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G = C2×C42⋊C3order 96 = 25·3

Direct product of C2 and C42⋊C3

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

Aliases: C2×C42⋊C3, C423C6, C23.4A4, (C2×C42)⋊C3, C22.1(C2×A4), SmallGroup(96,68)

Series: Derived Chief Lower central Upper central

C1C42 — C2×C42⋊C3
C1C22C42C42⋊C3 — C2×C42⋊C3
C42 — C2×C42⋊C3
C1C2

Generators and relations for C2×C42⋊C3
 G = < a,b,c,d | a2=b4=c4=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, dcd-1=b-1c2 >

3C2
3C2
16C3
3C4
3C22
3C4
3C4
3C4
3C22
16C6
3C2×C4
3C2×C4
3C2×C4
3C2×C4
3C2×C4
3C2×C4
4A4
3C22×C4
3C42
4C2×A4

Character table of C2×C42⋊C3

 class 12A2B2C3A3B4A4B4C4D4E4F4G4H6A6B
 size 11331616333333331616
ρ11111111111111111    trivial
ρ21-1-11111111-1-1-1-1-1-1    linear of order 2
ρ31-1-11ζ3ζ321111-1-1-1-1ζ6ζ65    linear of order 6
ρ41111ζ3ζ3211111111ζ32ζ3    linear of order 3
ρ51111ζ32ζ311111111ζ3ζ32    linear of order 3
ρ61-1-11ζ32ζ31111-1-1-1-1ζ65ζ6    linear of order 6
ρ73-3-3300-1-1-1-1111100    orthogonal lifted from C2×A4
ρ8333300-1-1-1-1-1-1-1-100    orthogonal lifted from A4
ρ933-1-100-1-2i1-1+2i1-1-2i11-1+2i00    complex lifted from C42⋊C3
ρ1033-1-1001-1-2i1-1+2i1-1+2i-1-2i100    complex lifted from C42⋊C3
ρ1133-1-1001-1+2i1-1-2i1-1-2i-1+2i100    complex lifted from C42⋊C3
ρ123-31-100-1+2i1-1-2i11-2i-1-11+2i00    complex faithful
ρ133-31-1001-1+2i1-1-2i-11+2i1-2i-100    complex faithful
ρ143-31-100-1-2i1-1+2i11+2i-1-11-2i00    complex faithful
ρ153-31-1001-1-2i1-1+2i-11-2i1+2i-100    complex faithful
ρ1633-1-100-1+2i1-1-2i1-1+2i11-1-2i00    complex lifted from C42⋊C3

Permutation representations of C2×C42⋊C3
On 12 points - transitive group 12T55
Generators in S12
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)
(5 6 7 8)(9 10 11 12)
(1 4 3 2)(5 7)(6 8)(9 10 11 12)
(1 6 11)(2 7 10)(3 8 9)(4 5 12)

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

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

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

G:=TransitiveGroup(12,55);

On 24 points - transitive group 24T173
Generators in S24
(1 4)(2 3)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 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)
(1 5 4 7)(2 6 3 8)(9 16 11 14)(10 13 12 15)(17 21)(18 22)(19 23)(20 24)
(1 20 9)(2 22 15)(3 24 13)(4 18 11)(5 21 10)(6 19 16)(7 23 12)(8 17 14)

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

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

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

G:=TransitiveGroup(24,173);

On 24 points - transitive group 24T174
Generators in S24
(1 7)(2 8)(3 6)(4 5)(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4 2 3)(5 8 6 7)(9 12 11 10)(13 16 15 14)
(1 17 10)(2 19 12)(3 20 9)(4 18 11)(5 23 16)(6 21 14)(7 22 15)(8 24 13)

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

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

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

G:=TransitiveGroup(24,174);

On 24 points - transitive group 24T175
Generators in S24
(1 2)(3 4)(5 6)(7 8)(9 23)(10 24)(11 21)(12 22)(13 17)(14 18)(15 19)(16 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)
(1 6 3 7)(2 5 4 8)(9 24 11 22)(10 21 12 23)(13 19)(14 20)(15 17)(16 18)
(1 17 12)(2 13 22)(3 19 10)(4 15 24)(5 20 23)(6 16 9)(7 14 11)(8 18 21)

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

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

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

G:=TransitiveGroup(24,175);

C2×C42⋊C3 is a maximal subgroup of   C23.9S4  C42⋊C12  C422C12
C2×C42⋊C3 is a maximal quotient of   C424C4⋊C3

Polynomial with Galois group C2×C42⋊C3 over ℚ
actionf(x)Disc(f)
12T55x12-38x10+538x8-3458x6+9659x4-8788x2+169224·312·712·1314·2234

Matrix representation of C2×C42⋊C3 in GL3(𝔽5) generated by

400
040
004
,
213
301
003
,
040
414
120
,
141
034
031
G:=sub<GL(3,GF(5))| [4,0,0,0,4,0,0,0,4],[2,3,0,1,0,0,3,1,3],[0,4,1,4,1,2,0,4,0],[1,0,0,4,3,3,1,4,1] >;

C2×C42⋊C3 in GAP, Magma, Sage, TeX

C_2\times C_4^2\rtimes C_3
% in TeX

G:=Group("C2xC4^2:C3");
// GroupNames label

G:=SmallGroup(96,68);
// by ID

G=gap.SmallGroup(96,68);
# by ID

G:=PCGroup([6,-2,-3,-2,2,-2,2,116,230,801,69,730,1307]);
// Polycyclic

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

Export

Subgroup lattice of C2×C42⋊C3 in TeX
Character table of C2×C42⋊C3 in TeX

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