direct product, non-abelian, soluble, monomial
Aliases: C4×PSU3(𝔽2), C32⋊(C4×Q8), (C3×C12)⋊1Q8, C3⋊Dic3⋊2Q8, C2.1(C2×PSU3(𝔽2)), (C2×PSU3(𝔽2)).3C2, C2.PSU3(𝔽2).4C2, (C3×C6).2(C2×Q8), (C4×C32⋊C4).9C2, C32⋊C4.5(C2×C4), C3⋊S3.6(C4○D4), C3⋊S3.2(C22×C4), (C2×C3⋊S3).13C23, (C4×C3⋊S3).63C22, (C2×C32⋊C4).19C22, SmallGroup(288,892)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×PSU3(𝔽2)
G = < a,b,c,d,e | a4=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ece-1=bc=cb, dbd-1=c-1, ebe-1=b-1c, dcd-1=b, ede-1=d-1 >
Subgroups: 428 in 78 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C2×C4, Q8, C32, Dic3, C12, D6, C42, C4⋊C4, C2×Q8, C3⋊S3, C3×C6, C4×S3, C4×Q8, C3⋊Dic3, C3×C12, C32⋊C4, C32⋊C4, C2×C3⋊S3, C4×C3⋊S3, PSU3(𝔽2), C2×C32⋊C4, C2.PSU3(𝔽2), C4×C32⋊C4, C2×PSU3(𝔽2), C4×PSU3(𝔽2)
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, C22×C4, C2×Q8, C4○D4, C4×Q8, PSU3(𝔽2), C2×PSU3(𝔽2), C4×PSU3(𝔽2)
Character table of C4×PSU3(𝔽2)
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 6 | 12A | 12B | |
| size | 1 | 1 | 9 | 9 | 8 | 1 | 1 | 9 | 9 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ9 | 1 | -1 | -1 | 1 | 1 | -i | i | -i | i | -i | -1 | -i | i | -1 | 1 | i | -i | -1 | 1 | i | 1 | -1 | -i | i | linear of order 4 |
| ρ10 | 1 | -1 | -1 | 1 | 1 | -i | i | -i | i | i | 1 | i | -i | -1 | 1 | i | -i | 1 | -1 | -i | -1 | -1 | -i | i | linear of order 4 |
| ρ11 | 1 | -1 | -1 | 1 | 1 | -i | i | -i | i | -i | 1 | i | -i | 1 | -1 | -i | i | -1 | 1 | i | -1 | -1 | -i | i | linear of order 4 |
| ρ12 | 1 | -1 | -1 | 1 | 1 | -i | i | -i | i | i | -1 | -i | i | 1 | -1 | -i | i | 1 | -1 | -i | 1 | -1 | -i | i | linear of order 4 |
| ρ13 | 1 | -1 | -1 | 1 | 1 | i | -i | i | -i | i | -1 | i | -i | -1 | 1 | -i | i | -1 | 1 | -i | 1 | -1 | i | -i | linear of order 4 |
| ρ14 | 1 | -1 | -1 | 1 | 1 | i | -i | i | -i | -i | 1 | -i | i | -1 | 1 | -i | i | 1 | -1 | i | -1 | -1 | i | -i | linear of order 4 |
| ρ15 | 1 | -1 | -1 | 1 | 1 | i | -i | i | -i | i | 1 | -i | i | 1 | -1 | i | -i | -1 | 1 | -i | -1 | -1 | i | -i | linear of order 4 |
| ρ16 | 1 | -1 | -1 | 1 | 1 | i | -i | i | -i | -i | -1 | i | -i | 1 | -1 | i | -i | 1 | -1 | i | 1 | -1 | i | -i | linear of order 4 |
| ρ17 | 2 | 2 | -2 | -2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | symplectic lifted from Q8, Schur index 2 |
| ρ18 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | symplectic lifted from Q8, Schur index 2 |
| ρ19 | 2 | -2 | 2 | -2 | 2 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2i | -2i | complex lifted from C4○D4 |
| ρ20 | 2 | -2 | 2 | -2 | 2 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2i | 2i | complex lifted from C4○D4 |
| ρ21 | 8 | 8 | 0 | 0 | -1 | -8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | orthogonal lifted from C2×PSU3(𝔽2) |
| ρ22 | 8 | 8 | 0 | 0 | -1 | 8 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from PSU3(𝔽2) |
| ρ23 | 8 | -8 | 0 | 0 | -1 | -8i | 8i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | i | -i | complex faithful |
| ρ24 | 8 | -8 | 0 | 0 | -1 | 8i | -8i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -i | i | complex faithful |
