direct product, non-abelian, soluble
Aliases: C2×He3⋊D4, He3⋊(C2×D4), (C2×He3)⋊D4, C6.6S3≀C2, He3⋊C2⋊D4, He3⋊C4⋊C22, C32⋊D6⋊C22, He3⋊C2.2C23, C3.(C2×S3≀C2), (C2×He3⋊C4)⋊3C2, (C2×C32⋊D6)⋊5C2, (C2×He3⋊C2).6C22, SmallGroup(432,530)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×He3⋊D4
G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf=bc=cb, dbd-1=ede-1=bc-1, ebe-1=cd-1, cd=dc, ce=ec, fcf=c-1, fdf=d-1, fef=e-1 >
Subgroups: 1439 in 145 conjugacy classes, 23 normal (11 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, D4, C23, C32, C12, D6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, D12, C2×C12, C22×S3, He3, S32, S3×C6, C2×C3⋊S3, C2×D12, C32⋊C6, He3⋊C2, C2×He3, C2×S32, He3⋊C4, C32⋊D6, C32⋊D6, C2×C32⋊C6, C2×He3⋊C2, He3⋊D4, C2×He3⋊C4, C2×C32⋊D6, C2×He3⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, S3≀C2, C2×S3≀C2, He3⋊D4, C2×He3⋊D4
Character table of C2×He3⋊D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 9 | 9 | 18 | 18 | 18 | 18 | 2 | 12 | 12 | 18 | 18 | 2 | 12 | 12 | 18 | 18 | 36 | 36 | 36 | 36 | 18 | 18 | 18 | 18 | |
| 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 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ6 | 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 |
| ρ7 | 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 |
| ρ8 | 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 |
| ρ9 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 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 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 2 | 4 | -2 | 1 | 0 | 0 | -4 | 2 | -1 | 0 | 0 | 0 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S3≀C2 |
| ρ12 | 4 | 4 | 0 | 0 | 2 | 0 | 0 | 2 | 4 | -2 | 1 | 0 | 0 | 4 | -2 | 1 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ13 | 4 | 4 | 0 | 0 | 0 | 2 | 2 | 0 | 4 | 1 | -2 | 0 | 0 | 4 | 1 | -2 | 0 | 0 | -1 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ14 | 4 | -4 | 0 | 0 | 0 | -2 | 2 | 0 | 4 | 1 | -2 | 0 | 0 | -4 | -1 | 2 | 0 | 0 | 1 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S3≀C2 |
| ρ15 | 4 | -4 | 0 | 0 | 0 | 2 | -2 | 0 | 4 | 1 | -2 | 0 | 0 | -4 | -1 | 2 | 0 | 0 | -1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S3≀C2 |
| ρ16 | 4 | -4 | 0 | 0 | 2 | 0 | 0 | -2 | 4 | -2 | 1 | 0 | 0 | -4 | 2 | -1 | 0 | 0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S3≀C2 |
| ρ17 | 4 | 4 | 0 | 0 | 0 | -2 | -2 | 0 | 4 | 1 | -2 | 0 | 0 | 4 | 1 | -2 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ18 | 4 | 4 | 0 | 0 | -2 | 0 | 0 | -2 | 4 | -2 | 1 | 0 | 0 | 4 | -2 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ19 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 2 | 2 | -3 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from He3⋊D4 |
| ρ20 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 2 | -2 | 3 | 0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | orthogonal faithful |
| ρ21 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | -2 | 2 | 3 | 0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | orthogonal faithful |
| ρ22 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | -2 | -2 | -3 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from He3⋊D4 |
| ρ23 | 6 | 6 | 2 | 2 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | -√3 | √3 | √3 | -√3 | orthogonal lifted from He3⋊D4 |
| ρ24 | 6 | 6 | 2 | 2 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | √3 | -√3 | -√3 | √3 | orthogonal lifted from He3⋊D4 |
