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G = C3×A4⋊C4order 144 = 24·32

Direct product of C3 and A4⋊C4

direct product, non-abelian, soluble, monomial

Aliases: C3×A4⋊C4, A4⋊C12, C6.10S4, (C2×A4).C6, (C3×A4)⋊1C4, C2.1(C3×S4), C23.(C3×S3), (C6×A4).1C2, (C2×C6)⋊1Dic3, C22⋊(C3×Dic3), (C22×C6).1S3, SmallGroup(144,123)

Series: Derived Chief Lower central Upper central

C1C22A4 — C3×A4⋊C4
C1C22A4C2×A4C6×A4 — C3×A4⋊C4
A4 — C3×A4⋊C4
C1C6

Generators and relations for C3×A4⋊C4
 G = < a,b,c,d,e | a3=b2=c2=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=d-1 >

3C2
3C2
4C3
8C3
3C22
3C22
6C4
6C4
3C6
3C6
4C6
8C6
4C32
3C2×C4
3C2×C4
2A4
3C2×C6
3C2×C6
4Dic3
6C12
6C12
4C3×C6
3C22⋊C4
2C2×A4
3C2×C12
3C2×C12
4C3×Dic3
3C3×C22⋊C4

Character table of C3×A4⋊C4

 class 12A2B2C3A3B3C3D3E4A4B4C4D6A6B6C6D6E6F6G6H6I12A12B12C12D12E12F12G12H
 size 113311888666611333388866666666
ρ1111111111111111111111111111111    trivial
ρ2111111111-1-1-1-1111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111ζ3ζ32ζ31ζ321111ζ32ζ3ζ3ζ32ζ32ζ31ζ32ζ3ζ32ζ3ζ3ζ3ζ32ζ32ζ32ζ3    linear of order 3
ρ41111ζ32ζ3ζ321ζ3-1-1-1-1ζ3ζ32ζ32ζ3ζ3ζ321ζ3ζ32ζ65ζ6ζ6ζ6ζ65ζ65ζ65ζ6    linear of order 6
ρ51111ζ3ζ32ζ31ζ32-1-1-1-1ζ32ζ3ζ3ζ32ζ32ζ31ζ32ζ3ζ6ζ65ζ65ζ65ζ6ζ6ζ6ζ65    linear of order 6
ρ61111ζ32ζ3ζ321ζ31111ζ3ζ32ζ32ζ3ζ3ζ321ζ3ζ32ζ3ζ32ζ32ζ32ζ3ζ3ζ3ζ32    linear of order 3
ρ71-1-1111111i-i-ii-1-1-1-111-1-1-1i-iii-i-ii-i    linear of order 4
ρ81-1-1111111-iii-i-1-1-1-111-1-1-1-ii-i-iii-ii    linear of order 4
ρ91-1-11ζ32ζ3ζ321ζ3-iii-iζ65ζ6ζ6ζ65ζ3ζ32-1ζ65ζ6ζ43ζ3ζ4ζ32ζ43ζ32ζ43ζ32ζ4ζ3ζ4ζ3ζ43ζ3ζ4ζ32    linear of order 12
ρ101-1-11ζ3ζ32ζ31ζ32i-i-iiζ6ζ65ζ65ζ6ζ32ζ3-1ζ6ζ65ζ4ζ32ζ43ζ3ζ4ζ3ζ4ζ3ζ43ζ32ζ43ζ32ζ4ζ32ζ43ζ3    linear of order 12
ρ111-1-11ζ32ζ3ζ321ζ3i-i-iiζ65ζ6ζ6ζ65ζ3ζ32-1ζ65ζ6ζ4ζ3ζ43ζ32ζ4ζ32ζ4ζ32ζ43ζ3ζ43ζ3ζ4ζ3ζ43ζ32    linear of order 12
ρ121-1-11ζ3ζ32ζ31ζ32-iii-iζ6ζ65ζ65ζ6ζ32ζ3-1ζ6ζ65ζ43ζ32ζ4ζ3ζ43ζ3ζ43ζ3ζ4ζ32ζ4ζ32ζ43ζ32ζ4ζ3    linear of order 12
