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G = C3×CSU2(𝔽3)  order 144 = 24·32

Direct product of C3 and CSU2(𝔽3)

direct product, non-abelian, soluble

Aliases: C3×CSU2(𝔽3), C6.8S4, SL2(𝔽3).C6, Q8.(C3×S3), C2.2(C3×S4), (C3×Q8).4S3, (C3×SL2(𝔽3)).2C2, SmallGroup(144,121)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(𝔽3) — C3×CSU2(𝔽3)
C1C2Q8SL2(𝔽3)C3×SL2(𝔽3) — C3×CSU2(𝔽3)
SL2(𝔽3) — C3×CSU2(𝔽3)
C1C6

Generators and relations for C3×CSU2(𝔽3)
 G = < a,b,c,d,e | a3=b4=d3=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece-1=b-1, dbd-1=bc, ebe-1=b2c, dcd-1=b, ede-1=d-1 >

4C3
8C3
3C4
6C4
4C6
8C6
4C32
3C8
3Q8
3C12
4Dic3
6C12
4C3×C6
3Q16
2SL2(𝔽3)
3C24
3C3×Q8
4C3×Dic3
3C3×Q16

Character table of C3×CSU2(𝔽3)

 class 123A3B3C3D3E4A4B6A6B6C6D6E8A8B12A12B12C12D24A24B24C24D
 size 111188861211888666612126666
ρ1111111111111111111111111    trivial
ρ211111111-111111-1-111-1-1-1-1-1-1    linear of order 2
ρ311ζ32ζ3ζ3ζ3211-1ζ32ζ31ζ32ζ3-1-1ζ3ζ32ζ6ζ65ζ6ζ6ζ65ζ65    linear of order 6
ρ411ζ3ζ32ζ32ζ3111ζ3ζ321ζ3ζ3211ζ32ζ3ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ511ζ3ζ32ζ32ζ311-1ζ3ζ321ζ3ζ32-1-1ζ32ζ3ζ65ζ6ζ65ζ65ζ6ζ6    linear of order 6
ρ611ζ32ζ3ζ3ζ32111ζ32ζ31ζ32ζ311ζ3ζ32ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ72222-1-1-12022-1-1-10022000000    orthogonal lifted from S3
ρ82-222-1-1-100-2-2111-220000-222-2    symplectic lifted from CSU2(𝔽3), Schur index 2
ρ92-222-1-1-100-2-21112-200002-2-22    symplectic lifted from CSU2(𝔽3), Schur index 2
ρ1022-1--3-1+-3ζ65ζ6-120-1--3-1+-3-1ζ6ζ6500-1+-3-1--3000000    complex lifted from C3×S3
ρ1122-1+-3-1--3ζ6ζ65-120-1+-3-1--3-1ζ65ζ600-1--3-1+-3000000    complex lifted from C3×S3
ρ122-2-1--3-1+-3ζ65ζ6-1001+-31--31ζ32ζ3-220000ζ83ζ328ζ32ζ87ζ3285ζ32ζ87ζ385ζ3ζ83ζ38ζ3    complex faithful
ρ132-2-1--3-1+-3ζ65ζ6-1001+-31--31ζ32ζ32-20000ζ87ζ3285ζ32ζ83ζ328ζ32ζ83ζ38ζ3ζ87ζ385ζ3    complex faithful
ρ142-2-1+-3-1--3ζ6ζ65-1001--31+-31ζ3ζ32-220000ζ83ζ38ζ3ζ87ζ385ζ3ζ87ζ3285ζ32ζ83ζ328ζ32    complex faithful
ρ152-2-1+-3-1--3ζ6ζ65-1001--31+-31ζ3ζ322-20000ζ87ζ385ζ3ζ83ζ38ζ3ζ83ζ328ζ32ζ87ζ3285ζ32    complex faithful
ρ163333000-1133000-1-1-1-111-1-1-1-1    orthogonal lifted from S4
ρ173333000-1-13300011-1-1-1-11111    orthogonal lifted from S4
ρ1833-3-3-3/2-3+3-3/2000-11-3-3-3/2-3+3-3/2000-1-1ζ65ζ6ζ32ζ3ζ6ζ6ζ65ζ65    complex lifted from C3×S4
ρ1933-3+3-3/2-3-3-3/2000-1-1-3+3-3/2-3-3-3/200011ζ6ζ65ζ65ζ6ζ3ζ3ζ32ζ32    complex lifted from C3×S4
ρ2033-3+3-3/2-3-3-3/2000-11-3+3-3/2-3-3-3/2000-1-1ζ6ζ65ζ3ζ32ζ65ζ65ζ6ζ6    complex lifted from C3×S4
ρ2133-3-3-3/2-3+3-3/2000-1-1-3-3-3/2-3+3-3/200011ζ65ζ6ζ6ζ65ζ32ζ32ζ3ζ3    complex lifted from C3×S4
ρ224-44411100-4-4-1-1-10000000000    symplectic lifted from CSU2(𝔽3), Schur index 2
ρ234-4-2-2-3-2+2-3ζ3ζ321002+2-32-2-3-1ζ6ζ650000000000    complex faithful
ρ244-4-2+2-3-2-2-3ζ32ζ31002-2-32+2-3-1ζ65ζ60000000000    complex faithful

