(6,13),(7,11),(9,15),(12,14),(16,19),(18,22),(20,24)], [(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(24,177);

On 24 points - transitive group 24T178
Generators in S24
(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 10)(2 16)(3 14)(4 22)(5 9)(6 19)(7 24)(8 20)(11 13)(12 17)(15 18)(21 23)
(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 15)(2 16)(3 12)(4 20)(5 9)(6 24)(7 19)(8 22)(10 18)(11 13)(14 17)(21 23)
(1 10)(2 13)(3 17)(4 22)(5 21)(6 7)(8 20)(9 23)(11 16)(12 14)(15 18)(19 24)
(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)(7,17)(8,18)(9,16)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20), (1,10)(2,16)(3,14)(4,22)(5,9)(6,19)(7,24)(8,20)(11,13)(12,17)(15,18)(21,23), (1,15)(2,11)(3,17)(4,20)(5,23)(6,7)(8,22)(9,21)(10,18)(12,14)(13,16)(19,24), (1,15)(2,16)(3,12)(4,20)(5,9)(6,24)(7,19)(8,22)(10,18)(11,13)(14,17)(21,23), (1,10)(2,13)(3,17)(4,22)(5,21)(6,7)(8,20)(9,23)(11,16)(12,14)(15,18)(19,24), (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)(7,17)(8,18)(9,16)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20), (1,10)(2,16)(3,14)(4,22)(5,9)(6,19)(7,24)(8,20)(11,13)(12,17)(15,18)(21,23), (1,15)(2,11)(3,17)(4,20)(5,23)(6,7)(8,22)(9,21)(10,18)(12,14)(13,16)(19,24), (1,15)(2,16)(3,12)(4,20)(5,9)(6,24)(7,19)(8,22)(10,18)(11,13)(14,17)(21,23), (1,10)(2,13)(3,17)(4,22)(5,21)(6,7)(8,20)(9,23)(11,16)(12,14)(15,18)(19,24), (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),(7,17),(8,18),(9,16),(10,22),(11,23),(12,24),(13,21),(14,19),(15,20)], [(1,10),(2,16),(3,14),(4,22),(5,9),(6,19),(7,24),(8,20),(11,13),(12,17),(15,18),(21,23)], [(1,15),(2,11),(3,17),(4,20),(5,23),(6,7),(8,22),(9,21),(10,18),(12,14),(13,16),(19,24)], [(1,15),(2,16),(3,12),(4,20),(5,9),(6,24),(7,19),(8,22),(10,18),(11,13),(14,17),(21,23)], [(1,10),(2,13),(3,17),(4,22),(5,21),(6,7),(8,20),(9,23),(11,16),(12,14),(15,18),(19,24)], [(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(24,178);

C2×C22⋊A4 is a maximal subgroup of   C24⋊C12  C244Dic3  C2×A42
C2×C22⋊A4 is a maximal quotient of   C4○D4⋊A4  2+ 1+4.3C6

Polynomial with Galois group C2×C22⋊A4 over ℚ
actionf(x)Disc(f)
12T56x12-14x10+70x8-150x6+132x4-35x2+1212·54·138·532

Matrix representation of C2×C22⋊A4 in GL6(ℤ)

100000
010000
001000
000-100
0000-10
00000-1
,
-100000
0-10000
001000
000-100
000010
00000-1
,
-100000
010000
00-1000
000100
0000-10
00000-1
,
-100000
0-10000
001000
000100
000010
000001
,
-100000
010000
00-1000
000100
000010
000001
,
010000
001000
100000
000010
000001
000100

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],[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] >;

C2×C22⋊A4 in GAP, Magma, Sage, TeX

C_2\times C_2^2\rtimes A_4
% in TeX

G:=Group("C2xC2^2:A4");
// GroupNames label

G:=SmallGroup(96,229);
// by ID

G=gap.SmallGroup(96,229);
# by ID

G:=PCGroup([6,-2,-3,-2,2,-2,2,116,225,730,1307]);
// Polycyclic

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

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Character table of C2×C22⋊A4 in TeX

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