ρ13222222-1-1-10000222222-1-1-100000000    orthogonal lifted from S3
ρ142-2-2222-1-1-10000-2-2-2-22211100000000    symplectic lifted from Dic3, Schur index 2
ρ152-2-22-1--3-1+-3ζ6-1ζ6500001--31+-31+-31--3-1+-3-1--31ζ3ζ3200000000    complex lifted from C3×Dic3
ρ162222-1+-3-1--3ζ65-1ζ60000-1--3-1+-3-1+-3-1--3-1--3-1+-3-1ζ6ζ6500000000    complex lifted from C3×S3
ρ172-2-22-1+-3-1--3ζ65-1ζ600001+-31--31--31+-3-1--3-1+-31ζ32ζ300000000    complex lifted from C3×Dic3
ρ182222-1--3-1+-3ζ6-1ζ650000-1+-3-1--3-1--3-1+-3-1+-3-1--3-1ζ65ζ600000000    complex lifted from C3×S3
ρ1933-1-1330001-11-133-1-1-1-100011-11-11-1-1    orthogonal lifted from S4
ρ2033-1-133000-11-1133-1-1-1-1000-1-11-11-111    orthogonal lifted from S4
ρ2133-1-1-3+3-3/2-3-3-3/2000-11-11-3-3-3/2-3+3-3/2ζ65ζ6ζ6ζ65000ζ6ζ65ζ3ζ65ζ32ζ6ζ32ζ3    complex lifted from C3×S4
ρ2233-1-1-3-3-3/2-3+3-3/2000-11-11-3+3-3/2-3-3-3/2ζ6ζ65ζ65ζ6000ζ65ζ6ζ32ζ6ζ3ζ65ζ3ζ32    complex lifted from C3×S4
ρ2333-1-1-3+3-3/2-3-3-3/20001-11-1-3-3-3/2-3+3-3/2ζ65ζ6ζ6ζ65000ζ32ζ3ζ65ζ3ζ6ζ32ζ6ζ65    complex lifted from C3×S4
ρ2433-1-1-3-3-3/2-3+3-3/20001-11-1-3+3-3/2-3-3-3/2ζ6ζ65ζ65ζ6000ζ3ζ32ζ6ζ32ζ65ζ3ζ65ζ6    complex lifted from C3×S4
ρ253-31-133000-i-iii-3-311-1-1000-iii-i-iii-i    complex lifted from A4⋊C4
ρ263-31-133000ii-i-i-3-311-1-1000i-i-iii-i-ii    complex lifted from A4⋊C4
ρ273-31-1-3+3-3/2-3-3-3/2000-i-iii3+3-3/23-3-3/2ζ3ζ32ζ6ζ65000ζ43ζ32ζ4ζ3ζ4ζ3ζ43ζ3ζ43ζ32ζ4ζ32ζ4ζ32ζ43ζ3    complex faithful
ρ283-31-1-3-3-3/2-3+3-3/2000-i-iii3-3-3/23+3-3/2ζ32ζ3ζ65ζ6000ζ43ζ3ζ4ζ32ζ4ζ32ζ43ζ32ζ43ζ3ζ4ζ3ζ4ζ3ζ43ζ32    complex faithful
ρ293-31-1-3+3-3/2-3-3-3/2000ii-i-i3+3-3/23-3-3/2ζ3ζ32ζ6ζ65000ζ4ζ32ζ43ζ3ζ43ζ3ζ4ζ3ζ4ζ32ζ43ζ32ζ43ζ32ζ4ζ3    complex faithful
ρ303-31-1-3-3-3/2-3+3-3/2000ii-i-i3-3-3/23+3-3/2ζ32ζ3ζ65ζ6000ζ4ζ3ζ43ζ32ζ43ζ32ζ4ζ32ζ4ζ3ζ43ζ3ζ43ζ3ζ4ζ32    complex faithful

