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G = C42⋊C12order 192 = 26·3

1st semidirect product of C42 and C12 acting via C12/C2=C6

metabelian, soluble, monomial

Aliases: C421C12, C425C4⋊C3, C42⋊C32C4, C22.3(C4×A4), (C2×C42).1C6, (C22×C4).1A4, C23.12(C2×A4), C2.1(C42⋊C6), (C2×C42⋊C3).1C2, SmallGroup(192,192)

Series: Derived Chief Lower central Upper central

C1C42 — C42⋊C12
C1C22C42C2×C42C2×C42⋊C3 — C42⋊C12
C42 — C42⋊C12
C1C2

Generators and relations for C42⋊C12
 G = < a,b,c | a4=b4=c12=1, ab=ba, cac-1=a-1b-1, cbc-1=a-1 >

3C2
3C2
16C3
3C22
3C22
4C4
6C4
6C4
12C4
16C6
3C2×C4
3C2×C4
6C2×C4
6C2×C4
6C2×C4
6C2×C4
6C2×C4
6C2×C4
4A4
16C12
3C22×C4
3C42
3C22×C4
4C2×A4
3C2.C42
3C2.C42
4C4×A4

Character table of C42⋊C12

 class 12A2B2C3A3B4A4B4C4D4E4F4G4H6A6B12A12B12C12D
 size 113316164466661212161616161616
ρ111111111111111111111    trivial
ρ2111111-1-11111-1-111-1-1-1-1    linear of order 2
ρ31111ζ3ζ32-1-11111-1-1ζ32ζ3ζ65ζ6ζ65ζ6    linear of order 6
ρ41111ζ3ζ3211111111ζ32ζ3ζ3ζ32ζ3ζ32    linear of order 3
ρ51111ζ32ζ3-1-11111-1-1ζ3ζ32ζ6ζ65ζ6ζ65    linear of order 6
ρ61111ζ32ζ311111111ζ3ζ32ζ32ζ3ζ32ζ3    linear of order 3
ρ71-11-111-ii-11-11i-i-1-1-i-iii    linear of order 4
ρ81-11-111i-i-11-11-ii-1-1ii-i-i    linear of order 4
ρ91-11-1ζ32ζ3-ii-11-11i-iζ65ζ6ζ43ζ32ζ43ζ3ζ4ζ32ζ4ζ3    linear of order 12
ρ101-11-1ζ3ζ32-ii-11-11i-iζ6ζ65ζ43ζ3ζ43ζ32ζ4ζ3ζ4ζ32    linear of order 12
ρ111-11-1ζ32ζ3i-i-11-11-iiζ65ζ6ζ4ζ32ζ4ζ3ζ43ζ32ζ43ζ3    linear of order 12
ρ121-11-1ζ3ζ32i-i-11-11-iiζ6ζ65ζ4ζ3ζ4ζ32ζ43ζ3ζ43ζ32    linear of order 12
ρ1333330033-1-1-1-1-1-1000000    orthogonal lifted from A4
ρ14333300-3-3-1-1-1-111000000    orthogonal lifted from C2×A4
ρ153-33-3003i-3i1-11-1i-i000000    complex lifted from C4×A4
ρ163-33-300-3i3i1-11-1-ii000000    complex lifted from C4×A4
ρ1766-2-200002i-2i-2i2i00000000    complex lifted from C42⋊C6
ρ186-6-2200002i2i-2i-2i00000000    complex faithful
ρ196-6-220000-2i-2i2i2i00000000    complex faithful
ρ2066-2-20000-2i2i2i-2i00000000    complex lifted from C42⋊C6

