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G = C422C12order 192 = 26·3

2nd semidirect product of C42 and C12 acting via C12/C2=C6

metabelian, soluble, monomial

Aliases: C422C12, C429C4⋊C3, C42⋊C34C4, C22.4(C4×A4), (C2×C42).2C6, (C22×C4).2A4, C23.13(C2×A4), C2.1(C23.A4), (C2×C42⋊C3).2C2, SmallGroup(192,193)

Series: Derived Chief Lower central Upper central

C1C42 — C422C12
C1C22C42C2×C42C2×C42⋊C3 — C422C12
C42 — C422C12
C1C2

Generators and relations for C422C12
 G = < a,b,c | a4=b4=c12=1, ab=ba, cac-1=a-1b, cbc-1=ab2 >

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

Character table of C422C12

 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
ρ1766-2-20000-222-200000000    orthogonal lifted from C23.A4
ρ1866-2-200002-2-2200000000    orthogonal lifted from C23.A4
ρ196-6-22000022-2-200000000    symplectic faithful, Schur index 2
ρ206-6-220000-2-22200000000    symplectic faithful, Schur index 2

Permutation representations of C422C12
On 24 points - transitive group 24T311
Generators in S24
(2 24 8 18)(3 13 9 19)(5 21 11 15)(6 22 12 16)
(1 23 7 17)(2 8)(3 13 9 19)(4 20 10 14)(5 11)(6 22 12 16)(15 21)(18 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)| (2,24,8,18)(3,13,9,19)(5,21,11,15)(6,22,12,16), (1,23,7,17)(2,8)(3,13,9,19)(4,20,10,14)(5,11)(6,22,12,16)(15,21)(18,24), (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( (2,24,8,18)(3,13,9,19)(5,21,11,15)(6,22,12,16), (1,23,7,17)(2,8)(3,13,9,19)(4,20,10,14)(5,11)(6,22,12,16)(15,21)(18,24), (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([(2,24,8,18),(3,13,9,19),(5,21,11,15),(6,22,12,16)], [(1,23,7,17),(2,8),(3,13,9,19),(4,20,10,14),(5,11),(6,22,12,16),(15,21),(18,24)], [(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(24,311);

Matrix representation of C422C12 in GL6(𝔽3)

021010
101020
212022
212000
111120
110020
,
002022
202201
202202
200101
200200
020102
,
111202
121122
221112
200202
121201
022220

G:=sub<GL(6,GF(3))| [0,1,2,2,1,1,2,0,1,1,1,1,1,1,2,2,1,0,0,0,0,0,1,0,1,2,2,0,2,2,0,0,2,0,0,0],[0,2,2,2,2,0,0,0,0,0,0,2,2,2,2,0,0,0,0,2,2,1,2,1,2,0,0,0,0,0,2,1,2,1,0,2],[1,1,2,2,1,0,1,2,2,0,2,2,1,1,1,0,1,2,2,1,1,2,2,2,0,2,1,0,0,2,2,2,2,2,1,0] >;

C422C12 in GAP, Magma, Sage, TeX

C_4^2\rtimes_2C_{12}
% in TeX

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

G:=SmallGroup(192,193);
// by ID

G=gap.SmallGroup(192,193);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,2,-2,2,42,4371,346,360,2524,2321,102,2028,3541]);
// Polycyclic

G:=Group<a,b,c|a^4=b^4=c^12=1,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=a*b^2>;
// generators/relations

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Subgroup lattice of C422C12 in TeX
Character table of C422C12 in TeX

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