Copied to
clipboard

G = C33⋊D8order 432 = 24·33

2nd semidirect product of C33 and D8 acting via D8/C2=D4

non-abelian, soluble, monomial

Aliases: C332D8, C6.15S3≀C2, C32(C32⋊D8), D6⋊S31S3, C334C81C2, C339D46C2, (C32×C6).9D4, C323(D4⋊S3), C3⋊Dic3.10D6, C2.4(C33⋊D4), (C3×D6⋊S3)⋊1C2, (C3×C6).15(C3⋊D4), (C3×C3⋊Dic3).7C22, SmallGroup(432,582)

Series: Derived Chief Lower central Upper central

C1C32C3×C3⋊Dic3 — C33⋊D8
C1C3C33C32×C6C3×C3⋊Dic3C339D4 — C33⋊D8
C33C32×C6C3×C3⋊Dic3 — C33⋊D8
C1C2

Generators and relations for C33⋊D8
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=b-1, eae=b, bc=cb, dbd-1=ebe=a, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 684 in 96 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C4, C22 [×2], S3 [×5], C6, C6 [×9], C8, D4 [×2], C32, C32 [×4], Dic3 [×2], C12, D6 [×4], C2×C6 [×4], D8, C3×S3 [×8], C3⋊S3, C3×C6, C3×C6 [×5], C3⋊C8, D12, C3⋊D4 [×2], C3×D4, C33, C3×Dic3 [×2], C3⋊Dic3, S3×C6 [×6], C2×C3⋊S3, C62, D4⋊S3, S3×C32, C3×C3⋊S3, C32×C6, C322C8, D6⋊S3, D6⋊S3, C3⋊D12, C3×C3⋊D4, C3×C3⋊Dic3, S3×C3×C6, C6×C3⋊S3, C32⋊D8, C334C8, C3×D6⋊S3, C339D4, C33⋊D8
Quotients: C1, C2 [×3], C22, S3, D4, D6, D8, C3⋊D4, D4⋊S3, S3≀C2, C32⋊D8, C33⋊D4, C33⋊D8

Character table of C33⋊D8

 class 12A2B2C3A3B3C3D3E3F46A6B6C6D6E6F6G6H6I6J6K6L6M6N6O6P8A8B12
 size 1112362444481824444812121212121212123636545436
ρ1111111111111111111111111111111    trivial
ρ211-111111111111111-1-1-1-1-1-1-1-111-1-11    linear of order 2
ρ311-1-11111111111111-1-1-1-1-1-1-1-1-1-1111    linear of order 2
ρ4111-1111111111111111111111-1-1-1-11    linear of order 2
ρ52220-1-122-1-12-12-1-12-1-12-1-12-1-1-10000-1    orthogonal lifted from S3
ρ62200222222-2222222000000000000-2    orthogonal lifted from D4
ρ722-20-1-122-1-12-12-1-12-11-211-21110000-1    orthogonal lifted from D6
ρ82-2002222220-2-2-2-2-2-200000000002-20    orthogonal lifted from D8
ρ92-2002222220-2-2-2-2-2-20000000000-220    orthogonal lifted from D8
ρ102200-1-122-1-1-2-12-1-12-1--30-3--30-3--3-300001    complex lifted from C3⋊D4
ρ112200-1-122-1-1-2-12-1-12-1-30--3-30--3-3--300001    complex lifted from C3⋊D4
ρ124-400-2-244-2-202-422-420000000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ13440-24-21-2-21041-2-2-210000000011000    orthogonal lifted from S3≀C2
ρ1444024-21-2-21041-2-2-2100000000-1-1000    orthogonal lifted from S3≀C2
ρ1544-2041-211-204-2111-2-2111111-200000    orthogonal lifted from S3≀C2
ρ16442041-211-204-2111-22-1-1-1-1-1-1200000    orthogonal lifted from S3≀C2
ρ174-400-2-1+3-3/2-21-1-3-3/210221+3-3/21-3-3/2-1-10--3-3+-3/23+-3/2-33--3/2-3--3/2000000    complex faithful
ρ184-40041-211-20-42-1-1-120--3--3--3-3-3-3000000    complex lifted from C32⋊D8
ρ1944-20-2-1-3-3/2-21-1+3-3/210-2-2-1+3-3/2-1-3-3/2111+-31ζ3ζ321ζ3ζ321--300000    complex lifted from C33⋊D4
ρ204420-2-1-3-3/2-21-1+3-3/210-2-2-1+3-3/2-1-3-3/211-1--3-1ζ65ζ6-1ζ65ζ6-1+-300000    complex lifted from C33⋊D4
ρ214-400-2-1-3-3/2-21-1+3-3/210221-3-3/21+3-3/2-1-10-3-3--3/23--3/2--33+-3/2-3+-3/2000000    complex faithful
ρ224-400-2-1-3-3/2-21-1+3-3/210221-3-3/21+3-3/2-1-10--33+-3/2-3+-3/2-3-3--3/23--3/2000000    complex faithful
ρ234420-2-1+3-3/2-21-1-3-3/210-2-2-1-3-3/2-1+3-3/211-1+-3-1ζ6ζ65-1ζ6ζ65-1--300000    complex lifted from C33⋊D4
ρ244-4004-21-2-210-4-1222-100000000--3-3000    complex lifted from C32⋊D8
ρ254-4004-21-2-210-4-1222-100000000-3--3000    complex lifted from C32⋊D8
ρ264-400-2-1+3-3/2-21-1-3-3/210221+3-3/21-3-3/2-1-10-33--3/2-3--3/2--3-3+-3/23+-3/2000000    complex faithful
ρ274-40041-211-20-42-1-1-120-3-3-3--3--3--3000000    complex lifted from C32⋊D8
ρ2844-20-2-1+3-3/2-21-1-3-3/210-2-2-1-3-3/2-1+3-3/2111--31ζ32ζ31ζ32ζ31+-300000    complex lifted from C33⋊D4
ρ298800-422-42-10-4222-4-10000000000000    orthogonal lifted from C33⋊D4
ρ308-800-422-42-104-2-2-2410000000000000    orthogonal faithful

