Aliases: GL2(𝔽3)⋊1C4, C2.6(C4×S4), (C2×C4).3S4, (C4×Q8)⋊3S3, Q8.3(C4×S3), Q8⋊Dic3⋊5C2, (C2×Q8).10D6, C22.12(C2×S4), C2.1(C4.3S4), (C4×SL2(𝔽3))⋊1C2, SL2(𝔽3)⋊2(C2×C4), C2.2(Q8.D6), (C2×GL2(𝔽3)).1C2, (C2×SL2(𝔽3)).10C22, SmallGroup(192,953)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C2×GL2(𝔽3) — GL2(𝔽3)⋊C4 |
| SL2(𝔽3) — GL2(𝔽3)⋊C4 |
Generators and relations for GL2(𝔽3)⋊C4
G = < a,b,c,d,e | a4=c3=d2=e4=1, b2=a2, bab-1=dbd=a-1, cac-1=ab, dad=a2b, ae=ea, cbc-1=a, be=eb, dcd=c-1, ce=ec, ede-1=a2d >
Subgroups: 327 in 77 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, SL2(𝔽3), C2×Dic3, C2×C12, C22×S3, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, D6⋊C4, GL2(𝔽3), C2×SL2(𝔽3), SD16⋊C4, Q8⋊Dic3, C4×SL2(𝔽3), C2×GL2(𝔽3), GL2(𝔽3)⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, C4×S3, S4, C2×S4, C4×S4, Q8.D6, C4.3S4, GL2(𝔽3)⋊C4
Character table of GL2(𝔽3)⋊C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 8 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | 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 | 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 | 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 | -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 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | -1 | -i | 1 | i | i | -i | -1 | 1 | -1 | i | -i | 1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | i | -1 | -i | 1 | i | -i | i | -1 | 1 | -1 | -i | i | -1 | 1 | i | i | -i | -i | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | i | -i | -1 | i | 1 | -i | -i | i | -1 | 1 | -1 | -i | i | 1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | i | -i | -1 | i | 1 | -i | i | -i | -1 | 1 | -1 | i | -i | -1 | 1 | -i | -i | i | i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2i | -2i | -2 | 2i | 2 | -2i | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | i | i | -i | -i | complex lifted from C4×S3 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2i | 2i | -2 | -2i | 2 | 2i | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | -i | -i | i | i | complex lifted from C4×S3 |
| ρ13 | 3 | 3 | 3 | 3 | 1 | 1 | 0 | -3 | -3 | -1 | 1 | -1 | 1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ14 | 3 | 3 | 3 | 3 | 1 | 1 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ15 | 3 | 3 | 3 | 3 | -1 | -1 | 0 | -3 | -3 | -1 | 1 | -1 | 1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ16 | 3 | 3 | 3 | 3 | -1 | -1 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ17 | 3 | 3 | -3 | -3 | -1 | 1 | 0 | -3i | 3i | 1 | i | -1 | -i | -i | i | 0 | 0 | 0 | i | -i | 1 | -1 | 0 | 0 | 0 | 0 | complex lifted from C4×S4 |
| ρ18 | 3 | 3 | -3 | -3 | -1 | 1 | 0 | 3i | -3i | 1 | -i | -1 | i | i | -i | 0 | 0 | 0 | -i | i | 1 | -1 | 0 | 0 | 0 | 0 | complex lifted from C4×S4 |
| ρ19 | 3 | 3 | -3 | -3 | 1 | -1 | 0 | 3i | -3i | 1 | -i | -1 | i | -i | i | 0 | 0 | 0 | i | -i | -1 | 1 | 0 | 0 | 0 | 0 | complex lifted from C4×S4 |
| ρ20 | 3 | 3 | -3 | -3 | 1 | -1 | 0 | -3i | 3i | 1 | i | -1 | -i | i | -i | 0 | 0 | 0 | -i | i | -1 | 1 | 0 | 0 | 0 | 0 | complex lifted from C4×S4 |
| ρ21 | 4 | -4 | -4 | 4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | -√3 | √3 | √3 | -√3 | orthogonal lifted from C4.3S4 |
| ρ22 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.3S4 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | √3 | -√3 | -√3 | √3 | orthogonal lifted from C4.3S4 |