Smallest permutation representation of C3×A4⋊C4
On 36 points
Generators in S36
(1 26 14)(2 27 15)(3 28 16)(4 25 13)(5 29 17)(6 30 18)(7 31 19)(8 32 20)(9 33 21)(10 34 22)(11 35 23)(12 36 24)
(1 4)(2 3)(5 11)(6 8)(7 9)(10 12)(13 14)(15 16)(17 23)(18 20)(19 21)(22 24)(25 26)(27 28)(29 35)(30 32)(31 33)(34 36)
(1 3)(2 4)(5 9)(6 10)(7 11)(8 12)(13 15)(14 16)(17 21)(18 22)(19 23)(20 24)(25 27)(26 28)(29 33)(30 34)(31 35)(32 36)
(1 5 10)(2 11 6)(3 7 12)(4 9 8)(13 21 20)(14 17 22)(15 23 18)(16 19 24)(25 33 32)(26 29 34)(27 35 30)(28 31 36)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)

G:=sub<Sym(36)| (1,26,14)(2,27,15)(3,28,16)(4,25,13)(5,29,17)(6,30,18)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24), (1,4)(2,3)(5,11)(6,8)(7,9)(10,12)(13,14)(15,16)(17,23)(18,20)(19,21)(22,24)(25,26)(27,28)(29,35)(30,32)(31,33)(34,36), (1,3)(2,4)(5,9)(6,10)(7,11)(8,12)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24)(25,27)(26,28)(29,33)(30,34)(31,35)(32,36), (1,5,10)(2,11,6)(3,7,12)(4,9,8)(13,21,20)(14,17,22)(15,23,18)(16,19,24)(25,33,32)(26,29,34)(27,35,30)(28,31,36), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)>;

G:=Group( (1,26,14)(2,27,15)(3,28,16)(4,25,13)(5,29,17)(6,30,18)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24), (1,4)(2,3)(5,11)(6,8)(7,9)(10,12)(13,14)(15,16)(17,23)(18,20)(19,21)(22,24)(25,26)(27,28)(29,35)(30,32)(31,33)(34,36), (1,3)(2,4)(5,9)(6,10)(7,11)(8,12)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24)(25,27)(26,28)(29,33)(30,34)(31,35)(32,36), (1,5,10)(2,11,6)(3,7,12)(4,9,8)(13,21,20)(14,17,22)(15,23,18)(16,19,24)(25,33,32)(26,29,34)(27,35,30)(28,31,36), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36) );

G=PermutationGroup([(1,26,14),(2,27,15),(3,28,16),(4,25,13),(5,29,17),(6,30,18),(7,31,19),(8,32,20),(9,33,21),(10,34,22),(11,35,23),(12,36,24)], [(1,4),(2,3),(5,11),(6,8),(7,9),(10,12),(13,14),(15,16),(17,23),(18,20),(19,21),(22,24),(25,26),(27,28),(29,35),(30,32),(31,33),(34,36)], [(1,3),(2,4),(5,9),(6,10),(7,11),(8,12),(13,15),(14,16),(17,21),(18,22),(19,23),(20,24),(25,27),(26,28),(29,33),(30,34),(31,35),(32,36)], [(1,5,10),(2,11,6),(3,7,12),(4,9,8),(13,21,20),(14,17,22),(15,23,18),(16,19,24),(25,33,32),(26,29,34),(27,35,30),(28,31,36)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36)])

C3×A4⋊C4 is a maximal subgroup of   Dic3.S4  Dic32S4  A4⋊D12  C12×S4  C626Dic3
C3×A4⋊C4 is a maximal quotient of   C62.Dic3  C625Dic3

Matrix representation of C3×A4⋊C4 in GL3(𝔽13) generated by

900
090
009
,
1200
010
0012
,
100
0120
0012
,
020
001
700
,
500
005
050
G:=sub<GL(3,GF(13))| [9,0,0,0,9,0,0,0,9],[12,0,0,0,1,0,0,0,12],[1,0,0,0,12,0,0,0,12],[0,0,7,2,0,0,0,1,0],[5,0,0,0,0,5,0,5,0] >;

C3×A4⋊C4 in GAP, Magma, Sage, TeX

C_3\times A_4\rtimes C_4
% in TeX

G:=Group("C3xA4:C4");
// GroupNames label

G:=SmallGroup(144,123);
// by ID

G=gap.SmallGroup(144,123);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-2,2,36,579,2164,202,1301,347]);
// Polycyclic

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

Export

Subgroup lattice of C3×A4⋊C4 in TeX
Character table of C3×A4⋊C4 in TeX

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