non-abelian, soluble, monomial
Aliases: C22⋊D36, C12.2S4, C23.3D18, C3.(C4⋊S4), C4⋊(C3.S4), (C2×C6).D12, C3.A4⋊1D4, C6.18(C2×S4), (C22×C4)⋊2D9, (C22×C12).3S3, (C22×C6).15D6, (C2×C3.S4)⋊1C2, (C4×C3.A4)⋊1C2, C2.4(C2×C3.S4), (C2×C3.A4).3C22, SmallGroup(288,334)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊D36
G = < a,b,c,d | a2=b2=c36=d2=1, dad=cbc-1=ab=ba, cac-1=b, bd=db, dcd=c-1 >
Subgroups: 668 in 96 conjugacy classes, 18 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C9, Dic3, C12, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D9, C18, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C4⋊D4, C36, C3.A4, D18, C4⋊Dic3, D6⋊C4, C2×D12, C2×C3⋊D4, C22×C12, D36, C3.S4, C2×C3.A4, C12⋊7D4, C4×C3.A4, C2×C3.S4, C22⋊D36
Quotients: C1, C2, C22, S3, D4, D6, D9, D12, S4, D18, C2×S4, D36, C3.S4, C4⋊S4, C2×C3.S4, C22⋊D36
Character table of C22⋊D36
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | 18B | 18C | 36A | 36B | 36C | 36D | 36E | 36F | |
| size | 1 | 1 | 3 | 3 | 36 | 36 | 2 | 2 | 6 | 36 | 36 | 2 | 6 | 6 | 8 | 8 | 8 | 2 | 2 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| 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 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -2 | -2 | -2 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ7 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ8 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | 1 | 1 | 1 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | 1 | 1 | 1 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | -1 | -1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | -1 | -1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | -1 | -1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | 1 | 1 | 1 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ14 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -√3 | -√3 | √3 | √3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | √3 | √3 | -√3 | -√3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | √3 | -√3 | √3 | -√3 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | ζ4ζ97-ζ4ζ92 | ζ43ζ98-ζ43ζ9 | -ζ4ζ97+ζ4ζ92 | ζ4ζ95-ζ4ζ94 | -ζ43ζ98+ζ43ζ9 | -ζ4ζ95+ζ4ζ94 | orthogonal lifted from D36 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -√3 | √3 | -√3 | √3 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ4ζ97+ζ4ζ92 | -ζ43ζ98+ζ43ζ9 | ζ4ζ97-ζ4ζ92 | -ζ4ζ95+ζ4ζ94 | ζ43ζ98-ζ43ζ9 | ζ4ζ95-ζ4ζ94 | orthogonal lifted from D36 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | √3 | -√3 | √3 | -√3 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ4ζ95+ζ4ζ94 | ζ4ζ97-ζ4ζ92 | ζ4ζ95-ζ4ζ94 | -ζ43ζ98+ζ43ζ9 | -ζ4ζ97+ζ4ζ92 | ζ43ζ98-ζ43ζ9 | orthogonal lifted from D36 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | √3 | -√3 | √3 | -√3 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | ζ43ζ98-ζ43ζ9 | -ζ4ζ95+ζ4ζ94 | -ζ43ζ98+ζ43ζ9 | -ζ4ζ97+ζ4ζ92 | ζ4ζ95-ζ4ζ94 | ζ4ζ97-ζ4ζ92 | orthogonal lifted from D36 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -√3 | √3 | -√3 | √3 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ43ζ98+ζ43ζ9 | ζ4ζ95-ζ4ζ94 | ζ43ζ98-ζ43ζ9 | ζ4ζ97-ζ4ζ92 | -ζ4ζ95+ζ4ζ94 | -ζ4ζ97+ζ4ζ92 | orthogonal lifted from D36 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -√3 | √3 | -√3 | √3 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | ζ4ζ95-ζ4ζ94 | -ζ4ζ97+ζ4ζ92 | -ζ4ζ95+ζ4ζ94 | ζ43ζ98-ζ43ζ9 | ζ4ζ97-ζ4ζ92 | -ζ43ζ98+ζ43ζ9 | orthogonal lifted from D36 |
