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G = C8.A4order 96 = 25·3

The central extension by C8 of A4

non-abelian, soluble

Aliases: C8.A4, Q8.C12, C8SL2(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

C1C2Q8 — C8.A4
C1C2Q8C4○D4C4.A4 — C8.A4
Q8 — C8.A4
C1C8

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 >

6C2
4C3
3C22
3C4
4C6
3C2×C4
3C8
3D4
4C12
3M4(2)
3C2×C8
4C24

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-2i011885838700-i-iiiζ83ζ8ζ87ζ83ζ85ζ8ζ87ζ85    complex faithful
ρ142-20-1-1-2i2i011838788500ii-i-iζ8ζ83ζ85ζ8ζ87ζ83ζ85ζ87    complex faithful
ρ152-20-1-1-2i2i011878385800ii-i-iζ85ζ87ζ8ζ85ζ83ζ87ζ8ζ83    complex faithful
ρ162-20-1-12i-2i011858878300-i-iiiζ87ζ85ζ83ζ87ζ8ζ85ζ83ζ8    complex faithful
ρ172-20ζ6ζ65-2i2i0ζ3ζ32878385800ζ82ζ32ζ82ζ3ζ86ζ32ζ86ζ3ζ85ζ32ζ87ζ3ζ8ζ3ζ85ζ3ζ83ζ32ζ87ζ32ζ8ζ32ζ83ζ3    complex faithful
ρ182-20ζ65ζ6-2i2i0ζ32ζ3838788500ζ82ζ3ζ82ζ32ζ86ζ3ζ86ζ32ζ8ζ3ζ83ζ32ζ85ζ32ζ8ζ32ζ87ζ3ζ83ζ3ζ85ζ3ζ87ζ32    complex faithful
ρ192-20ζ65ζ6-2i2i0ζ32ζ3878385800ζ82ζ3ζ82ζ32ζ86ζ3ζ86ζ32ζ85ζ3ζ87ζ32ζ8ζ32ζ85ζ32ζ83ζ3ζ87ζ3ζ8ζ3ζ83ζ32    complex faithful
ρ202-20ζ6ζ65-2i2i0ζ3ζ32838788500ζ82ζ32ζ82ζ3ζ86ζ32ζ86ζ3ζ8ζ32ζ83ζ3ζ85ζ3ζ8ζ3ζ87ζ32ζ83ζ32ζ85ζ32ζ87ζ3    complex faithful
ρ212-20ζ65ζ62i-2i0ζ32ζ3858878300ζ86ζ3ζ86ζ32ζ82ζ3ζ82ζ32ζ87ζ3ζ85ζ32ζ83ζ32ζ87ζ32ζ8ζ3ζ85ζ3ζ83ζ3ζ8ζ32    complex faithful
ρ222-20ζ6ζ652i-2i0ζ3ζ32885838700ζ86ζ32ζ86ζ3ζ82ζ32ζ82ζ3ζ83ζ32ζ8ζ3ζ87ζ3ζ83ζ3ζ85ζ32ζ8ζ32ζ87ζ32ζ85ζ3    complex faithful
ρ232-20ζ65ζ62i-2i0ζ32ζ3885838700ζ86ζ3ζ86ζ32ζ82ζ3ζ82ζ32ζ83ζ3ζ8ζ32ζ87ζ32ζ83ζ32ζ85ζ3ζ8ζ3ζ87ζ3ζ85ζ32    complex faithful
ρ242-20ζ6ζ652i-2i0ζ3ζ32858878300ζ86ζ32ζ86ζ3ζ82ζ32ζ82ζ3ζ87ζ32ζ85ζ3ζ83ζ3ζ87ζ3ζ8ζ32ζ85ζ32ζ83ζ32ζ8ζ3    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 15 5 11)(2 16 6 12)(3 9 7 13)(4 10 8 14)(17 27 21 31)(18 28 22 32)(19 29 23 25)(20 30 24 26)
(1 23 5 19)(2 24 6 20)(3 17 7 21)(4 18 8 22)(9 31 13 27)(10 32 14 28)(11 25 15 29)(12 26 16 30)
(9 17 27)(10 18 28)(11 19 29)(12 20 30)(13 21 31)(14 22 32)(15 23 25)(16 24 26)

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

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

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

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

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