Copied to
clipboard

G = C23.A4order 96 = 25·3

2nd non-split extension by C23 of A4 acting faithfully

metabelian, soluble, monomial

Aliases: C422C6, C23.2A4, C41D4⋊C3, C42⋊C33C2, C22.4(C2×A4), SmallGroup(96,72)

Series: Derived Chief Lower central Upper central

C1C42 — C23.A4
C1C22C42C42⋊C3 — C23.A4
C42 — C23.A4
C1

Generators and relations for C23.A4
 G = < a,b,c,d,e,f | a2=b2=c2=f3=1, d2=cb=fbf-1=bc, e2=fcf-1=b, eae-1=ab=ba, ac=ca, dad-1=abc, af=fa, bd=db, be=eb, fef-1=cd=dc, ce=ec, de=ed, fdf-1=bde >

3C2
4C2
12C2
16C3
3C4
3C4
6C22
6C22
6C22
6C22
16C6
3C2×C4
3C23
6D4
6D4
6D4
6D4
4A4
3C2×D4
3C2×D4
4C2×A4

Character table of C23.A4

 class 12A2B2C3A3B4A4B6A6B
 size 134121616661616
ρ11111111111    trivial
ρ211-1-11111-1-1    linear of order 2
ρ31111ζ32ζ311ζ3ζ32    linear of order 3
ρ41111ζ3ζ3211ζ32ζ3    linear of order 3
ρ511-1-1ζ3ζ3211ζ6ζ65    linear of order 6
ρ611-1-1ζ32ζ311ζ65ζ6    linear of order 6
ρ7333-100-1-100    orthogonal lifted from A4
ρ833-3100-1-100    orthogonal lifted from C2×A4
ρ96-200002-200    orthogonal faithful
ρ106-20000-2200    orthogonal faithful

Permutation representations of C23.A4
On 12 points - transitive group 12T60
Generators in S12
(1 3)(2 4)(5 8)(6 7)(9 10)(11 12)
(1 2)(3 4)(5 7)(6 8)
(1 2)(3 4)(9 11)(10 12)
(5 6 7 8)(9 10 11 12)
(1 4 2 3)(5 6 7 8)(9 11)(10 12)
(1 9 5)(2 11 7)(3 10 8)(4 12 6)

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

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

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

G:=TransitiveGroup(12,60);

On 12 points - transitive group 12T61
Generators in S12
(3 4)(6 8)(10 12)
(1 2)(3 4)(5 7)(6 8)
(1 2)(3 4)(9 11)(10 12)
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 4 2 3)(5 8 7 6)
(1 11 5)(2 9 7)(3 10 8)(4 12 6)

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

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

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

G:=TransitiveGroup(12,61);

On 16 points - transitive group 16T185
Generators in S16
(2 4)(5 12)(6 11)(7 10)(8 9)(14 16)
(1 13)(2 14)(3 15)(4 16)(5 12)(6 9)(7 10)(8 11)
(1 15)(2 16)(3 13)(4 14)(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 12 13 5)(2 9 14 6)(3 10 15 7)(4 11 16 8)
(2 10 6)(3 13 15)(4 7 11)(5 8 14)(9 16 12)

G:=sub<Sym(16)| (2,4)(5,12)(6,11)(7,10)(8,9)(14,16), (1,13)(2,14)(3,15)(4,16)(5,12)(6,9)(7,10)(8,11), (1,15)(2,16)(3,13)(4,14)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,12,13,5)(2,9,14,6)(3,10,15,7)(4,11,16,8), (2,10,6)(3,13,15)(4,7,11)(5,8,14)(9,16,12)>;

G:=Group( (2,4)(5,12)(6,11)(7,10)(8,9)(14,16), (1,13)(2,14)(3,15)(4,16)(5,12)(6,9)(7,10)(8,11), (1,15)(2,16)(3,13)(4,14)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,12,13,5)(2,9,14,6)(3,10,15,7)(4,11,16,8), (2,10,6)(3,13,15)(4,7,11)(5,8,14)(9,16,12) );

G=PermutationGroup([(2,4),(5,12),(6,11),(7,10),(8,9),(14,16)], [(1,13),(2,14),(3,15),(4,16),(5,12),(6,9),(7,10),(8,11)], [(1,15),(2,16),(3,13),(4,14),(5,10),(6,11),(7,12),(8,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,12,13,5),(2,9,14,6),(3,10,15,7),(4,11,16,8)], [(2,10,6),(3,13,15),(4,7,11),(5,8,14),(9,16,12)])