Permutation representations of C33⋊D8
On 24 points - transitive group 24T1290
Generators in S24
(2 22 11)(4 13 24)(6 18 15)(8 9 20)
(1 10 21)(3 23 12)(5 14 17)(7 19 16)
(1 21 10)(2 11 22)(3 23 12)(4 13 24)(5 17 14)(6 15 18)(7 19 16)(8 9 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 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(17 24)(18 23)(19 22)(20 21)

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

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

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

G:=TransitiveGroup(24,1290);

On 24 points - transitive group 24T1314
Generators in S24
(1 24 10)(2 17 11)(3 12 18)(4 13 19)(5 20 14)(6 21 15)(7 16 22)(8 9 23)
(1 10 24)(2 17 11)(3 18 12)(4 13 19)(5 14 20)(6 21 15)(7 22 16)(8 9 23)
(1 24 10)(2 11 17)(3 18 12)(4 13 19)(5 20 14)(6 15 21)(7 22 16)(8 9 23)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 8)(3 7)(4 6)(9 17)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)

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

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

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

G:=TransitiveGroup(24,1314);

Matrix representation of C33⋊D8 in GL4(𝔽7) generated by

1040
5614
4406
0001
,
6211
2661
0010
0002
,
3145
1335
0040
0002
,
6501
0034
1632
1165
,
6111
0344
1632
6612
G:=sub<GL(4,GF(7))| [1,5,4,0,0,6,4,0,4,1,0,0,0,4,6,1],[6,2,0,0,2,6,0,0,1,6,1,0,1,1,0,2],[3,1,0,0,1,3,0,0,4,3,4,0,5,5,0,2],[6,0,1,1,5,0,6,1,0,3,3,6,1,4,2,5],[6,0,1,6,1,3,6,6,1,4,3,1,1,4,2,2] >;

C33⋊D8 in GAP, Magma, Sage, TeX

C_3^3\rtimes D_8
% in TeX

G:=Group("C3^3:D8");
// GroupNames label

G:=SmallGroup(432,582);
// by ID

G=gap.SmallGroup(432,582);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,254,135,58,1684,571,298,677,1027,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=b^-1,e*a*e=b,b*c=c*b,d*b*d^-1=e*b*e=a,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

Export

Character table of C33⋊D8 in TeX

׿
×
𝔽