| ρ24 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8.D6, Schur index 2 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | -√-3 | √-3 | -√-3 | √-3 | complex lifted from Q8.D6 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | √-3 | -√-3 | √-3 | -√-3 | complex lifted from Q8.D6 |
►On 32 points
Generators in S32
(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)
(1 11 3 9)(2 10 4 12)(5 32 7 30)(6 31 8 29)(13 20 15 18)(14 19 16 17)(21 28 23 26)(22 27 24 25)
(2 11 10)(4 9 12)(5 8 31)(6 29 7)(13 17 20)(15 19 18)(21 25 28)(23 27 26)
(1 14)(2 17)(3 16)(4 19)(5 23)(6 28)(7 21)(8 26)(9 15)(10 20)(11 13)(12 18)(22 32)(24 30)(25 29)(27 31)
(1 30 14 22)(2 31 15 23)(3 32 16 24)(4 29 13 21)(5 19 27 11)(6 20 28 12)(7 17 25 9)(8 18 26 10)
G:=sub<Sym(32)| (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), (1,11,3,9)(2,10,4,12)(5,32,7,30)(6,31,8,29)(13,20,15,18)(14,19,16,17)(21,28,23,26)(22,27,24,25), (2,11,10)(4,9,12)(5,8,31)(6,29,7)(13,17,20)(15,19,18)(21,25,28)(23,27,26), (1,14)(2,17)(3,16)(4,19)(5,23)(6,28)(7,21)(8,26)(9,15)(10,20)(11,13)(12,18)(22,32)(24,30)(25,29)(27,31), (1,30,14,22)(2,31,15,23)(3,32,16,24)(4,29,13,21)(5,19,27,11)(6,20,28,12)(7,17,25,9)(8,18,26,10)>;
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), (1,11,3,9)(2,10,4,12)(5,32,7,30)(6,31,8,29)(13,20,15,18)(14,19,16,17)(21,28,23,26)(22,27,24,25), (2,11,10)(4,9,12)(5,8,31)(6,29,7)(13,17,20)(15,19,18)(21,25,28)(23,27,26), (1,14)(2,17)(3,16)(4,19)(5,23)(6,28)(7,21)(8,26)(9,15)(10,20)(11,13)(12,18)(22,32)(24,30)(25,29)(27,31), (1,30,14,22)(2,31,15,23)(3,32,16,24)(4,29,13,21)(5,19,27,11)(6,20,28,12)(7,17,25,9)(8,18,26,10) );
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)], [(1,11,3,9),(2,10,4,12),(5,32,7,30),(6,31,8,29),(13,20,15,18),(14,19,16,17),(21,28,23,26),(22,27,24,25)], [(2,11,10),(4,9,12),(5,8,31),(6,29,7),(13,17,20),(15,19,18),(21,25,28),(23,27,26)], [(1,14),(2,17),(3,16),(4,19),(5,23),(6,28),(7,21),(8,26),(9,15),(10,20),(11,13),(12,18),(22,32),(24,30),(25,29),(27,31)], [(1,30,14,22),(2,31,15,23),(3,32,16,24),(4,29,13,21),(5,19,27,11),(6,20,28,12),(7,17,25,9),(8,18,26,10)]])
Matrix representation of GL2(𝔽3)⋊C4 ►in GL7(𝔽73)
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 46 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 66 | 7 | 66 |
| 0 | 0 | 0 | 7 | 0 | 66 | 66 |
| 0 | 0 | 0 | 66 | 7 | 0 | 66 |
| 0 | 0 | 0 | 7 | 7 | 7 | 0 |
G:=sub<GL(7,GF(73))| [0,72,1,0,0,0,0,0,72,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,0,1,0,0],[0,1,72,0,0,0,0,1,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,1,0],[1,0,72,0,0,0,0,0,0,72,0,0,0,0,0,1,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1],[46,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,7,66,7,0,0,0,66,0,7,7,0,0,0,7,66,0,7,0,0,0,66,66,66,0] >;
GL2(𝔽3)⋊C4 in GAP, Magma, Sage, TeX
{\rm GL}_2({\mathbb F}_3)\rtimes C_4 % in TeX
G:=Group("GL(2,3):C4"); // GroupNames label
G:=SmallGroup(192,953);
// by ID
G=gap.SmallGroup(192,953);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,1373,36,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^3=d^2=e^4=1,b^2=a^2,b*a*b^-1=d*b*d=a^-1,c*a*c^-1=a*b,d*a*d=a^2*b,a*e=e*a,c*b*c^-1=a,b*e=e*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=a^2*d>;
// generators/relations
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