non-abelian, soluble, monomial
Aliases: C32:D8:3C2, C4.14S3wrC2, D6:D6:6C2, (C3xC12).11D4, D6.D6:2C2, C32:1(C8:C22), D6:S3:8C22, C32:2C8:1C22, C32:2Q8:7C22, C32:2SD16:1C2, C3:Dic3.2C23, C32:M4(2):4C2, C2.8(C2xS3wrC2), (C3xC6).5(C2xD4), (C2xC3:S3).29D4, (C4xC3:S3).30C22, SmallGroup(288,872)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32:D8:C2
G = < a,b,c,d,e | a3=b3=c8=d2=e2=1, ab=ba, cac-1=b, dad=eae=cbc-1=a-1, bd=db, ebe=b-1, dcd=c-1, ece=c5, ede=c4d >
Subgroups: 688 in 115 conjugacy classes, 23 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C2xC4, D4, Q8, C23, C32, Dic3, C12, D6, C2xC6, M4(2), D8, SD16, C2xD4, C4oD4, C3xS3, C3:S3, C3xC6, Dic6, C4xS3, D12, C3:D4, C2xC12, C3xD4, C22xS3, C8:C22, C3xDic3, C3:Dic3, C3xC12, S32, S3xC6, C2xC3:S3, C4oD12, S3xD4, C32:2C8, D6:S3, D6:S3, C3:D12, C32:2Q8, S3xC12, C3xD12, C4xC3:S3, C2xS32, C32:D8, C32:2SD16, C32:M4(2), D6.D6, D6:D6, C32:D8:C2
Quotients: C1, C2, C22, D4, C23, C2xD4, C8:C22, S3wrC2, C2xS3wrC2, C32:D8:C2
Character table of C32:D8:C2
class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 12A | 12B | 12C | 12D | 12E | |
size | 1 | 1 | 12 | 12 | 12 | 18 | 4 | 4 | 2 | 12 | 18 | 4 | 4 | 12 | 12 | 24 | 24 | 36 | 36 | 4 | 4 | 8 | 12 | 12 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
ρ2 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ3 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
ρ4 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
ρ5 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
ρ6 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
ρ7 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
ρ8 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ9 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | orthogonal lifted from D4 |
ρ10 | 2 | 2 | 0 | 0 | 0 | -2 | 2 | 2 | 2 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | orthogonal lifted from D4 |
ρ11 | 4 | 4 | -2 | -2 | 0 | 0 | -2 | 1 | 4 | 0 | 0 | 1 | -2 | 0 | 0 | 1 | 1 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | orthogonal lifted from S3wrC2 |
ρ12 | 4 | 4 | 2 | -2 | 0 | 0 | -2 | 1 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | -1 | 1 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | orthogonal lifted from C2xS3wrC2 |
ρ13 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8:C22 |
ρ14 | 4 | 4 | 2 | 2 | 0 | 0 | -2 | 1 | 4 | 0 | 0 | 1 | -2 | 0 | 0 | -1 | -1 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | orthogonal lifted from S3wrC2 |
ρ15 | 4 | 4 | 0 | 0 | -2 | 0 | 1 | -2 | -4 | 2 | 0 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | -1 | orthogonal lifted from C2xS3wrC2 |
ρ16 | 4 | 4 | 0 | 0 | 2 | 0 | 1 | -2 | 4 | 2 | 0 | -2 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -2 | -1 | -1 | orthogonal lifted from S3wrC2 |
ρ17 | 4 | 4 | -2 | 2 | 0 | 0 | -2 | 1 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 1 | -1 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | orthogonal lifted from C2xS3wrC2 |
ρ18 | 4 | 4 | 0 | 0 | -2 | 0 | 1 | -2 | 4 | -2 | 0 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | -2 | 1 | 1 | orthogonal lifted from S3wrC2 |
ρ19 | 4 | 4 | 0 | 0 | 2 | 0 | 1 | -2 | -4 | -2 | 0 | -2 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | 1 | 1 | orthogonal lifted from C2xS3wrC2 |
ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 2 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | -3i | 3i | 0 | -√3 | √3 | complex faithful |
ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 2 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 3i | -3i | 0 | √3 | -√3 | complex faithful |
ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 2 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | -3i | 3i | 0 | √3 | -√3 | complex faithful |
ρ23 | 4 | -4 | 0 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 2 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 3i | -3i | 0 | -√3 | √3 | complex faithful |
ρ24 | 8 | -8 | 0 | 0 | 0 | 0 | -4 | 2 | 0 | 0 | 0 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
(1 15 21)(2 16 22)(3 23 9)(4 24 10)(5 11 17)(6 12 18)(7 19 13)(8 20 14)
(1 15 21)(2 22 16)(3 23 9)(4 10 24)(5 11 17)(6 18 12)(7 19 13)(8 14 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 12)(10 11)(13 16)(14 15)(17 24)(18 23)(19 22)(20 21)
(1 5)(3 7)(9 19)(10 24)(11 21)(12 18)(13 23)(14 20)(15 17)(16 22)
G:=sub<Sym(24)| (1,15,21)(2,16,22)(3,23,9)(4,24,10)(5,11,17)(6,12,18)(7,19,13)(8,20,14), (1,15,21)(2,22,16)(3,23,9)(4,10,24)(5,11,17)(6,18,12)(7,19,13)(8,14,20), (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)(9,12)(10,11)(13,16)(14,15)(17,24)(18,23)(19,22)(20,21), (1,5)(3,7)(9,19)(10,24)(11,21)(12,18)(13,23)(14,20)(15,17)(16,22)>;
G:=Group( (1,15,21)(2,16,22)(3,23,9)(4,24,10)(5,11,17)(6,12,18)(7,19,13)(8,20,14), (1,15,21)(2,22,16)(3,23,9)(4,10,24)(5,11,17)(6,18,12)(7,19,13)(8,14,20), (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)(9,12)(10,11)(13,16)(14,15)(17,24)(18,23)(19,22)(20,21), (1,5)(3,7)(9,19)(10,24)(11,21)(12,18)(13,23)(14,20)(15,17)(16,22) );
G=PermutationGroup([[(1,15,21),(2,16,22),(3,23,9),(4,24,10),(5,11,17),(6,12,18),(7,19,13),(8,20,14)], [(1,15,21),(2,22,16),(3,23,9),(4,10,24),(5,11,17),(6,18,12),(7,19,13),(8,14,20)], [(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),(9,12),(10,11),(13,16),(14,15),(17,24),(18,23),(19,22),(20,21)], [(1,5),(3,7),(9,19),(10,24),(11,21),(12,18),(13,23),(14,20),(15,17),(16,22)]])
G:=TransitiveGroup(24,658);
Matrix representation of C32:D8:C2 ►in GL8(Z)
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
-1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
-1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
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 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
0 | 0 | 0 | 0 | -1 | 0 | 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 | 0 | 0 |
0 | 0 | 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 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
-1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
G:=sub<GL(8,Integers())| [0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1] >;
C32:D8:C2 in GAP, Magma, Sage, TeX
C_3^2\rtimes D_8\rtimes C_2
% in TeX
G:=Group("C3^2:D8:C2");
// GroupNames label
G:=SmallGroup(288,872);
// by ID
G=gap.SmallGroup(288,872);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,219,100,675,346,80,2693,2028,362,797,1203]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^8=d^2=e^2=1,a*b=b*a,c*a*c^-1=b,d*a*d=e*a*e=c*b*c^-1=a^-1,b*d=d*b,e*b*e=b^-1,d*c*d=c^-1,e*c*e=c^5,e*d*e=c^4*d>;
// generators/relations
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