Aliases: C42⋊2C18, C4⋊1D4⋊C9, C42⋊C9⋊3C2, (C4×C12).2C6, (C22×C6).3A4, C3.(C23.A4), C23.2(C3.A4), (C2×C6).8(C2×A4), (C3×C4⋊1D4).C3, C22.4(C2×C3.A4), SmallGroup(288,75)
Series: Derived ►Chief ►Lower central ►Upper central
C1 — C22 — C42 — C4×C12 — C42⋊C9 — C42⋊2C18 |
C42 — C42⋊2C18 |
Generators and relations for C42⋊2C18
G = < a,b,c | a4=b4=c18=1, ab=ba, cac-1=a2b-1, cbc-1=a-1b-1 >
Character table of C42⋊2C18
class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 18A | 18B | 18C | 18D | 18E | 18F | |
size | 1 | 3 | 4 | 12 | 1 | 1 | 6 | 6 | 3 | 3 | 4 | 4 | 12 | 12 | 16 | 16 | 16 | 16 | 16 | 16 | 6 | 6 | 6 | 6 | 16 | 16 | 16 | 16 | 16 | 16 | |
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 | 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 | -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 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | linear of order 6 |
ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
ρ6 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | linear of order 6 |
ρ7 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | ζ65 | ζ97 | ζ92 | ζ98 | ζ94 | ζ9 | ζ95 | ζ32 | ζ32 | ζ3 | ζ3 | -ζ9 | -ζ95 | -ζ92 | -ζ97 | -ζ94 | -ζ98 | linear of order 18 |
ρ8 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | ζ65 | ζ94 | ζ95 | ζ92 | ζ9 | ζ97 | ζ98 | ζ32 | ζ32 | ζ3 | ζ3 | -ζ97 | -ζ98 | -ζ95 | -ζ94 | -ζ9 | -ζ92 | linear of order 18 |
ρ9 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | ζ65 | ζ9 | ζ98 | ζ95 | ζ97 | ζ94 | ζ92 | ζ32 | ζ32 | ζ3 | ζ3 | -ζ94 | -ζ92 | -ζ98 | -ζ9 | -ζ97 | -ζ95 | linear of order 18 |
ρ10 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ92 | ζ97 | ζ9 | ζ95 | ζ98 | ζ94 | ζ3 | ζ3 | ζ32 | ζ32 | ζ98 | ζ94 | ζ97 | ζ92 | ζ95 | ζ9 | linear of order 9 |
ρ11 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | ζ6 | ζ95 | ζ94 | ζ97 | ζ98 | ζ92 | ζ9 | ζ3 | ζ3 | ζ32 | ζ32 | -ζ92 | -ζ9 | -ζ94 | -ζ95 | -ζ98 | -ζ97 | linear of order 18 |
ρ12 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ94 | ζ95 | ζ92 | ζ9 | ζ97 | ζ98 | ζ32 | ζ32 | ζ3 | ζ3 | ζ97 | ζ98 | ζ95 | ζ94 | ζ9 | ζ92 | linear of order 9 |
ρ13 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ9 | ζ98 | ζ95 | ζ97 | ζ94 | ζ92 | ζ32 | ζ32 | ζ3 | ζ3 | ζ94 | ζ92 | ζ98 | ζ9 | ζ97 | ζ95 | linear of order 9 |
ρ14 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | ζ6 | ζ92 | ζ97 | ζ9 | ζ95 | ζ98 | ζ94 | ζ3 | ζ3 | ζ32 | ζ32 | -ζ98 | -ζ94 | -ζ97 | -ζ92 | -ζ95 | -ζ9 | linear of order 18 |
ρ15 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | ζ6 | ζ98 | ζ9 | ζ94 | ζ92 | ζ95 | ζ97 | ζ3 | ζ3 | ζ32 | ζ32 | -ζ95 | -ζ97 | -ζ9 | -ζ98 | -ζ92 | -ζ94 | linear of order 18 |
ρ16 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ95 | ζ94 | ζ97 | ζ98 | ζ92 | ζ9 | ζ3 | ζ3 | ζ32 | ζ32 | ζ92 | ζ9 | ζ94 | ζ95 | ζ98 | ζ97 | linear of order 9 |
ρ17 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ98 | ζ9 | ζ94 | ζ92 | ζ95 | ζ97 | ζ3 | ζ3 | ζ32 | ζ32 | ζ95 | ζ97 | ζ9 | ζ98 | ζ92 | ζ94 | linear of order 9 |
ρ18 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ97 | ζ92 | ζ98 | ζ94 | ζ9 | ζ95 | ζ32 | ζ32 | ζ3 | ζ3 | ζ9 | ζ95 | ζ92 | ζ97 | ζ94 | ζ98 | linear of order 9 |
ρ19 | 3 | 3 | 3 | -1 | 3 | 3 | -1 | -1 | 3 | 3 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
ρ20 | 3 | 3 | -3 | 1 | 3 | 3 | -1 | -1 | 3 | 3 | -3 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
ρ21 | 3 | 3 | -3 | 1 | -3-3√-3/2 | -3+3√-3/2 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 3-3√-3/2 | 3+3√-3/2 | ζ32 | ζ3 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ6 | ζ65 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C2×C3.A4 |
ρ22 | 3 | 3 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | -3-3√-3/2 | -3+3√-3/2 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ65 | ζ6 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3.A4 |
