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G = C232D4⋊C3order 192 = 26·3

The semidirect product of C232D4 and C3 acting faithfully

non-abelian, soluble

Aliases: C232D4⋊C3, (C22×C4).3A4, C23.14(C2×A4), C23.3A41C2, C22.2(C4.A4), C2.2(C23.A4), C2.C42.2C6, SmallGroup(192,194)

Series: Derived Chief Lower central Upper central

C1C2C2.C42 — C232D4⋊C3
C1C2C23C2.C42C23.3A4 — C232D4⋊C3
C2.C42 — C232D4⋊C3
C1C2

Generators and relations for C232D4⋊C3
 G = < a,b,c,d,e,f | a2=b2=c2=d4=e2=f3=1, eae=ab=ba, ac=ca, dad-1=abc, faf-1=ec=ce, bc=cb, bd=db, be=eb, fbf-1=d2, cd=dc, fcf-1=b, ede=d-1, fdf-1=abcd-1e, fef-1=cde >

Subgroups: 379 in 67 conjugacy classes, 9 normal (all characteristic)
C1, C2, C2 [×4], C3, C4 [×3], C22, C22 [×12], C6, C2×C4 [×5], D4 [×8], C23, C23 [×8], C12, A4, C22⋊C4 [×2], C22×C4, C22×C4, C2×D4 [×6], C24, C2×A4, C2.C42, C2×C22⋊C4, C22×D4, C4×A4, C232D4, C23.3A4, C232D4⋊C3
Quotients: C1, C2, C3, C6, A4, C2×A4, C4.A4, C23.A4, C232D4⋊C3

Character table of C232D4⋊C3

 class 12A2B2C2D2E3A3B4A4B4C4D6A6B12A12B12C12D
 size 113312121616441212161616161616
ρ1111111111111111111    trivial
ρ21111-1-111-1-11111-1-1-1-1    linear of order 2
ρ3111111ζ32ζ31111ζ3ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ41111-1-1ζ32ζ3-1-111ζ3ζ32ζ6ζ6ζ65ζ65    linear of order 6
ρ51111-1-1ζ3ζ32-1-111ζ32ζ3ζ65ζ65ζ6ζ6    linear of order 6
ρ6111111ζ3ζ321111ζ32ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ72-22-200-1-1-2i2i0011i-ii-i    complex lifted from C4.A4
ρ82-22-200-1-12i-2i0011-ii-ii    complex lifted from C4.A4
ρ92-22-200ζ6ζ65-2i2i00ζ3ζ32ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3    complex lifted from C4.A4
ρ102-22-200ζ6ζ652i-2i00ζ3ζ32ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3    complex lifted from C4.A4
ρ112-22-200ζ65ζ62i-2i00ζ32ζ3ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32    complex lifted from C4.A4
ρ122-22-200ζ65ζ6-2i2i00ζ32ζ3ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32    complex lifted from C4.A4
ρ1333331100-3-3-1-1000000    orthogonal lifted from C2×A4
ρ143333-1-10033-1-1000000    orthogonal lifted from A4
ρ156-6-22-22000000000000    orthogonal faithful
ρ1666-2-2000000-22000000    orthogonal lifted from C23.A4
ρ176-6-222-2000000000000    orthogonal faithful
ρ1866-2-20000002-2000000    orthogonal lifted from C23.A4

Permutation representations of C232D4⋊C3
On 12 points - transitive group 12T104
Generators in S12
(1 4)(2 3)(5 7)(9 11)(10 12)
(1 2)(3 4)
(5 7)(6 8)
(3 4)(5 6)(7 8)(9 10 11 12)
(3 4)(9 10)(11 12)
(1 8 11)(2 6 9)(3 7 10)(4 5 12)

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

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

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

G:=TransitiveGroup(12,104);

