Copied to
clipboard

G = C8.A4  order 96 = 25·3

The central extension by C8 of A4

non-abelian, soluble

Aliases: C8.A4, Q8.C12, C8○SL2(𝔽3), SL2(𝔽3).C4, C8○D4⋊C3, C8○(C4.A4), C4.5(C2×A4), C2.3(C4×A4), C4.A4.3C2, C4○D4.2C6, SmallGroup(96,74)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — Q8 — C8.A4
Chief series
C1 — C2 — Q8 — C4○D4 — C4.A4 — C8.A4
Lower central
Q8 — C8.A4
Upper central
C1 — C8

Generators and relations for C8.A4
 G = < a,b,c,d | a8=d3=1, b2=c2=a4, ab=ba, ac=ca, ad=da, cbc-1=a4b, dbd-1=a4bc, dcd-1=b >

C1
C2
6C2
4C3
C4
3C22
3C4
4C6
C8
Q8
3C2×C4
3C8
3D4
4C12
C4○D4
3M4(2)
3C2×C8
SL2(𝔽3)
4C24
C8○D4
C4.A4
C8.A4

Character table of C8.A4

 class 12A2B3A3B4A4B4C6A6B8A8B8C8D8E8F12A12B12C12D24A24B24C24D24E24F24G24H
 size 1164411644111166444444444444
ρ11111111111111111111111111111    trivial
ρ21111111111-1-1-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111ζ3ζ32111ζ32ζ3111111ζ3ζ32ζ3ζ32ζ3ζ32ζ32ζ32ζ3ζ3ζ3ζ32    linear of order 3
ρ4111ζ3ζ32111ζ32ζ3-1-1-1-1-1-1ζ3ζ32ζ3ζ32ζ65ζ6ζ6ζ6ζ65ζ65ζ65ζ6    linear of order 6
ρ5111ζ32ζ3111ζ3ζ32111111ζ32ζ3ζ32ζ3ζ32ζ3ζ3ζ3ζ32ζ32ζ32ζ3    linear of order 3
ρ6111ζ32ζ3111ζ3ζ32-1-1-1-1-1-1ζ32ζ3ζ32ζ3ζ6ζ65ζ65ζ65ζ6ζ6ζ6ζ65    linear of order 6
ρ711-111-1-1111-i-iiii-i-1-1-1-1i-iii-i-ii-i    linear of order 4
ρ811-111-1-1111ii-i-i-ii-1-1-1-1-ii-i-iii-ii    linear of order 4
ρ911-1ζ3ζ32-1-11ζ32ζ3-i-iiii-iζ65ζ6ζ65ζ6ζ4ζ3ζ43ζ32ζ4ζ32ζ4ζ32ζ43ζ3ζ43ζ3ζ4ζ3ζ43ζ32    linear of order 12
ρ1011-1ζ32ζ3-1-11ζ3ζ32ii-i-i-iiζ6ζ65ζ6ζ65ζ43ζ32ζ4ζ3ζ43ζ3ζ43ζ3ζ4ζ32ζ4ζ32ζ43ζ32ζ4ζ3    linear of order 12
ρ1111-1ζ32ζ3-1-11ζ3ζ32-i-iiii-iζ6ζ65ζ6ζ65ζ4ζ32ζ43ζ3ζ4ζ3ζ4ζ3ζ43ζ32ζ43ζ32ζ4ζ32ζ43ζ3    linear of order 12
ρ1211-1ζ3ζ32-1-11ζ32ζ3ii-i-i-iiζ65ζ6ζ65ζ6ζ43ζ3ζ4ζ32ζ43ζ32ζ43ζ32ζ4ζ3ζ4ζ3ζ43ζ3ζ4ζ32    linear of order 12
ρ132-20-1-12i-2i0112ζ82ζ852ζ832ζ8700-i-iiiζ83ζ8ζ87ζ83ζ85ζ8ζ87ζ85    complex faithful
ρ142-20-1-1-2i2i0112ζ832ζ872ζ82ζ8500ii-i-iζ8ζ83ζ85ζ8ζ87ζ83ζ85ζ87    complex faithful
ρ152-20-1-1-2i2i0112ζ872ζ832ζ852ζ800ii-i-iζ85ζ87ζ8ζ85ζ83ζ87ζ8ζ83    complex faithful
ρ162-20-1-12i-2i0112ζ852ζ82ζ872ζ8300-i-iiiζ87ζ85ζ83ζ87ζ8ζ85ζ83ζ8    complex faithful
ρ172-20ζ6ζ65-2i2i0ζ3ζ322ζ872ζ832ζ852ζ800ζ82ζ32ζ82ζ3ζ86ζ32ζ86ζ3ζ85ζ32ζ87ζ3ζ8ζ3ζ85ζ3ζ83ζ32ζ87ζ32ζ8ζ32ζ83ζ3    complex faithful
ρ182-20ζ65ζ6-2i2i0ζ32ζ32ζ832ζ872ζ82ζ8500ζ82ζ3ζ82ζ32ζ86ζ3ζ86ζ32ζ8ζ3ζ83ζ32ζ85ζ32ζ8ζ32ζ87ζ3ζ83ζ3ζ85ζ3ζ87ζ32    complex faithful
ρ192-20ζ65ζ6-2i2i0ζ32ζ32ζ872ζ832ζ852ζ800ζ82ζ3ζ82ζ32ζ86ζ3ζ86ζ32ζ85ζ3ζ87ζ32ζ8ζ32ζ85ζ32ζ83ζ3ζ87ζ3ζ8ζ3ζ83ζ32    complex faithful
ρ202-20ζ6ζ65-2i2i0ζ3ζ322ζ832ζ872ζ82ζ8500ζ82ζ32ζ82ζ3ζ86ζ32ζ86ζ3ζ8ζ32ζ83ζ3ζ85ζ3ζ8ζ3ζ87ζ32ζ83ζ32ζ85ζ32ζ87ζ3    complex faithful
ρ212-20ζ65ζ62i-2i0ζ32ζ32ζ852ζ82ζ872ζ8300ζ86ζ3ζ86ζ32ζ82ζ3ζ82ζ32ζ87ζ3ζ85ζ32ζ83ζ32ζ87ζ32ζ8ζ3ζ85ζ3ζ83ζ3ζ8ζ32    complex faithful
ρ222-20ζ6ζ652i-2i0ζ3ζ322ζ82ζ852ζ832ζ8700ζ86ζ32ζ86ζ3ζ82ζ32ζ82ζ3ζ83ζ32ζ8ζ3ζ87ζ3ζ83ζ3ζ85ζ32ζ8ζ32ζ87ζ32ζ85ζ3    complex faithful
ρ232-20ζ65ζ62i-2i0ζ32ζ32ζ82ζ852ζ832ζ8700ζ86ζ3ζ86ζ32ζ82ζ3ζ82ζ32ζ83ζ3ζ8ζ32ζ87ζ32ζ83ζ32ζ85ζ3ζ8ζ3ζ87ζ3ζ85ζ32    complex faithful
ρ242-20ζ6ζ652i-2i0ζ3ζ322ζ852ζ82ζ872ζ8300ζ86ζ32ζ86ζ3ζ82ζ32ζ82ζ3ζ87ζ32ζ85ζ3ζ83ζ3ζ87ζ3ζ8ζ32ζ85ζ32ζ83ζ32ζ8ζ3    complex faithful
ρ2533-10033-1003333-1-1000000000000    orthogonal lifted from A4
ρ2633-10033-100-3-3-3-311000000000000    orthogonal lifted from C2×A4
ρ2733100-3-3-1003i3i-3i-3ii-i000000000000    complex lifted from C4×A4
ρ2833100-3-3-100-3i-3i3i3i-ii000000000000    complex lifted from C4×A4

