metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C9⋊2D4, Dic9⋊C2, D18⋊2C2, C6.10D6, C2.5D18, C22⋊2D9, C18.5C22, (C2×C18)⋊2C2, (C2×C6).3S3, C3.(C3⋊D4), SmallGroup(72,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9⋊D4
G = < a,b,c | a9=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Character table of C9⋊D4
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 9A | 9B | 9C | 18A | 18B | 18C | 18D | 18E | 18F | 18G | 18H | 18I | |
| size | 1 | 1 | 2 | 18 | 2 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 0 | 2 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
| ρ7 | 2 | 2 | -2 | 0 | 2 | 0 | -2 | 2 | -2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | -2 | 0 | -1 | 0 | 1 | -1 | 1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ98+ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D18 |
| ρ9 | 2 | 2 | -2 | 0 | -1 | 0 | 1 | -1 | 1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ97+ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D18 |
| ρ10 | 2 | 2 | 2 | 0 | -1 | 0 | -1 | -1 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ11 | 2 | 2 | 2 | 0 | -1 | 0 | -1 | -1 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ12 | 2 | 2 | -2 | 0 | -1 | 0 | 1 | -1 | 1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ95+ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D18 |
| ρ13 | 2 | 2 | 2 | 0 | -1 | 0 | -1 | -1 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | -1 | -1 | -1 | 1 | -√-3 | -√-3 | -√-3 | √-3 | √-3 | √-3 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ15 | 2 | -2 | 0 | 0 | -1 | 0 | -√-3 | 1 | √-3 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ95-ζ94 | ζ98-ζ9 | -ζ97+ζ92 | ζ95-ζ94 | ζ97-ζ92 | -ζ98+ζ9 | -ζ95+ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | complex faithful |
| ρ16 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | -1 | -1 | -1 | 1 | √-3 | √-3 | √-3 | -√-3 | -√-3 | -√-3 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ17 | 2 | -2 | 0 | 0 | -1 | 0 | -√-3 | 1 | √-3 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ98-ζ9 | -ζ97+ζ92 | ζ95-ζ94 | ζ98-ζ9 | -ζ95+ζ94 | ζ97-ζ92 | -ζ98+ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | complex faithful |
| ρ18 | 2 | -2 | 0 | 0 | -1 | 0 | -√-3 | 1 | √-3 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ97-ζ92 | ζ95-ζ94 | ζ98-ζ9 | -ζ97+ζ92 | -ζ98+ζ9 | -ζ95+ζ94 | ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | complex faithful |
| ρ19 | 2 | -2 | 0 | 0 | -1 | 0 | √-3 | 1 | -√-3 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ95-ζ94 | -ζ98+ζ9 | ζ97-ζ92 | -ζ95+ζ94 | -ζ97+ζ92 | ζ98-ζ9 | ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | complex faithful |
| ρ20 | 2 | -2 | 0 | 0 | -1 | 0 | √-3 | 1 | -√-3 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ98-ζ9 | ζ97-ζ92 | -ζ95+ζ94 | -ζ98+ζ9 | ζ95-ζ94 | -ζ97+ζ92 | ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | complex faithful |
| ρ21 | 2 | -2 | 0 | 0 | -1 | 0 | √-3 | 1 | -√-3 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ97-ζ92 | -ζ95+ζ94 | -ζ98+ζ9 | ζ97-ζ92 | ζ98-ζ9 | ζ95-ζ94 | -ζ97+ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | complex faithful |
►On 36 points
Generators in S36
(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)
(1 20 11 29)(2 19 12 28)(3 27 13 36)(4 26 14 35)(5 25 15 34)(6 24 16 33)(7 23 17 32)(8 22 18 31)(9 21 10 30)
(2 9)(3 8)(4 7)(5 6)(10 12)(13 18)(14 17)(15 16)(19 30)(20 29)(21 28)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)
G:=sub<Sym(36)| (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), (1,20,11,29)(2,19,12,28)(3,27,13,36)(4,26,14,35)(5,25,15,34)(6,24,16,33)(7,23,17,32)(8,22,18,31)(9,21,10,30), (2,9)(3,8)(4,7)(5,6)(10,12)(13,18)(14,17)(15,16)(19,30)(20,29)(21,28)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)>;
G:=Group( (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), (1,20,11,29)(2,19,12,28)(3,27,13,36)(4,26,14,35)(5,25,15,34)(6,24,16,33)(7,23,17,32)(8,22,18,31)(9,21,10,30), (2,9)(3,8)(4,7)(5,6)(10,12)(13,18)(14,17)(15,16)(19,30)(20,29)(21,28)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31) );
G=PermutationGroup([[(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)], [(1,20,11,29),(2,19,12,28),(3,27,13,36),(4,26,14,35),(5,25,15,34),(6,24,16,33),(7,23,17,32),(8,22,18,31),(9,21,10,30)], [(2,9),(3,8),(4,7),(5,6),(10,12),(13,18),(14,17),(15,16),(19,30),(20,29),(21,28),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31)]])
C9⋊D4 is a maximal subgroup of
D36⋊5C2 D4×D9 D4⋊2D9 C27⋊D4 C9.S4 D6⋊D9 C9⋊D12 Dic9⋊C6 C6.D18 C9⋊S4 Q8.D18 C23.D18 C45⋊D4 C9⋊D20 C45⋊7D4
C9⋊D4 is a maximal quotient of
Dic9⋊C4 D18⋊C4 D4.D9 D4⋊D9 C9⋊Q16 Q8⋊2D9 C18.D4 C27⋊D4 D6⋊D9 C9⋊D12 C6.D18 C23.D18 C45⋊D4 C9⋊D20 C45⋊7D4
Matrix representation of C9⋊D4 ►in GL2(𝔽19) generated by
| 8 | 15 |
| 15 | 14 |
| 12 | 4 |
| 16 | 7 |
| 18 | 11 |
| 0 | 1 |
G:=sub<GL(2,GF(19))| [8,15,15,14],[12,16,4,7],[18,0,11,1] >;
C9⋊D4 in GAP, Magma, Sage, TeX
C_9\rtimes D_4
% in TeX
G:=Group("C9:D4"); // GroupNames label
G:=SmallGroup(72,8);
// by ID
G=gap.SmallGroup(72,8);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-3,61,803,138,1204]);
// Polycyclic
G:=Group<a,b,c|a^9=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C9⋊D4 in TeX
Character table of C9⋊D4 in TeX