non-abelian, soluble, monomial
Aliases: C24⋊3D9, (C2×C6).4S4, C22⋊(C3.S4), C3.(C22⋊S4), C24⋊C9⋊2C2, (C23×C6).5S3, SmallGroup(288,836)
Series: Derived ►Chief ►Lower central ►Upper central
| C24⋊C9 — C24⋊D9 |
Generators and relations for C24⋊D9
G = < a,b,c,d,e,f | a2=b2=c2=d2=e9=f2=1, eae-1=fbf=ab=ba, ac=ca, ad=da, af=fa, bc=cb, bd=db, ebe-1=a, fcf=ede-1=cd=dc, ece-1=d, df=fd, fef=e-1 >
Subgroups: 668 in 101 conjugacy classes, 12 normal (6 characteristic)
C1, C2, C3, C4, C22, C22, S3, C6, C2×C4, D4, C23, C9, Dic3, D6, C2×C6, C2×C6, C22⋊C4, C2×D4, C24, D9, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C22≀C2, C3.A4, C6.D4, C2×C3⋊D4, C23×C6, C3.S4, C24⋊4S3, C24⋊C9, C24⋊D9
Quotients: C1, C2, S3, D9, S4, C3.S4, C22⋊S4, C24⋊D9
Character table of C24⋊D9
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 9A | 9B | 9C | |
| size | 1 | 3 | 3 | 3 | 6 | 36 | 2 | 36 | 36 | 36 | 6 | 6 | 6 | 6 | 6 | 32 | 32 | 32 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ4 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ5 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ6 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ7 | 3 | 3 | -1 | -1 | -1 | 1 | 3 | -1 | -1 | 1 | -1 | 3 | -1 | -1 | -1 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ8 | 3 | -1 | -1 | 3 | -1 | -1 | 3 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ9 | 3 | -1 | 3 | -1 | -1 | 1 | 3 | -1 | 1 | -1 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ10 | 3 | -1 | 3 | -1 | -1 | -1 | 3 | 1 | -1 | 1 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ11 | 3 | -1 | -1 | 3 | -1 | 1 | 3 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ12 | 3 | 3 | -1 | -1 | -1 | -1 | 3 | 1 | 1 | -1 | -1 | 3 | -1 | -1 | -1 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ13 | 6 | -2 | -2 | -2 | 2 | 0 | 6 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | orthogonal lifted from C22⋊S4 |
| ρ14 | 6 | -2 | 6 | -2 | -2 | 0 | -3 | 0 | 0 | 0 | 1 | 1 | -3 | 1 | 1 | 0 | 0 | 0 | orthogonal lifted from C3.S4 |
| ρ15 | 6 | 6 | -2 | -2 | -2 | 0 | -3 | 0 | 0 | 0 | 1 | -3 | 1 | 1 | 1 | 0 | 0 | 0 | orthogonal lifted from C3.S4 |
| ρ16 | 6 | -2 | -2 | 6 | -2 | 0 | -3 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -3 | 0 | 0 | 0 | orthogonal lifted from C3.S4 |
| ρ17 | 6 | -2 | -2 | -2 | 2 | 0 | -3 | 0 | 0 | 0 | -1-2√-3 | 1 | 1 | -1+2√-3 | 1 | 0 | 0 | 0 | complex faithful |
| ρ18 | 6 | -2 | -2 | -2 | 2 | 0 | -3 | 0 | 0 | 0 | -1+2√-3 | 1 | 1 | -1-2√-3 | 1 | 0 | 0 | 0 | complex faithful |
►On 36 points
Generators in S36
(1 23)(2 35)(3 17)(4 26)(5 29)(6 11)(7 20)(8 32)(9 14)(10 27)(12 31)(13 21)(15 34)(16 24)(18 28)(19 30)(22 33)(25 36)
(1 15)(2 24)(3 36)(4 18)(5 27)(6 30)(7 12)(8 21)(9 33)(10 29)(11 19)(13 32)(14 22)(16 35)(17 25)(20 31)(23 34)(26 28)
(1 23)(2 16)(3 36)(4 26)(5 10)(6 30)(7 20)(8 13)(9 33)(11 19)(12 31)(14 22)(15 34)(17 25)(18 28)(21 32)(24 35)(27 29)
(1 15)(2 35)(3 25)(4 18)(5 29)(6 19)(7 12)(8 32)(9 22)(10 27)(11 30)(13 21)(14 33)(16 24)(17 36)(20 31)(23 34)(26 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)
(1 9)(2 8)(3 7)(4 6)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(28 30)(31 36)(32 35)(33 34)
G:=sub<Sym(36)| (1,23)(2,35)(3,17)(4,26)(5,29)(6,11)(7,20)(8,32)(9,14)(10,27)(12,31)(13,21)(15,34)(16,24)(18,28)(19,30)(22,33)(25,36), (1,15)(2,24)(3,36)(4,18)(5,27)(6,30)(7,12)(8,21)(9,33)(10,29)(11,19)(13,32)(14,22)(16,35)(17,25)(20,31)(23,34)(26,28), (1,23)(2,16)(3,36)(4,26)(5,10)(6,30)(7,20)(8,13)(9,33)(11,19)(12,31)(14,22)(15,34)(17,25)(18,28)(21,32)(24,35)(27,29), (1,15)(2,35)(3,25)(4,18)(5,29)(6,19)(7,12)(8,32)(9,22)(10,27)(11,30)(13,21)(14,33)(16,24)(17,36)(20,31)(23,34)(26,28), (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), (1,9)(2,8)(3,7)(4,6)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(28,30)(31,36)(32,35)(33,34)>;
