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G = C2×C4.A4order 96 = 25·3

Direct product of C2 and C4.A4

direct product, non-abelian, soluble

Aliases: C2×C4.A4, SL2(𝔽3)⋊3C22, C4(C4.A4), C4○D42C6, C4.6(C2×A4), (C2×C4).2A4, Q8.1(C2×C6), (C2×Q8).2C6, C22.9(C2×A4), C2.5(C22×A4), (C2×C4)SL2(𝔽3), C4(C2×SL2(𝔽3)), (C2×SL2(𝔽3))⋊4C2, (C2×C4○D4)⋊C3, (C2×C4)(C2×SL2(𝔽3)), SmallGroup(96,200)

Series: Derived Chief Lower central Upper central

C1C2Q8 — C2×C4.A4
C1C2Q8SL2(𝔽3)C2×SL2(𝔽3) — C2×C4.A4
Q8 — C2×C4.A4
C1C2×C4

Generators and relations for C2×C4.A4
 G = < a,b,c,d,e | a2=b4=e3=1, c2=d2=b2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=b2c, ece-1=b2cd, ede-1=c >

6C2
6C2
4C3
3C22
3C4
3C22
3C4
6C22
6C22
4C6
4C6
4C6
3D4
3D4
3Q8
3C2×C4
3C2×C4
3C2×C4
3C2×C4
3C23
3C2×C4
3D4
3D4
4C12
4C12
4C2×C6
3C4○D4
3C4○D4
3C2×D4
3C22×C4
4C2×C12

Character table of C2×C4.A4

 class 12A2B2C2D2E3A3B4A4B4C4D4E4F6A6B6C6D6E6F12A12B12C12D12E12F12G12H
 size 1111664411116644444444444444
ρ11111111111111111111111111111    trivial
ρ21-11-1-1111-11-111-11-1-11-1-11111-1-1-1-1    linear of order 2
ρ31-11-11-1111-11-11-11-1-11-1-1-1-1-1-11111    linear of order 2
ρ41111-1-111-1-1-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-1-1ζ32ζ3-1-1-1-111ζ3ζ32ζ32ζ32ζ3ζ3ζ6ζ6ζ65ζ65ζ65ζ65ζ6ζ6    linear of order 6
ρ6111111ζ32ζ3111111ζ3ζ32ζ32ζ32ζ3ζ3ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ71-11-11-1ζ3ζ321-11-11-1ζ32ζ65ζ65ζ3ζ6ζ6ζ65ζ65ζ6ζ6ζ32ζ32ζ3ζ3    linear of order 6
ρ81-11-1-11ζ3ζ32-11-111-1ζ32ζ65ζ65ζ3ζ6ζ6ζ3ζ3ζ32ζ32ζ6ζ6ζ65ζ65    linear of order 6
ρ91-11-1-11ζ32ζ3-11-111-1ζ3ζ6ζ6ζ32ζ65ζ65ζ32ζ32ζ3ζ3ζ65ζ65ζ6ζ6    linear of order 6
ρ101-11-11-1ζ32ζ31-11-11-1ζ3ζ6ζ6ζ32ζ65ζ65ζ6ζ6ζ65ζ65ζ3ζ3ζ32ζ32    linear of order 6
ρ11111111ζ3ζ32111111ζ32ζ3ζ3ζ3ζ32ζ32ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ121111-1-1ζ3ζ32-1-1-1-111ζ32ζ3ζ3ζ3ζ32ζ32ζ65ζ65ζ6ζ6ζ6ζ6ζ65ζ65    linear of order 6
ρ132-2-2200-1-12i-2i-2i2i0011-111-1-ii-iii-ii-i    complex lifted from C4.A4
ρ1422-2-200-1-1-2i-2i2i2i001-111-11-ii-ii-ii-ii    complex lifted from C4.A4
ρ152-2-2200-1-1-2i2i2i-2i0011-111-1i-ii-i-ii-ii    complex lifted from C4.A4
ρ1622-2-200-1-12i2i-2i-2i001-111-11i-ii-ii-ii-i    complex lifted from C4.A4
ρ172-2-2200ζ6ζ652i-2i-2i2i00ζ3ζ32ζ6ζ32ζ3ζ65ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32    complex lifted from C4.A4
ρ1822-2-200ζ65ζ6-2i-2i2i2i00ζ32ζ65ζ3ζ3ζ6ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3    complex lifted from C4.A4
ρ192-2-2200ζ65ζ62i-2i-2i2i00ζ32ζ3ζ65ζ3ζ32ζ6ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3    complex lifted from C4.A4
ρ202-2-2200ζ6ζ65-2i2i2i-2i00ζ3ζ32ζ6ζ32ζ3ζ65ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32    complex lifted from C4.A4
ρ2122-2-200ζ6ζ652i2i-2i-2i00ζ3ζ6ζ32ζ32ζ65ζ3ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32    complex lifted from C4.A4
ρ222-2-2200ζ65ζ6-2i2i2i-2i00ζ32ζ3ζ65ζ3ζ32ζ6ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3    complex lifted from C4.A4
ρ2322-2-200ζ65ζ62i2i-2i-2i00ζ32ζ65ζ3ζ3ζ6ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3    complex lifted from C4.A4
ρ2422-2-200ζ6ζ65-2i-2i2i2i00ζ3ζ6ζ32ζ32ζ65ζ3ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32    complex lifted from C4.A4
ρ253-33-3-11003-33-3-1100000000000000    orthogonal lifted from C2×A4
ρ263333-1-1003333-1-100000000000000    orthogonal lifted from A4
ρ273-33-31-100-33-33-1100000000000000    orthogonal lifted from C2×A4
ρ2833331100-3-3-3-3-1-100000000000000    orthogonal lifted from C2×A4