On 24 points - transitive group 24T496
Generators in S24
(1 5)(2 16)(3 6)(4 15)(7 9)(8 11)(10 13)(12 14)(17 21)(18 22)(19 23)(20 24)
(1 3)(2 4)(5 6)(15 16)
(7 14)(8 13)(9 12)(10 11)
(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 18 19 20)(21 22 23 24)
(1 2)(3 4)(5 15)(6 16)(7 9)(8 10)(11 13)(12 14)(17 24)(18 23)(19 22)(20 21)
(1 7 17)(2 9 23)(3 14 19)(4 12 21)(5 11 24)(6 10 22)(8 20 15)(13 18 16)

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

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

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

G:=TransitiveGroup(24,496);

On 24 points - transitive group 24T497
Generators in S24
(1 7)(2 10)(3 14)(4 16)(5 13)(6 15)(8 12)(9 11)(17 19)(18 20)(21 23)(22 24)
(1 11)(2 12)(7 9)(8 10)
(3 6)(4 5)(13 16)(14 15)
(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 2)(7 8)(9 10)(11 12)(17 22)(18 21)(19 24)(20 23)
(1 15 20)(2 6 22)(3 24 12)(4 17 10)(5 19 8)(7 13 23)(9 16 21)(11 14 18)

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

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

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

G:=TransitiveGroup(24,497);

On 24 points - transitive group 24T498
Generators in S24
(1 13)(3 10)(4 12)(5 9)(6 11)(7 15)(17 24)(18 21)(19 22)(20 23)
(1 13)(2 14)(7 15)(8 16)
(3 6)(4 5)(9 12)(10 11)
(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 16)(2 15)(3 4)(5 6)(7 14)(8 13)(9 11)(10 12)(18 20)(21 23)
(1 6 23)(2 9 19)(3 21 13)(4 20 7)(5 18 15)(8 10 22)(11 24 16)(12 17 14)

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

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

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

G:=TransitiveGroup(24,498);

On 24 points - transitive group 24T499
Generators in S24
(1 8)(2 5)(3 6)(4 7)(10 19)(12 17)(13 23)(15 21)
(1 2)(3 4)(5 8)(6 7)(9 18)(10 19)(11 20)(12 17)
(1 3)(2 4)(5 7)(6 8)(13 23)(14 24)(15 21)(16 22)
(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 7)(6 8)(9 17)(10 20)(11 19)(12 18)(13 23)(14 22)(15 21)(16 24)
(1 23 11)(2 13 9)(3 21 20)(4 15 18)(5 16 17)(6 24 10)(7 14 12)(8 22 19)

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

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

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

G:=TransitiveGroup(24,499);

On 24 points - transitive group 24T500
Generators in S24
(1 7)(2 3)(4 5)(6 8)(9 11)(10 20)(12 18)(13 22)(15 24)(17 19)
(1 6)(2 5)(3 4)(7 8)(9 17)(10 18)(11 19)(12 20)
(1 2)(3 7)(4 8)(5 6)(13 22)(14 23)(15 24)(16 21)
(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 2)(3 8)(4 7)(5 6)(9 20)(10 19)(11 18)(12 17)(14 16)(21 23)
(1 16 10)(2 14 18)(3 24 11)(4 15 9)(5 23 20)(6 21 12)(7 22 19)(8 13 17)

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

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

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

G:=TransitiveGroup(24,500);

Polynomial with Galois group C232D4⋊C3 over ℚ
actionf(x)Disc(f)
12T104x12-6x10+12x8-8x6-3x4+6x2-1-224·316

Matrix representation of C232D4⋊C3 in GL6(ℤ)

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

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

C232D4⋊C3 in GAP, Magma, Sage, TeX

C_2^3\rtimes_2D_4\rtimes C_3
% in TeX

G:=Group("C2^3:2D4:C3");
// GroupNames label

G:=SmallGroup(192,194);
// by ID

G=gap.SmallGroup(192,194);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,1640,135,604,1011,934,521,304,851,1524]);
// Polycyclic

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

Export

Character table of C232D4⋊C3 in TeX

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