Smallest permutation representation of C3×CSU2(𝔽3)
On 48 points
Generators in S48
(1 13 25)(2 14 26)(3 15 27)(4 16 28)(5 34 39)(6 35 40)(7 36 37)(8 33 38)(9 17 21)(10 18 22)(11 19 23)(12 20 24)(29 44 48)(30 41 45)(31 42 46)(32 43 47)
(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)(37 38 39 40)(41 42 43 44)(45 46 47 48)
(1 11 3 9)(2 10 4 12)(5 45 7 47)(6 48 8 46)(13 19 15 17)(14 18 16 20)(21 25 23 27)(22 28 24 26)(29 33 31 35)(30 36 32 34)(37 43 39 41)(38 42 40 44)
(1 25 13)(2 23 18)(3 27 15)(4 21 20)(5 32 40)(6 34 43)(7 30 38)(8 36 41)(9 24 16)(10 26 19)(11 22 14)(12 28 17)(29 44 48)(31 42 46)(33 37 45)(35 39 47)
(1 29 3 31)(2 35 4 33)(5 24 7 22)(6 28 8 26)(9 30 11 32)(10 34 12 36)(13 44 15 42)(14 40 16 38)(17 41 19 43)(18 39 20 37)(21 45 23 47)(25 48 27 46)

G:=sub<Sym(48)| (1,13,25)(2,14,26)(3,15,27)(4,16,28)(5,34,39)(6,35,40)(7,36,37)(8,33,38)(9,17,21)(10,18,22)(11,19,23)(12,20,24)(29,44,48)(30,41,45)(31,42,46)(32,43,47), (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)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,11,3,9)(2,10,4,12)(5,45,7,47)(6,48,8,46)(13,19,15,17)(14,18,16,20)(21,25,23,27)(22,28,24,26)(29,33,31,35)(30,36,32,34)(37,43,39,41)(38,42,40,44), (1,25,13)(2,23,18)(3,27,15)(4,21,20)(5,32,40)(6,34,43)(7,30,38)(8,36,41)(9,24,16)(10,26,19)(11,22,14)(12,28,17)(29,44,48)(31,42,46)(33,37,45)(35,39,47), (1,29,3,31)(2,35,4,33)(5,24,7,22)(6,28,8,26)(9,30,11,32)(10,34,12,36)(13,44,15,42)(14,40,16,38)(17,41,19,43)(18,39,20,37)(21,45,23,47)(25,48,27,46)>;

G:=Group( (1,13,25)(2,14,26)(3,15,27)(4,16,28)(5,34,39)(6,35,40)(7,36,37)(8,33,38)(9,17,21)(10,18,22)(11,19,23)(12,20,24)(29,44,48)(30,41,45)(31,42,46)(32,43,47), (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)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,11,3,9)(2,10,4,12)(5,45,7,47)(6,48,8,46)(13,19,15,17)(14,18,16,20)(21,25,23,27)(22,28,24,26)(29,33,31,35)(30,36,32,34)(37,43,39,41)(38,42,40,44), (1,25,13)(2,23,18)(3,27,15)(4,21,20)(5,32,40)(6,34,43)(7,30,38)(8,36,41)(9,24,16)(10,26,19)(11,22,14)(12,28,17)(29,44,48)(31,42,46)(33,37,45)(35,39,47), (1,29,3,31)(2,35,4,33)(5,24,7,22)(6,28,8,26)(9,30,11,32)(10,34,12,36)(13,44,15,42)(14,40,16,38)(17,41,19,43)(18,39,20,37)(21,45,23,47)(25,48,27,46) );

G=PermutationGroup([(1,13,25),(2,14,26),(3,15,27),(4,16,28),(5,34,39),(6,35,40),(7,36,37),(8,33,38),(9,17,21),(10,18,22),(11,19,23),(12,20,24),(29,44,48),(30,41,45),(31,42,46),(32,43,47)], [(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),(37,38,39,40),(41,42,43,44),(45,46,47,48)], [(1,11,3,9),(2,10,4,12),(5,45,7,47),(6,48,8,46),(13,19,15,17),(14,18,16,20),(21,25,23,27),(22,28,24,26),(29,33,31,35),(30,36,32,34),(37,43,39,41),(38,42,40,44)], [(1,25,13),(2,23,18),(3,27,15),(4,21,20),(5,32,40),(6,34,43),(7,30,38),(8,36,41),(9,24,16),(10,26,19),(11,22,14),(12,28,17),(29,44,48),(31,42,46),(33,37,45),(35,39,47)], [(1,29,3,31),(2,35,4,33),(5,24,7,22),(6,28,8,26),(9,30,11,32),(10,34,12,36),(13,44,15,42),(14,40,16,38),(17,41,19,43),(18,39,20,37),(21,45,23,47),(25,48,27,46)])

C3×CSU2(𝔽3) is a maximal subgroup of   CSU2(𝔽3)⋊S3  Dic3.5S4  D6.2S4  C322CSU2(𝔽3)
C3×CSU2(𝔽3) is a maximal quotient of   C32.CSU2(𝔽3)  C32⋊CSU2(𝔽3)

Matrix representation of C3×CSU2(𝔽3) in GL2(𝔽7) generated by

20
02
,
65
11
,
56
52
,
20
54
,
25
65
G:=sub<GL(2,GF(7))| [2,0,0,2],[6,1,5,1],[5,5,6,2],[2,5,0,4],[2,6,5,5] >;

C3×CSU2(𝔽3) in GAP, Magma, Sage, TeX

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

G:=Group("C3xCSU(2,3)");
// GroupNames label

G:=SmallGroup(144,121);
// by ID

G=gap.SmallGroup(144,121);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,2,-2,432,218,867,447,117,544,286,202,88]);
// Polycyclic

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

Export

Subgroup lattice of C3×CSU2(𝔽3) in TeX
Character table of C3×CSU2(𝔽3) in TeX

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