►On 36 points
Generators in S36
(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)
(1 7 16)(2 8 13)(3 5 14)(4 6 15)(9 18 29)(10 19 30)(11 20 31)(12 17 32)(21 33 27)(22 34 28)(23 35 25)(24 36 26)
(1 29 21)(2 30 22)(3 31 23)(4 32 24)(5 11 35)(6 12 36)(7 9 33)(8 10 34)(13 19 28)(14 20 25)(15 17 26)(16 18 27)
(1 3)(2 4)(5 29 14 21)(6 30 15 22)(7 31 16 23)(8 32 13 24)(9 20 27 35)(10 17 28 36)(11 18 25 33)(12 19 26 34)
(5 35 14 20)(6 36 15 17)(7 33 16 18)(8 34 13 19)(9 29 27 21)(10 30 28 22)(11 31 25 23)(12 32 26 24)
G:=sub<Sym(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), (1,7,16)(2,8,13)(3,5,14)(4,6,15)(9,18,29)(10,19,30)(11,20,31)(12,17,32)(21,33,27)(22,34,28)(23,35,25)(24,36,26), (1,29,21)(2,30,22)(3,31,23)(4,32,24)(5,11,35)(6,12,36)(7,9,33)(8,10,34)(13,19,28)(14,20,25)(15,17,26)(16,18,27), (1,3)(2,4)(5,29,14,21)(6,30,15,22)(7,31,16,23)(8,32,13,24)(9,20,27,35)(10,17,28,36)(11,18,25,33)(12,19,26,34), (5,35,14,20)(6,36,15,17)(7,33,16,18)(8,34,13,19)(9,29,27,21)(10,30,28,22)(11,31,25,23)(12,32,26,24)>;
G:=Group( (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), (1,7,16)(2,8,13)(3,5,14)(4,6,15)(9,18,29)(10,19,30)(11,20,31)(12,17,32)(21,33,27)(22,34,28)(23,35,25)(24,36,26), (1,29,21)(2,30,22)(3,31,23)(4,32,24)(5,11,35)(6,12,36)(7,9,33)(8,10,34)(13,19,28)(14,20,25)(15,17,26)(16,18,27), (1,3)(2,4)(5,29,14,21)(6,30,15,22)(7,31,16,23)(8,32,13,24)(9,20,27,35)(10,17,28,36)(11,18,25,33)(12,19,26,34), (5,35,14,20)(6,36,15,17)(7,33,16,18)(8,34,13,19)(9,29,27,21)(10,30,28,22)(11,31,25,23)(12,32,26,24) );
G=PermutationGroup([[(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)], [(1,7,16),(2,8,13),(3,5,14),(4,6,15),(9,18,29),(10,19,30),(11,20,31),(12,17,32),(21,33,27),(22,34,28),(23,35,25),(24,36,26)], [(1,29,21),(2,30,22),(3,31,23),(4,32,24),(5,11,35),(6,12,36),(7,9,33),(8,10,34),(13,19,28),(14,20,25),(15,17,26),(16,18,27)], [(1,3),(2,4),(5,29,14,21),(6,30,15,22),(7,31,16,23),(8,32,13,24),(9,20,27,35),(10,17,28,36),(11,18,25,33),(12,19,26,34)], [(5,35,14,20),(6,36,15,17),(7,33,16,18),(8,34,13,19),(9,29,27,21),(10,30,28,22),(11,31,25,23),(12,32,26,24)]])
Matrix representation of C4×PSU3(𝔽2) ►in GL8(𝔽13)
| 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 0 | 12 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 12 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
G:=sub<GL(8,GF(13))| [5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5],[0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,12,12,12,12,12,12,12,12,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,12,12,12,12,12,12,12,12,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0] >;
C4×PSU3(𝔽2) in GAP, Magma, Sage, TeX
C_4\times {\rm PSU}_3({\mathbb F}_2) % in TeX
G:=Group("C4xPSU(3,2)"); // GroupNames label
G:=SmallGroup(288,892);
// by ID
G=gap.SmallGroup(288,892);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,141,64,100,9413,2028,362,12550,1581,1203]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*c*e^-1=b*c=c*b,d*b*d^-1=c^-1,e*b*e^-1=b^-1*c,d*c*d^-1=b,e*d*e^-1=d^-1>;
// generators/relations
Export