Smallest permutation representation of C8.A4
►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 10 5 14)(2 11 6 15)(3 12 7 16)(4 13 8 9)(17 31 21 27)(18 32 22 28)(19 25 23 29)(20 26 24 30)

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

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

C8.A4 is a maximal subgroup of
 C8.7S4  C16.A4  C8.S4  CU2(𝔽3)  C8.5S4  C8.4S4  C8.3S4  M4(2).A4  Q16.A4  SD16.A4  D8.A4  SL2(𝔽3).Dic3  C8.A5  SL2(𝔽3).Dic5  SL2(𝔽3).F5
C8.A4 is a maximal quotient of
 C8×SL2(𝔽3)  Q8.C36  SL2(𝔽3).Dic3  SL2(𝔽3).Dic5  SL2(𝔽3).F5

Matrix representation of C8.A4 ►in GL2(𝔽17) generated by

20
02
,
40
313
,
135
04
,
1613
130
G:=sub<GL(2,GF(17))| [2,0,0,2],[4,3,0,13],[13,0,5,4],[16,13,13,0] >;
 

C8.A4 in GAP, Magma, Sage, TeX 

C_8.A_4
 
% in TeX
 
G:=Group("C8.A4");
 
// GroupNames label
 
G:=SmallGroup(96,74);
 
// by ID
 
G=gap.SmallGroup(96,74);
 
# by ID
 
G:=PCGroup([6,-2,-3,-2,-2,2,-2,36,158,297,117,550,202,88]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=d^3=1,b^2=c^2=a^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^4*b,d*b*d^-1=a^4*b*c,d*c*d^-1=b>;
 
// generators/relations
 

Export

Subgroup lattice of C8.A4 in TeX
Character table of C8.A4 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