Aliases: C32⋊D6⋊C4, C6.5S3≀C2, He3⋊(C22⋊C4), C2.2(He3⋊D4), (C2×He3).5D4, He3⋊C2.3D4, C3.(S32⋊C4), (C2×He3⋊C4)⋊1C2, He3⋊(C2×C4)⋊4C2, (C2×C32⋊D6).C2, He3⋊C2.4(C2×C4), (C2×He3⋊C2).2C22, SmallGroup(432,238)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊D6⋊C4
G = < a,b,c,d,e | a3=b3=c6=d2=e4=1, ab=ba, cac-1=dad=a-1b-1, eae-1=c4, cbc-1=ebe-1=b-1, bd=db, dcd=c-1, ece-1=ac-1d, ede-1=ac2d >
Subgroups: 907 in 99 conjugacy classes, 15 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C22⋊C4, C3×S3, C3⋊S3, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, He3, C3×Dic3, C3⋊Dic3, S32, S3×C6, C2×C3⋊S3, D6⋊C4, C32⋊C6, He3⋊C2, C2×He3, S3×Dic3, C2×S32, C32⋊C12, He3⋊C4, C32⋊D6, C32⋊D6, C2×C32⋊C6, C2×He3⋊C2, He3⋊(C2×C4), C2×He3⋊C4, C2×C32⋊D6, C32⋊D6⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, S3≀C2, S32⋊C4, He3⋊D4, C32⋊D6⋊C4
Character table of C32⋊D6⋊C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 9 | 9 | 18 | 18 | 2 | 12 | 12 | 18 | 18 | 18 | 18 | 2 | 12 | 12 | 18 | 18 | 36 | 36 | 18 | 18 | 18 | 18 | 36 | 36 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -i | -i | i | i | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -i | i | -i | i | i | -i | linear of order 4 |
| ρ6 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | i | -i | -i | i | -1 | -1 | -1 | 1 | -1 | 1 | -1 | i | -i | i | -i | i | -i | linear of order 4 |
| ρ7 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -i | i | i | -i | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -i | i | -i | i | -i | i | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | i | i | -i | -i | -1 | -1 | -1 | 1 | -1 | -1 | 1 | i | -i | i | -i | -i | i | linear of order 4 |
| ρ9 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | 4 | 0 | 0 | 0 | 0 | 4 | -2 | 1 | 0 | 2 | 0 | 2 | 4 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3≀C2 |
| ρ12 | 4 | -4 | 0 | 0 | -2 | 2 | 4 | 1 | -2 | 0 | 0 | 0 | 0 | -4 | 2 | -1 | 0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32⋊C4 |
| ρ13 | 4 | 4 | 0 | 0 | 2 | 2 | 4 | 1 | -2 | 0 | 0 | 0 | 0 | 4 | -2 | 1 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ14 | 4 | -4 | 0 | 0 | 2 | -2 | 4 | 1 | -2 | 0 | 0 | 0 | 0 | -4 | 2 | -1 | 0 | 0 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32⋊C4 |
| ρ15 | 4 | 4 | 0 | 0 | 0 | 0 | 4 | -2 | 1 | 0 | -2 | 0 | -2 | 4 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from S3≀C2 |
| ρ16 | 4 | 4 | 0 | 0 | -2 | -2 | 4 | 1 | -2 | 0 | 0 | 0 | 0 | 4 | -2 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | -2 | 1 | 0 | 2i | 0 | -2i | -4 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | i | -i | complex lifted from S32⋊C4 |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | -2 | 1 | 0 | -2i | 0 | 2i | -4 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -i | i | complex lifted from S32⋊C4 |
| ρ19 | 6 | 6 | -2 | -2 | 0 | 0 | -3 | 0 | 0 | 2 | 0 | 2 | 0 | -3 | 0 | 0 | 1 | 1 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from He3⋊D4 |
| ρ20 | 6 | 6 | -2 | -2 | 0 | 0 | -3 | 0 | 0 | -2 | 0 | -2 | 0 | -3 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | orthogonal lifted from He3⋊D4 |
| ρ21 | 6 | 6 | 2 | 2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | -1 | -1 | 0 | 0 | √3 | -√3 | -√3 | √3 | 0 | 0 | orthogonal lifted from He3⋊D4 |
| ρ22 | 6 | 6 | 2 | 2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | -1 | -1 | 0 | 0 | -√3 | √3 | √3 | -√3 | 0 | 0 | orthogonal lifted from He3⋊D4 |
| ρ23 | 6 | -6 | 2 | -2 | 0 | 0 | -3 | 0 | 0 | -2i | 0 | 2i | 0 | 3 | 0 | 0 | 1 | -1 | 0 | 0 | i | -i | i | -i | 0 | 0 | complex faithful |
| ρ24 | 6 | -6 | 2 | -2 | 0 | 0 | -3 | 0 | 0 | 2i | 0 | -2i | 0 | 3 | 0 | 0 | 1 | -1 | 0 | 0 | -i | i | -i | i | 0 | 0 | complex faithful |
| ρ25 | 6 | -6 | -2 | 2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | -1 | 1 | 0 | 0 | √-3 | √-3 | -√-3 | -√-3 | 0 | 0 | complex faithful |