Smallest permutation representation of C2×C4.A4
On 32 points
Generators in S32
(1 13)(2 14)(3 15)(4 16)(5 26)(6 27)(7 28)(8 25)(9 17)(10 18)(11 19)(12 20)(21 32)(22 29)(23 30)(24 31)
(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 27 3 25)(2 28 4 26)(5 14 7 16)(6 15 8 13)(9 31 11 29)(10 32 12 30)(17 24 19 22)(18 21 20 23)
(1 29 3 31)(2 30 4 32)(5 18 7 20)(6 19 8 17)(9 27 11 25)(10 28 12 26)(13 22 15 24)(14 23 16 21)
(5 21 20)(6 22 17)(7 23 18)(8 24 19)(9 27 29)(10 28 30)(11 25 31)(12 26 32)

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

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

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

C2×C4.A4 is a maximal subgroup of
U2(𝔽3)⋊C2  C4.A4⋊C4  SL2(𝔽3).D4  (C2×C4).S4  SL2(𝔽3)⋊D4  C4○D4⋊C12  SL2(𝔽3)⋊5D4  SL2(𝔽3)⋊6D4  M4(2).A4  GL2(𝔽3)⋊C22  2- 1+43C6
C2×C4.A4 is a maximal quotient of
C2×C4×SL2(𝔽3)  SL2(𝔽3)⋊5D4  SL2(𝔽3)⋊6D4  SL2(𝔽3)⋊3Q8

Matrix representation of C2×C4.A4 in GL3(𝔽13) generated by

1200
010
001
,
100
050
005
,
100
093
034
,
100
001
0120
,
300
010
093
G:=sub<GL(3,GF(13))| [12,0,0,0,1,0,0,0,1],[1,0,0,0,5,0,0,0,5],[1,0,0,0,9,3,0,3,4],[1,0,0,0,0,12,0,1,0],[3,0,0,0,1,9,0,0,3] >;

C2×C4.A4 in GAP, Magma, Sage, TeX

C_2\times C_4.A_4
% in TeX

G:=Group("C2xC4.A4");
// GroupNames label

G:=SmallGroup(96,200);
// by ID

G=gap.SmallGroup(96,200);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,2,-2,295,159,117,286,202,88]);
// Polycyclic

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

Export

Subgroup lattice of C2×C4.A4 in TeX
Character table of C2×C4.A4 in TeX

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