G:=Group( (1,23)(2,35)(3,17)(4,26)(5,29)(6,11)(7,20)(8,32)(9,14)(10,27)(12,31)(13,21)(15,34)(16,24)(18,28)(19,30)(22,33)(25,36), (1,15)(2,24)(3,36)(4,18)(5,27)(6,30)(7,12)(8,21)(9,33)(10,29)(11,19)(13,32)(14,22)(16,35)(17,25)(20,31)(23,34)(26,28), (1,23)(2,16)(3,36)(4,26)(5,10)(6,30)(7,20)(8,13)(9,33)(11,19)(12,31)(14,22)(15,34)(17,25)(18,28)(21,32)(24,35)(27,29), (1,15)(2,35)(3,25)(4,18)(5,29)(6,19)(7,12)(8,32)(9,22)(10,27)(11,30)(13,21)(14,33)(16,24)(17,36)(20,31)(23,34)(26,28), (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), (1,9)(2,8)(3,7)(4,6)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(28,30)(31,36)(32,35)(33,34) );
G=PermutationGroup([[(1,23),(2,35),(3,17),(4,26),(5,29),(6,11),(7,20),(8,32),(9,14),(10,27),(12,31),(13,21),(15,34),(16,24),(18,28),(19,30),(22,33),(25,36)], [(1,15),(2,24),(3,36),(4,18),(5,27),(6,30),(7,12),(8,21),(9,33),(10,29),(11,19),(13,32),(14,22),(16,35),(17,25),(20,31),(23,34),(26,28)], [(1,23),(2,16),(3,36),(4,26),(5,10),(6,30),(7,20),(8,13),(9,33),(11,19),(12,31),(14,22),(15,34),(17,25),(18,28),(21,32),(24,35),(27,29)], [(1,15),(2,35),(3,25),(4,18),(5,29),(6,19),(7,12),(8,32),(9,22),(10,27),(11,30),(13,21),(14,33),(16,24),(17,36),(20,31),(23,34),(26,28)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,27),(11,26),(12,25),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19),(28,30),(31,36),(32,35),(33,34)]])
Matrix representation of C24⋊D9 ►in GL8(𝔽37)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 | 36 | 36 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 36 | 36 | 36 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 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 |
| 17 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 26 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 1 | 0 | 0 | 0 |
| 0 | 0 | 32 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 | 36 | 36 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 17 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 31 | 20 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 29 | 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 | 36 | 36 | 36 |
G:=sub<GL(8,GF(37))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,0,29,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,1,0,36,0,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,15,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,15,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,0,29,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[17,26,0,0,0,0,0,0,11,6,0,0,0,0,0,0,0,0,22,16,32,0,0,0,0,0,35,15,29,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,36,0],[17,31,0,0,0,0,0,0,11,20,0,0,0,0,0,0,0,0,22,1,10,0,0,0,0,0,35,15,29,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,36,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,36] >;
C24⋊D9 in GAP, Magma, Sage, TeX
C_2^4\rtimes D_9
% in TeX
G:=Group("C2^4:D9"); // GroupNames label
G:=SmallGroup(288,836);
// by ID
G=gap.SmallGroup(288,836);
# by ID
G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,141,92,254,1011,514,634,956,6053,4548,3534,1777]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^9=f^2=1,e*a*e^-1=f*b*f=a*b=b*a,a*c=c*a,a*d=d*a,a*f=f*a,b*c=c*b,b*d=d*b,e*b*e^-1=a,f*c*f=e*d*e^-1=c*d=d*c,e*c*e^-1=d,d*f=f*d,f*e*f=e^-1>;
// generators/relations
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