Copied to
clipboard

G = C4×SL2(𝔽3)  order 96 = 25·3

Direct product of C4 and SL2(𝔽3)

direct product, non-abelian, soluble

Aliases: C4×SL2(𝔽3), Q8⋊C12, (C4×Q8)⋊C3, C2.2(C4×A4), (C2×C4).1A4, C2.(C4.A4), (C2×Q8).1C6, C22.6(C2×A4), C2.(C2×SL2(𝔽3)), (C2×SL2(𝔽3)).3C2, SmallGroup(96,69)

Series: Derived Chief Lower central Upper central

C1C2Q8 — C4×SL2(𝔽3)
C1C2Q8C2×Q8C2×SL2(𝔽3) — C4×SL2(𝔽3)
Q8 — C4×SL2(𝔽3)
C1C2×C4

Generators and relations for C4×SL2(𝔽3)
 G = < a,b,c,d | a4=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >

4C3
3C4
3C4
6C4
4C6
4C6
4C6
3C2×C4
3Q8
3C2×C4
4C2×C6
4C12
4C12
3C4⋊C4
3C42
4C2×C12

Character table of C4×SL2(𝔽3)

 class 12A2B2C3A3B4A4B4C4D4E4F4G4H6A6B6C6D6E6F12A12B12C12D12E12F12G12H
 size 1111441111666644444444444444
ρ11111111111111111111111111111    trivial
ρ2111111-1-1-1-111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111ζ32ζ311111111ζ3ζ32ζ3ζ32ζ32ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ3ζ32    linear of order 3
ρ41111ζ3ζ32-1-1-1-111-1-1ζ32ζ3ζ32ζ3ζ3ζ32ζ6ζ65ζ65ζ65ζ6ζ6ζ6ζ65    linear of order 6
ρ51111ζ3ζ3211111111ζ32ζ3ζ32ζ3ζ3ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ32ζ3    linear of order 3
ρ61111ζ32ζ3-1-1-1-111-1-1ζ3ζ32ζ3ζ32ζ32ζ3ζ65ζ6ζ6ζ6ζ65ζ65ζ65ζ6    linear of order 6
ρ71-1-1111ii-i-i-11i-i11-1-1-1-1-ii-i-iii-ii    linear of order 4
ρ81-1-1111-i-iii-11-ii11-1-1-1-1i-iii-i-ii-i    linear of order 4
ρ91-1-11ζ3ζ32-i-iii-11-iiζ32ζ3ζ6ζ65ζ65ζ6ζ4ζ32ζ43ζ3ζ4ζ3ζ4ζ3ζ43ζ32ζ43ζ32ζ4ζ32ζ43ζ3    linear of order 12
ρ101-1-11ζ3ζ32ii-i-i-11i-iζ32ζ3ζ6ζ65ζ65ζ6ζ43ζ32ζ4ζ3ζ43ζ3ζ43ζ3ζ4ζ32ζ4ζ32ζ43ζ32ζ4ζ3    linear of order 12
ρ111-1-11ζ32ζ3-i-iii-11-iiζ3ζ32ζ65ζ6ζ6ζ65ζ4ζ3ζ43ζ32ζ4ζ32ζ4ζ32ζ43ζ3ζ43ζ3ζ4ζ3ζ43ζ32    linear of order 12
ρ121-1-11ζ32ζ3ii-i-i-11i-iζ3ζ32ζ65ζ6ζ6ζ65ζ43ζ3ζ4ζ32ζ43ζ32ζ43ζ32ζ4ζ3ζ4ζ3ζ43ζ3ζ4ζ32    linear of order 12
ρ132-22-2-1-12-22-2000011-1-11111-11-11-1-1    symplectic lifted from SL2(𝔽3), Schur index 2
ρ142-22-2-1-1-22-22000011-1-111-1-11-11-111    symplectic lifted from SL2(𝔽3), Schur index 2
ρ152-22-2ζ65ζ6-22-220000ζ32ζ3ζ6ζ65ζ3ζ32ζ6ζ65ζ3ζ65ζ32ζ6ζ32ζ3    complex lifted from SL2(𝔽3)
ρ162-22-2ζ6ζ65-22-220000ζ3ζ32ζ65ζ6ζ32ζ3ζ65ζ6ζ32ζ6ζ3ζ65ζ3ζ32    complex lifted from SL2(𝔽3)
ρ172-22-2ζ6ζ652-22-20000ζ3ζ32ζ65ζ6ζ32ζ3ζ3ζ32ζ6ζ32ζ65ζ3ζ65ζ6    complex lifted from SL2(𝔽3)
ρ182-22-2ζ65ζ62-22-20000ζ32ζ3ζ6ζ65ζ3ζ32ζ32ζ3ζ65ζ3ζ6ζ32ζ6ζ65    complex lifted from SL2(𝔽3)
ρ1922-2-2-1-1-2i2i2i-2i00001111-1-1i-i-iii-i-ii    complex lifted from C4.A4
ρ2022-2-2-1-12i-2i-2i2i00001111-1-1-iii-i-iii-i    complex lifted from C4.A4
ρ2122-2-2ζ65ζ62i-2i-2i2i0000ζ32ζ3ζ32ζ3ζ65ζ6ζ43ζ32ζ4ζ3ζ4ζ3ζ43ζ3ζ43ζ32ζ4ζ32ζ4ζ32ζ43ζ3    complex lifted from C4.A4
ρ2222-2-2ζ65ζ6-2i2i2i-2i0000ζ32ζ3ζ32ζ3ζ65ζ6ζ4ζ32ζ43ζ3ζ43ζ3ζ4ζ3ζ4ζ32ζ43ζ32ζ43ζ32ζ4ζ3    complex lifted from C4.A4
ρ2322-2-2ζ6ζ652i-2i-2i2i0000ζ3ζ32ζ3ζ32ζ6ζ65ζ43ζ3ζ4ζ32ζ4ζ32ζ43ζ32ζ43ζ3ζ4ζ3ζ4ζ3ζ43ζ32    complex lifted from C4.A4
ρ2422-2-2ζ6ζ65-2i2i2i-2i0000ζ3ζ32ζ3ζ32ζ6ζ65ζ4ζ3ζ43ζ32ζ43ζ32ζ4ζ32ζ4ζ3ζ43ζ3ζ43ζ3ζ4ζ32    complex lifted from C4.A4
ρ25333300-3-3-3-3-1-11100000000000000    orthogonal lifted from C2×A4
ρ263333003333-1-1-1-100000000000000    orthogonal lifted from A4
ρ273-3-33003i3i-3i-3i1-1-ii00000000000000    complex lifted from C4×A4
ρ283-3-3300-3i-3i3i3i1-1i-i00000000000000    complex lifted from C4×A4