ρ23 | 3 | 3 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | -3+3√-3/2 | -3-3√-3/2 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ6 | ζ65 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3.A4 |
ρ24 | 3 | 3 | -3 | 1 | -3+3√-3/2 | -3-3√-3/2 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 3+3√-3/2 | 3-3√-3/2 | ζ3 | ζ32 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ65 | ζ6 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C2×C3.A4 |
ρ25 | 6 | -2 | 0 | 0 | 6 | 6 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23.A4 |
ρ26 | 6 | -2 | 0 | 0 | 6 | 6 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23.A4 |
ρ27 | 6 | -2 | 0 | 0 | -3-3√-3 | -3+3√-3 | -2 | 2 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-3 | 1+√-3 | -1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
ρ28 | 6 | -2 | 0 | 0 | -3-3√-3 | -3+3√-3 | 2 | -2 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√-3 | -1-√-3 | 1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
ρ29 | 6 | -2 | 0 | 0 | -3+3√-3 | -3-3√-3 | 2 | -2 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√-3 | -1+√-3 | 1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
ρ30 | 6 | -2 | 0 | 0 | -3+3√-3 | -3-3√-3 | -2 | 2 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-3 | 1-√-3 | -1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
(2 11 30 21)(3 12 31 22)(5 24 33 14)(6 25 34 15)(8 17 36 27)(9 18 19 28)
(1 20 29 10)(2 11 30 21)(3 31)(4 13 32 23)(5 24 33 14)(6 34)(7 26 35 16)(8 17 36 27)(9 19)(12 22)(15 25)(18 28)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36)
G:=sub<Sym(36)| (2,11,30,21)(3,12,31,22)(5,24,33,14)(6,25,34,15)(8,17,36,27)(9,18,19,28), (1,20,29,10)(2,11,30,21)(3,31)(4,13,32,23)(5,24,33,14)(6,34)(7,26,35,16)(8,17,36,27)(9,19)(12,22)(15,25)(18,28), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)>;
G:=Group( (2,11,30,21)(3,12,31,22)(5,24,33,14)(6,25,34,15)(8,17,36,27)(9,18,19,28), (1,20,29,10)(2,11,30,21)(3,31)(4,13,32,23)(5,24,33,14)(6,34)(7,26,35,16)(8,17,36,27)(9,19)(12,22)(15,25)(18,28), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36) );
G=PermutationGroup([[(2,11,30,21),(3,12,31,22),(5,24,33,14),(6,25,34,15),(8,17,36,27),(9,18,19,28)], [(1,20,29,10),(2,11,30,21),(3,31),(4,13,32,23),(5,24,33,14),(6,34),(7,26,35,16),(8,17,36,27),(9,19),(12,22),(15,25),(18,28)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)]])
Matrix representation of C42⋊2C18 ►in GL6(𝔽37)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
9 | 0 | 36 | 0 | 0 | 0 |
7 | 0 | 0 | 0 | 0 | 36 |
0 | 0 | 0 | 0 | 1 | 0 |
1 | 35 | 0 | 0 | 0 | 0 |
1 | 36 | 0 | 0 | 0 | 0 |
0 | 28 | 0 | 1 | 0 | 0 |
9 | 28 | 36 | 0 | 0 | 0 |
7 | 30 | 0 | 0 | 36 | 0 |
7 | 30 | 0 | 0 | 0 | 36 |
34 | 0 | 0 | 0 | 0 | 22 |
0 | 0 | 0 | 0 | 26 | 11 |
0 | 10 | 0 | 0 | 0 | 25 |
0 | 0 | 0 | 0 | 0 | 25 |
0 | 0 | 0 | 10 | 0 | 3 |
0 | 0 | 10 | 0 | 0 | 3 |
G:=sub<GL(6,GF(37))| [1,0,0,9,7,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,36,0],[1,1,0,9,7,7,35,36,28,28,30,30,0,0,0,36,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[34,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,10,0,0,26,0,0,0,0,22,11,25,25,3,3] >;
C42⋊2C18 in GAP, Magma, Sage, TeX
C_4^2\rtimes_2C_{18}
% in TeX
G:=Group("C4^2:2C18");
// GroupNames label
G:=SmallGroup(288,75);
// by ID
G=gap.SmallGroup(288,75);
# by ID
G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,50,6555,514,360,3784,3476,102,3036,5305]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^18=1,a*b=b*a,c*a*c^-1=a^2*b^-1,c*b*c^-1=a^-1*b^-1>;
// generators/relations
Export
Subgroup lattice of C42⋊2C18 in TeX
Character table of C42⋊2C18 in TeX