Permutation representations of C42⋊C12
On 24 points - transitive group 24T299
Generators in S24
(1 4 7 10)(2 21 8 15)(3 13)(5 18 11 24)(6 22)(9 19)(12 16)(14 23 20 17)
(1 17)(2 11 8 5)(3 16 9 22)(4 14)(6 13 12 19)(7 23)(10 20)(15 18 21 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)| (1,4,7,10)(2,21,8,15)(3,13)(5,18,11,24)(6,22)(9,19)(12,16)(14,23,20,17), (1,17)(2,11,8,5)(3,16,9,22)(4,14)(6,13,12,19)(7,23)(10,20)(15,18,21,24), (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( (1,4,7,10)(2,21,8,15)(3,13)(5,18,11,24)(6,22)(9,19)(12,16)(14,23,20,17), (1,17)(2,11,8,5)(3,16,9,22)(4,14)(6,13,12,19)(7,23)(10,20)(15,18,21,24), (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([(1,4,7,10),(2,21,8,15),(3,13),(5,18,11,24),(6,22),(9,19),(12,16),(14,23,20,17)], [(1,17),(2,11,8,5),(3,16,9,22),(4,14),(6,13,12,19),(7,23),(10,20),(15,18,21,24)], [(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(24,299);

On 24 points - transitive group 24T305
Generators in S24
(1 10)(2 19 8 13)(3 23 9 17)(4 7)(5 22 11 16)(6 20 12 14)(15 18)(21 24)
(1 21 7 15)(2 11)(3 14 9 20)(4 18 10 24)(5 8)(6 17 12 23)(13 22)(16 19)
(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)| (1,10)(2,19,8,13)(3,23,9,17)(4,7)(5,22,11,16)(6,20,12,14)(15,18)(21,24), (1,21,7,15)(2,11)(3,14,9,20)(4,18,10,24)(5,8)(6,17,12,23)(13,22)(16,19), (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( (1,10)(2,19,8,13)(3,23,9,17)(4,7)(5,22,11,16)(6,20,12,14)(15,18)(21,24), (1,21,7,15)(2,11)(3,14,9,20)(4,18,10,24)(5,8)(6,17,12,23)(13,22)(16,19), (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([(1,10),(2,19,8,13),(3,23,9,17),(4,7),(5,22,11,16),(6,20,12,14),(15,18),(21,24)], [(1,21,7,15),(2,11),(3,14,9,20),(4,18,10,24),(5,8),(6,17,12,23),(13,22),(16,19)], [(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(24,305);

On 24 points - transitive group 24T308
Generators in S24
(2 14 8 20)(3 21 9 15)(4 10)(5 17 11 23)(6 18 12 24)(16 22)
(1 19 7 13)(3 21 9 15)(4 16 10 22)(5 11)(6 24 12 18)(17 23)
(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)| (2,14,8,20)(3,21,9,15)(4,10)(5,17,11,23)(6,18,12,24)(16,22), (1,19,7,13)(3,21,9,15)(4,16,10,22)(5,11)(6,24,12,18)(17,23), (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( (2,14,8,20)(3,21,9,15)(4,10)(5,17,11,23)(6,18,12,24)(16,22), (1,19,7,13)(3,21,9,15)(4,16,10,22)(5,11)(6,24,12,18)(17,23), (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([(2,14,8,20),(3,21,9,15),(4,10),(5,17,11,23),(6,18,12,24),(16,22)], [(1,19,7,13),(3,21,9,15),(4,16,10,22),(5,11),(6,24,12,18),(17,23)], [(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(24,308);

Matrix representation of C42⋊C12 in GL6(𝔽13)

080000
500000
000100
0012000
000080
000008
,
010000
1200000
005000
000500
000005
000080
,
000500
005000
000005
000050
050000
500000

G:=sub<GL(6,GF(13))| [0,5,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,8,0,0,0,0,5,0],[0,0,0,0,0,5,0,0,0,0,5,0,0,5,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0] >;

C42⋊C12 in GAP, Magma, Sage, TeX

C_4^2\rtimes C_{12}
% in TeX

G:=Group("C4^2:C12");
// GroupNames label

G:=SmallGroup(192,192);
// by ID

G=gap.SmallGroup(192,192);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,2,-2,2,42,1683,346,360,4204,2321,102,2028,3541]);
// Polycyclic

G:=Group<a,b,c|a^4=b^4=c^12=1,a*b=b*a,c*a*c^-1=a^-1*b^-1,c*b*c^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C42⋊C12 in TeX
Character table of C42⋊C12 in TeX

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