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G = C3×GL2(𝔽3)  order 144 = 24·32

Direct product of C3 and GL2(𝔽3)

direct product, non-abelian, soluble

Aliases: C3×GL2(𝔽3), C6.9S4, SL2(𝔽3)⋊C6, Q8⋊(C3×S3), C2.3(C3×S4), (C3×Q8)⋊2S3, (C3×SL2(𝔽3))⋊4C2, SmallGroup(144,122)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(𝔽3) — C3×GL2(𝔽3)
C1C2Q8SL2(𝔽3)C3×SL2(𝔽3) — C3×GL2(𝔽3)
SL2(𝔽3) — C3×GL2(𝔽3)
C1C6

Generators and relations for C3×GL2(𝔽3)
 G = < a,b,c,d,e | a3=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece=b-1, dbd-1=bc, ebe=b2c, dcd-1=b, ede=d-1 >

12C2
4C3
8C3
3C4
6C22
4C6
4S3
4S3
8C6
12C6
4C32
3C8
3D4
3C12
4D6
6C2×C6
4C3×C6
4C3×S3
4C3×S3
3SD16
2SL2(𝔽3)
3C24
3C3×D4
4S3×C6
3C3×SD16

Character table of C3×GL2(𝔽3)

 class 12A2B3A3B3C3D3E46A6B6C6D6E6F6G8A8B12A12B24A24B24C24D
 size 111211888611888121266666666
ρ1111111111111111111111111    trivial
ρ211-111111111111-1-1-1-111-1-1-1-1    linear of order 2
ρ311-1ζ3ζ32ζ31ζ321ζ3ζ32ζ31ζ32ζ6ζ65-1-1ζ32ζ3ζ65ζ65ζ6ζ6    linear of order 6
ρ411-1ζ32ζ3ζ321ζ31ζ32ζ3ζ321ζ3ζ65ζ6-1-1ζ3ζ32ζ6ζ6ζ65ζ65    linear of order 6
ρ5111ζ32ζ3ζ321ζ31ζ32ζ3ζ321ζ3ζ3ζ3211ζ3ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ6111ζ3ζ32ζ31ζ321ζ3ζ32ζ31ζ32ζ32ζ311ζ32ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ722022-1-1-1222-1-1-10000220000    orthogonal lifted from S3
ρ8220-1+-3-1--3ζ65-1ζ62-1+-3-1--3ζ65-1ζ60000-1--3-1+-30000    complex lifted from C3×S3
ρ9220-1--3-1+-3ζ6-1ζ652-1--3-1+-3ζ6-1ζ650000-1+-3-1--30000    complex lifted from C3×S3
ρ102-2022-1-1-10-2-211100--2-200--2-2-2--2    complex lifted from GL2(𝔽3)
ρ112-2022-1-1-10-2-211100-2--200-2--2--2-2    complex lifted from GL2(𝔽3)
ρ122-20-1--3-1+-3ζ6-1ζ6501+-31--3ζ321ζ300--2-200ζ87ζ3285ζ32ζ83ζ328ζ32ζ83ζ38ζ3ζ87ζ385ζ3    complex faithful
ρ132-20-1+-3-1--3ζ65-1ζ601--31+-3ζ31ζ3200--2-200ζ87ζ385ζ3ζ83ζ38ζ3ζ83ζ328ζ32ζ87ζ3285ζ32    complex faithful
ρ142-20-1+-3-1--3ζ65-1ζ601--31+-3ζ31ζ3200-2--200ζ83ζ38ζ3ζ87ζ385ζ3ζ87ζ3285ζ32ζ83ζ328ζ32    complex faithful
ρ152-20-1--3-1+-3ζ6-1ζ6501+-31--3ζ321ζ300-2--200ζ83ζ328ζ32ζ87ζ3285ζ32ζ87ζ385ζ3ζ83ζ38ζ3    complex faithful
ρ1633-133000-133000-1-111-1-11111    orthogonal lifted from S4
ρ1733133000-13300011-1-1-1-1-1-1-1-1    orthogonal lifted from S4
ρ1833-1-3+3-3/2-3-3-3/2000-1-3+3-3/2-3-3-3/2000ζ6ζ6511ζ6ζ65ζ3ζ3ζ32ζ32    complex lifted from C3×S4
ρ19331-3-3-3/2-3+3-3/2000-1-3-3-3/2-3+3-3/2000ζ3ζ32-1-1ζ65ζ6ζ6ζ6ζ65ζ65    complex lifted from C3×S4
ρ2033-1-3-3-3/2-3+3-3/2000-1-3-3-3/2-3+3-3/2000ζ65ζ611ζ65ζ6ζ32ζ32ζ3ζ3    complex lifted from C3×S4
ρ21331-3+3-3/2-3-3-3/2000-1-3+3-3/2-3-3-3/2000ζ32ζ3-1-1ζ6ζ65ζ65ζ65ζ6ζ6    complex lifted from C3×S4
ρ224-40441110-4-4-1-1-10000000000    orthogonal lifted from GL2(𝔽3)
ρ234-40-2+2-3-2-2-3ζ31ζ3202-2-32+2-3ζ65-1ζ60000000000    complex faithful
ρ244-40-2-2-3-2+2-3ζ321ζ302+2-32-2-3ζ6-1ζ650000000000    complex faithful

Permutation representations of C3×GL2(𝔽3)
On 24 points - transitive group 24T253
Generators in S24
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 19 11)(6 20 12)(7 17 9)(8 18 10)
(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 11 3 9)(2 10 4 12)(5 22 7 24)(6 21 8 23)(13 18 15 20)(14 17 16 19)
(2 11 10)(4 9 12)(5 8 21)(6 23 7)(13 19 18)(15 17 20)
(1 3)(2 11)(4 9)(5 21)(7 23)(13 19)(14 16)(15 17)(22 24)

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

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

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

G:=TransitiveGroup(24,253);

C3×GL2(𝔽3) is a maximal subgroup of   Dic3.4S4  GL2(𝔽3)⋊S3  D6.S4  C323GL2(𝔽3)
C3×GL2(𝔽3) is a maximal quotient of   C32.GL2(𝔽3)  C322GL2(𝔽3)

Matrix representation of C3×GL2(𝔽3) in GL2(𝔽19) generated by

110
011
,
61
113
,
018
10
,
23
416
,
113
018
G:=sub<GL(2,GF(19))| [11,0,0,11],[6,1,1,13],[0,1,18,0],[2,4,3,16],[1,0,13,18] >;

C3×GL2(𝔽3) in GAP, Magma, Sage, TeX

C_3\times {\rm GL}_2({\mathbb F}_3)
% in TeX

G:=Group("C3xGL(2,3)");
// GroupNames label

G:=SmallGroup(144,122);
// by ID

G=gap.SmallGroup(144,122);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,2,-2,218,867,447,117,544,286,202,88]);
// Polycyclic

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

Export

Subgroup lattice of C3×GL2(𝔽3) in TeX
Character table of C3×GL2(𝔽3) in TeX

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