| ρ26 | 6 | -6 | -2 | 2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | -1 | 1 | 0 | 0 | -√-3 | -√-3 | √-3 | √-3 | 0 | 0 | complex faithful |
►On 36 points
Generators in S36
(1 34 31)(2 13 16)(3 15 14)(4 17 18)(5 32 33)(6 36 35)(8 29 26)(9 30 27)(10 22 19)(11 23 20)
(1 6 5)(2 3 4)(7 25 28)(8 29 26)(9 27 30)(10 22 19)(11 20 23)(12 24 21)(13 15 17)(14 18 16)(31 35 33)(32 34 36)
(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)
(7 9)(11 12)(13 18)(14 17)(15 16)(20 24)(21 23)(25 27)(28 30)(31 36)(32 35)(33 34)
(1 12 2 7)(3 28 6 21)(4 25 5 24)(8 34 10 13)(9 31 11 16)(14 30 35 23)(15 26 36 19)(17 29 32 22)(18 27 33 20)
G:=sub<Sym(36)| (1,34,31)(2,13,16)(3,15,14)(4,17,18)(5,32,33)(6,36,35)(8,29,26)(9,30,27)(10,22,19)(11,23,20), (1,6,5)(2,3,4)(7,25,28)(8,29,26)(9,27,30)(10,22,19)(11,20,23)(12,24,21)(13,15,17)(14,18,16)(31,35,33)(32,34,36), (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), (7,9)(11,12)(13,18)(14,17)(15,16)(20,24)(21,23)(25,27)(28,30)(31,36)(32,35)(33,34), (1,12,2,7)(3,28,6,21)(4,25,5,24)(8,34,10,13)(9,31,11,16)(14,30,35,23)(15,26,36,19)(17,29,32,22)(18,27,33,20)>;
G:=Group( (1,34,31)(2,13,16)(3,15,14)(4,17,18)(5,32,33)(6,36,35)(8,29,26)(9,30,27)(10,22,19)(11,23,20), (1,6,5)(2,3,4)(7,25,28)(8,29,26)(9,27,30)(10,22,19)(11,20,23)(12,24,21)(13,15,17)(14,18,16)(31,35,33)(32,34,36), (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), (7,9)(11,12)(13,18)(14,17)(15,16)(20,24)(21,23)(25,27)(28,30)(31,36)(32,35)(33,34), (1,12,2,7)(3,28,6,21)(4,25,5,24)(8,34,10,13)(9,31,11,16)(14,30,35,23)(15,26,36,19)(17,29,32,22)(18,27,33,20) );
G=PermutationGroup([[(1,34,31),(2,13,16),(3,15,14),(4,17,18),(5,32,33),(6,36,35),(8,29,26),(9,30,27),(10,22,19),(11,23,20)], [(1,6,5),(2,3,4),(7,25,28),(8,29,26),(9,27,30),(10,22,19),(11,20,23),(12,24,21),(13,15,17),(14,18,16),(31,35,33),(32,34,36)], [(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)], [(7,9),(11,12),(13,18),(14,17),(15,16),(20,24),(21,23),(25,27),(28,30),(31,36),(32,35),(33,34)], [(1,12,2,7),(3,28,6,21),(4,25,5,24),(8,34,10,13),(9,31,11,16),(14,30,35,23),(15,26,36,19),(17,29,32,22),(18,27,33,20)]])
Matrix representation of C32⋊D6⋊C4 ►in GL6(𝔽13)
| 0 | 0 | 0 | 0 | 12 | 1 |
| 1 | 1 | 0 | 0 | 11 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 1 | 0 | 12 | 0 |
| 0 | 0 | 0 | 1 | 12 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 12 | 12 | 0 | 0 |
| 0 | 12 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 12 | 12 |
| 0 | 12 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 1 | 12 | 12 | 0 | 0 |
| 4 | 6 | 9 | 11 | 9 | 11 |
| 2 | 0 | 11 | 2 | 11 | 2 |
| 2 | 2 | 9 | 11 | 11 | 2 |
| 0 | 2 | 11 | 2 | 0 | 0 |
| 2 | 2 | 11 | 2 | 9 | 11 |
| 0 | 2 | 0 | 0 | 11 | 2 |
G:=sub<GL(6,GF(13))| [0,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,11,12,12,12,12,1,12,0,0,0,0],[0,1,1,0,1,0,12,12,0,12,0,12,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[0,1,1,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,12,0,0,0,0,0,12,1,0,0],[1,0,1,0,0,1,0,1,1,0,0,1,0,0,0,0,0,12,0,0,0,0,1,12,0,0,12,1,0,0,0,0,12,0,0,0],[4,2,2,0,2,0,6,0,2,2,2,2,9,11,9,11,11,0,11,2,11,2,2,0,9,11,11,0,9,11,11,2,2,0,11,2] >;
C32⋊D6⋊C4 in GAP, Magma, Sage, TeX
C_3^2\rtimes D_6\rtimes C_4
% in TeX
G:=Group("C3^2:D6:C4"); // GroupNames label
G:=SmallGroup(432,238);
// by ID
G=gap.SmallGroup(432,238);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,64,1124,851,298,348,1027,537,14118,7069]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^6=d^2=e^4=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^-1,e*a*e^-1=c^4,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d=c^-1,e*c*e^-1=a*c^-1*d,e*d*e^-1=a*c^2*d>;
// generators/relations
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