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G = C23.3A4order 96 = 25·3

1st non-split extension by C23 of A4 acting via A4/C22=C3

non-abelian, soluble

Aliases: C23.3A4, C22.SL2(𝔽3), C2.(C42⋊C3), C2.C42⋊C3, SmallGroup(96,3)

Series: Derived Chief Lower central Upper central

C1C2C2.C42 — C23.3A4
C1C2C23C2.C42 — C23.3A4
C2.C42 — C23.3A4
C1C2

Generators and relations for C23.3A4
 G = < a,b,c,d,e,f | a2=b2=c2=f3=1, d2=faf-1=abc, e2=fbf-1=a, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede-1=cd=dc, ce=ec, cf=fc, fdf-1=ade, fef-1=bcd >

3C2
3C2
16C3
3C22
3C22
6C4
6C4
16C6
3C2×C4
3C2×C4
6C2×C4
6C2×C4
4A4
3C22×C4
4C2×A4

Character table of C23.3A4

 class 12A2B2C3A3B4A4B4C4D6A6B
 size 1133161666661616
ρ1111111111111    trivial
ρ21111ζ3ζ321111ζ32ζ3    linear of order 3
ρ31111ζ32ζ31111ζ3ζ32    linear of order 3
ρ42-2-22-1-1000011    symplectic lifted from SL2(𝔽3), Schur index 2
ρ52-2-22ζ6ζ650000ζ3ζ32    complex lifted from SL2(𝔽3)
ρ62-2-22ζ65ζ60000ζ32ζ3    complex lifted from SL2(𝔽3)
ρ7333300-1-1-1-100    orthogonal lifted from A4
ρ833-1-1001-1-2i1-1+2i00    complex lifted from C42⋊C3
ρ933-1-1001-1+2i1-1-2i00    complex lifted from C42⋊C3
ρ1033-1-100-1+2i1-1-2i100    complex lifted from C42⋊C3
ρ1133-1-100-1-2i1-1+2i100    complex lifted from C42⋊C3
ρ126-62-200000000    orthogonal faithful

Permutation representations of C23.3A4
On 12 points - transitive group 12T57
Generators in S12
(1 2)(3 4)
(5 7)(6 8)
(1 2)(3 4)(5 7)(6 8)(9 11)(10 12)
(3 4)(5 6)(7 8)(9 10 11 12)
(1 3 2 4)(6 8)(9 10)(11 12)
(1 8 9)(2 6 11)(3 5 12)(4 7 10)

G:=sub<Sym(12)| (1,2)(3,4), (5,7)(6,8), (1,2)(3,4)(5,7)(6,8)(9,11)(10,12), (3,4)(5,6)(7,8)(9,10,11,12), (1,3,2,4)(6,8)(9,10)(11,12), (1,8,9)(2,6,11)(3,5,12)(4,7,10)>;

G:=Group( (1,2)(3,4), (5,7)(6,8), (1,2)(3,4)(5,7)(6,8)(9,11)(10,12), (3,4)(5,6)(7,8)(9,10,11,12), (1,3,2,4)(6,8)(9,10)(11,12), (1,8,9)(2,6,11)(3,5,12)(4,7,10) );

G=PermutationGroup([(1,2),(3,4)], [(5,7),(6,8)], [(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)], [(3,4),(5,6),(7,8),(9,10,11,12)], [(1,3,2,4),(6,8),(9,10),(11,12)], [(1,8,9),(2,6,11),(3,5,12),(4,7,10)])

G:=TransitiveGroup(12,57);

On 24 points - transitive group 24T179
Generators in S24
(3 8)(4 7)(5 11)(6 12)
(1 13)(2 14)(9 15)(10 16)
(1 13)(2 14)(3 8)(4 7)(5 11)(6 12)(9 15)(10 16)(17 19)(18 20)(21 23)(22 24)
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 18 19 20)(21 22 23 24)
(1 15)(2 10)(3 12 8 6)(4 5 7 11)(9 13)(14 16)(17 23)(18 22)(19 21)(20 24)
(1 22 6)(2 18 4)(3 16 21)(5 9 17)(7 14 20)(8 10 23)(11 15 19)(12 13 24)

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

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

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

G:=TransitiveGroup(24,179);

On 24 points - transitive group 24T180
Generators in S24
(1 6)(2 5)(3 4)(7 8)(13 19)(14 20)(15 17)(16 18)
(1 2)(3 7)(4 8)(5 6)(9 21)(10 22)(11 23)(12 24)
(1 5)(2 6)(3 8)(4 7)(9 23)(10 24)(11 21)(12 22)(13 17)(14 18)(15 19)(16 20)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4 6 3)(2 8 5 7)(10 24)(12 22)(13 16 19 18)(14 17 20 15)
(1 22 14)(2 24 20)(3 21 17)(4 9 19)(5 12 18)(6 10 16)(7 23 15)(8 11 13)

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

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

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

G:=TransitiveGroup(24,180);

C23.3A4 is a maximal subgroup of   C23.7S4  C23.8S4  C424C4⋊C3  C232D4⋊C3  C24.2A4  C24.3A4  C23.19(C2×A4)
C23.3A4 is a maximal quotient of   C23.SL2(𝔽3)  C2.(C42⋊C9)

Polynomial with Galois group C23.3A4 over ℚ
actionf(x)Disc(f)
12T57x12-69x10-2091x8-7571x6+134691x4+960267x2+1545049212·328·712·1110·134·10912·1132·1278

Matrix representation of C23.3A4 in GL5(𝔽13)

120000
012000
001200
00010
000012
,
120000
012000
00100
000120
000012
,
120000
012000
00100
00010
00001
,
01000
120000
00500
00080
00001
,
43000
39000
00500
000120
00005
,
10000
39000
00010
00001
00100

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,12,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,8,0,0,0,0,0,1],[4,3,0,0,0,3,9,0,0,0,0,0,5,0,0,0,0,0,12,0,0,0,0,0,5],[1,3,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;

C23.3A4 in GAP, Magma, Sage, TeX

C_2^3._3A_4
% in TeX

G:=Group("C2^3.3A4");
// GroupNames label

G:=SmallGroup(96,3);
// by ID

G=gap.SmallGroup(96,3);
# by ID

G:=PCGroup([6,-3,-2,2,-2,2,-2,73,151,596,332,50,1731,2524]);
// Polycyclic

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

Export

Subgroup lattice of C23.3A4 in TeX
Character table of C23.3A4 in TeX

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