G:=TransitiveGroup(16,185);

On 24 points - transitive group 24T187
Generators in S24
(1 6)(2 5)(3 7)(4 8)(9 21)(10 24)(11 23)(12 22)(13 18)(14 17)(15 20)(16 19)
(1 2)(3 4)(5 6)(7 8)(9 11)(10 12)(21 23)(22 24)
(1 2)(3 4)(5 6)(7 8)(13 15)(14 16)(17 19)(18 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 2 3)(5 8 6 7)(9 12 11 10)(21 24 23 22)
(1 14 24)(2 16 22)(3 13 23)(4 15 21)(5 19 12)(6 17 10)(7 18 11)(8 20 9)

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

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

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

G:=TransitiveGroup(24,187);

On 24 points - transitive group 24T188
Generators in S24
(1 8)(2 7)(3 6)(4 5)(9 12)(10 11)(13 17)(14 20)(15 19)(16 18)(21 22)(23 24)
(1 6)(2 5)(3 8)(4 7)(9 11)(10 12)(21 23)(22 24)
(1 6)(2 5)(3 8)(4 7)(13 15)(14 16)(17 19)(18 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 3 6 8)(2 4 5 7)(9 23 11 21)(10 24 12 22)(13 20)(14 17)(15 18)(16 19)
(1 16 11)(2 17 24)(3 20 12)(4 15 21)(5 19 22)(6 14 9)(7 13 23)(8 18 10)

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

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

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

G:=TransitiveGroup(24,188);

On 24 points - transitive group 24T189
Generators in S24
(5 8)(6 7)(10 12)(14 16)(17 19)(21 23)
(1 4)(2 3)(5 8)(6 7)(9 11)(10 12)(17 19)(18 20)
(1 4)(2 3)(5 8)(6 7)(13 15)(14 16)(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 6 4 7)(2 5 3 8)(9 17 11 19)(10 18 12 20)(13 22)(14 23)(15 24)(16 21)
(1 24 20)(2 13 9)(3 15 11)(4 22 18)(5 23 10)(6 16 17)(7 14 19)(8 21 12)

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

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

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

G:=TransitiveGroup(24,189);

On 24 points - transitive group 24T190
Generators in S24
(1 2)(3 4)(5 8)(6 7)(9 15)(10 14)(11 13)(12 16)(17 21)(18 24)(19 23)(20 22)
(1 4)(2 3)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 4)(2 3)(5 7)(6 8)(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 7 4 5)(2 8 3 6)(9 16 11 14)(10 13 12 15)(17 23)(18 24)(19 21)(20 22)
(1 23 9)(2 19 15)(3 17 13)(4 21 11)(5 20 12)(6 24 14)(7 18 10)(8 22 16)

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

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

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

G:=TransitiveGroup(24,190);

C23.A4 is a maximal subgroup of   C42⋊Dic3  C42⋊D6  C24.6A4  (C4×C12)⋊C6  C204D4⋊C3
C23.A4 is a maximal quotient of   C422C12  C232D4⋊C3  C24.3A4  C422C18  (C4×C12)⋊C6  C204D4⋊C3

Polynomial with Galois group C23.A4 over ℚ
actionf(x)Disc(f)
12T60x12-14x8-7x4+4254·318
12T61x12-11x8+30x4-16-244·318

Matrix representation of C23.A4 in GL6(ℤ)

010000
100000
000100
001000
000001
000010
,
-100000
0-10000
00-1000
000-100
000010
000001
,
-100000
0-10000
001000
000100
0000-10
00000-1
,
-100000
0-10000
000100
00-1000
000001
0000-10
,
010000
-100000
000-100
001000
000010
000001
,
001000
000100
000010
000001
100000
010000

G:=sub<GL(6,Integers())| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0],[0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C23.A4 in GAP, Magma, Sage, TeX

C_2^3.A_4
% in TeX

G:=Group("C2^3.A4");
// GroupNames label

G:=SmallGroup(96,72);
// by ID

G=gap.SmallGroup(96,72);
# by ID

G:=PCGroup([6,-2,-3,-2,2,-2,2,1406,116,230,867,801,69,730,1307]);
// Polycyclic

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

Export

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

׿
×
𝔽