Smallest permutation representation of C4×SL2(𝔽3)
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 21 20 28)(2 22 17 25)(3 23 18 26)(4 24 19 27)(5 16 30 9)(6 13 31 10)(7 14 32 11)(8 15 29 12)
(1 15 20 12)(2 16 17 9)(3 13 18 10)(4 14 19 11)(5 25 30 22)(6 26 31 23)(7 27 32 24)(8 28 29 21)
(5 16 22)(6 13 23)(7 14 24)(8 15 21)(9 25 30)(10 26 31)(11 27 32)(12 28 29)

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

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

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

C4×SL2(𝔽3) is a maximal subgroup of
C2.U2(𝔽3)  Q8⋊Dic6  CSU2(𝔽3)⋊C4  Q8.Dic6  Q8.D12  SL2(𝔽3)⋊Q8  Q8⋊D12  GL2(𝔽3)⋊C4  Q8.2D12  (C2×Q8)⋊C12  C4○D4⋊C12  SL2(𝔽3)⋊5D4  SL2(𝔽3)⋊6D4  SL2(𝔽3)⋊3Q8

Matrix representation of C4×SL2(𝔽3) in GL3(𝔽13) generated by

500
080
008
,
100
0104
043
,
100
0012
010
,
300
010
043
G:=sub<GL(3,GF(13))| [5,0,0,0,8,0,0,0,8],[1,0,0,0,10,4,0,4,3],[1,0,0,0,0,1,0,12,0],[3,0,0,0,1,4,0,0,3] >;

C4×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_4\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C4xSL(2,3)");
// GroupNames label

G:=SmallGroup(96,69);
// by ID

G=gap.SmallGroup(96,69);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,2,-2,36,297,117,550,202,88]);
// Polycyclic

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

Export

Subgroup lattice of C4×SL2(𝔽3) in TeX
Character table of C4×SL2(𝔽3) in TeX

׿
×
𝔽