| ρ22 | 3 | 3 | -1 | -1 | -1 | 1 | 3 | -3 | 1 | 1 | -1 | 3 | -1 | -1 | 0 | 0 | 0 | -3 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ23 | 3 | 3 | -1 | -1 | 1 | 1 | 3 | 3 | -1 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | 0 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ24 | 3 | 3 | -1 | -1 | -1 | -1 | 3 | 3 | -1 | 1 | 1 | 3 | -1 | -1 | 0 | 0 | 0 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ25 | 3 | 3 | -1 | -1 | 1 | -1 | 3 | -3 | 1 | -1 | 1 | 3 | -1 | -1 | 0 | 0 | 0 | -3 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ26 | 6 | -6 | -2 | 2 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | -6 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4⋊S4 |
| ρ27 | 6 | 6 | -2 | -2 | 0 | 0 | -3 | 6 | -2 | 0 | 0 | -3 | 1 | 1 | 0 | 0 | 0 | -3 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C3.S4 |
| ρ28 | 6 | 6 | -2 | -2 | 0 | 0 | -3 | -6 | 2 | 0 | 0 | -3 | 1 | 1 | 0 | 0 | 0 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×C3.S4 |
| ρ29 | 6 | -6 | -2 | 2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 3 | -1 | 1 | 0 | 0 | 0 | -3√3 | 3√3 | √3 | -√3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ30 | 6 | -6 | -2 | 2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 3 | -1 | 1 | 0 | 0 | 0 | 3√3 | -3√3 | -√3 | √3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 36 points
Generators in S36
(1 19)(2 20)(4 22)(5 23)(7 25)(8 26)(10 28)(11 29)(13 31)(14 32)(16 34)(17 35)
(1 19)(3 21)(4 22)(6 24)(7 25)(9 27)(10 28)(12 30)(13 31)(15 33)(16 34)(18 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 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(13 15)(28 36)(29 35)(30 34)(31 33)
G:=sub<Sym(36)| (1,19)(2,20)(4,22)(5,23)(7,25)(8,26)(10,28)(11,29)(13,31)(14,32)(16,34)(17,35), (1,19)(3,21)(4,22)(6,24)(7,25)(9,27)(10,28)(12,30)(13,31)(15,33)(16,34)(18,36), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)(28,36)(29,35)(30,34)(31,33)>;
G:=Group( (1,19)(2,20)(4,22)(5,23)(7,25)(8,26)(10,28)(11,29)(13,31)(14,32)(16,34)(17,35), (1,19)(3,21)(4,22)(6,24)(7,25)(9,27)(10,28)(12,30)(13,31)(15,33)(16,34)(18,36), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)(28,36)(29,35)(30,34)(31,33) );
G=PermutationGroup([[(1,19),(2,20),(4,22),(5,23),(7,25),(8,26),(10,28),(11,29),(13,31),(14,32),(16,34),(17,35)], [(1,19),(3,21),(4,22),(6,24),(7,25),(9,27),(10,28),(12,30),(13,31),(15,33),(16,34),(18,36)], [(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)], [(1,27),(2,26),(3,25),(4,24),(5,23),(6,22),(7,21),(8,20),(9,19),(10,18),(11,17),(12,16),(13,15),(28,36),(29,35),(30,34),(31,33)]])
Matrix representation of C22⋊D36 ►in GL5(𝔽37)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 1 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 1 | 36 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 | 1 |
| 0 | 0 | 36 | 1 | 0 |
| 27 | 7 | 0 | 0 | 0 |
| 30 | 27 | 0 | 0 | 0 |
| 0 | 0 | 36 | 1 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 | 1 |
| 14 | 29 | 0 | 0 | 0 |
| 29 | 23 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 1 | 0 |
| 0 | 0 | 36 | 0 | 1 |
G:=sub<GL(5,GF(37))| [1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,36,36,36,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,36,36,36,0,0,0,0,1,0,0,0,1,0],[27,30,0,0,0,7,27,0,0,0,0,0,36,36,36,0,0,1,0,0,0,0,0,0,1],[14,29,0,0,0,29,23,0,0,0,0,0,36,36,36,0,0,0,1,0,0,0,0,0,1] >;
C22⋊D36 in GAP, Magma, Sage, TeX
C_2^2\rtimes D_{36} % in TeX
G:=Group("C2^2:D36"); // GroupNames label
G:=SmallGroup(288,334);
// by ID
G=gap.SmallGroup(288,334);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,85,36,1123,192,1684,6053,782,3534,1350]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^36=d^2=1,d*a*d=c*b*c^-1=a*b=b*a,c*a*c^-1=b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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