| ρ25 | 6 | -6 | 2 | -2 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 1 | -1 | 0 | 0 | 0 | 0 | √3 | -√3 | √3 | -√3 | orthogonal faithful |
| ρ26 | 6 | -6 | 2 | -2 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 1 | -1 | 0 | 0 | 0 | 0 | -√3 | √3 | -√3 | √3 | orthogonal faithful |
►On 36 points
Generators in S36
(1 4)(2 3)(5 7)(6 8)(9 12)(10 11)(13 29)(14 30)(15 31)(16 32)(17 25)(18 26)(19 27)(20 28)(21 35)(22 36)(23 33)(24 34)
(1 7 9)(2 14 18)(3 30 26)(4 5 12)(6 28 33)(8 20 23)(10 21 16)(11 35 32)(13 24 19)(27 29 34)
(1 7 9)(2 8 10)(3 6 11)(4 5 12)(13 19 24)(14 20 21)(15 17 22)(16 18 23)(25 36 31)(26 33 32)(27 34 29)(28 35 30)
(1 22 24)(4 36 34)(5 31 29)(7 15 13)(9 17 19)(12 25 27)(14 20 21)(16 23 18)(26 32 33)(28 35 30)
(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 10)(3 11)(4 12)(13 15)(17 24)(18 23)(19 22)(20 21)(25 34)(26 33)(27 36)(28 35)(29 31)
G:=sub<Sym(36)| (1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,29)(14,30)(15,31)(16,32)(17,25)(18,26)(19,27)(20,28)(21,35)(22,36)(23,33)(24,34), (1,7,9)(2,14,18)(3,30,26)(4,5,12)(6,28,33)(8,20,23)(10,21,16)(11,35,32)(13,24,19)(27,29,34), (1,7,9)(2,8,10)(3,6,11)(4,5,12)(13,19,24)(14,20,21)(15,17,22)(16,18,23)(25,36,31)(26,33,32)(27,34,29)(28,35,30), (1,22,24)(4,36,34)(5,31,29)(7,15,13)(9,17,19)(12,25,27)(14,20,21)(16,23,18)(26,32,33)(28,35,30), (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)(2,10)(3,11)(4,12)(13,15)(17,24)(18,23)(19,22)(20,21)(25,34)(26,33)(27,36)(28,35)(29,31)>;
G:=Group( (1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,29)(14,30)(15,31)(16,32)(17,25)(18,26)(19,27)(20,28)(21,35)(22,36)(23,33)(24,34), (1,7,9)(2,14,18)(3,30,26)(4,5,12)(6,28,33)(8,20,23)(10,21,16)(11,35,32)(13,24,19)(27,29,34), (1,7,9)(2,8,10)(3,6,11)(4,5,12)(13,19,24)(14,20,21)(15,17,22)(16,18,23)(25,36,31)(26,33,32)(27,34,29)(28,35,30), (1,22,24)(4,36,34)(5,31,29)(7,15,13)(9,17,19)(12,25,27)(14,20,21)(16,23,18)(26,32,33)(28,35,30), (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)(2,10)(3,11)(4,12)(13,15)(17,24)(18,23)(19,22)(20,21)(25,34)(26,33)(27,36)(28,35)(29,31) );
G=PermutationGroup([[(1,4),(2,3),(5,7),(6,8),(9,12),(10,11),(13,29),(14,30),(15,31),(16,32),(17,25),(18,26),(19,27),(20,28),(21,35),(22,36),(23,33),(24,34)], [(1,7,9),(2,14,18),(3,30,26),(4,5,12),(6,28,33),(8,20,23),(10,21,16),(11,35,32),(13,24,19),(27,29,34)], [(1,7,9),(2,8,10),(3,6,11),(4,5,12),(13,19,24),(14,20,21),(15,17,22),(16,18,23),(25,36,31),(26,33,32),(27,34,29),(28,35,30)], [(1,22,24),(4,36,34),(5,31,29),(7,15,13),(9,17,19),(12,25,27),(14,20,21),(16,23,18),(26,32,33),(28,35,30)], [(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),(2,10),(3,11),(4,12),(13,15),(17,24),(18,23),(19,22),(20,21),(25,34),(26,33),(27,36),(28,35),(29,31)]])
Matrix representation of C2×He3⋊D4 ►in GL6(ℤ)
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | -1 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | -1 | -2 |
| 0 | 0 | 0 | 0 | 1 | -1 |
| 1 | 0 | 1 | 0 | 0 | 1 |
| 0 | 1 | 1 | 0 | 0 | 1 |
| -1 | 1 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 0 | 1 |
| 0 | -1 | 0 | -1 | -1 | -1 |
| 1 | 1 | 1 | 1 | 2 | 1 |
| 1 | 1 | 1 | 1 | 1 | 2 |
| -1 | 1 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | -1 | -1 | -1 | -1 |
| 0 | -1 | 0 | -1 | -1 | -1 |
| 0 | -1 | 0 | 0 | -1 | 0 |
| 0 | -1 | -1 | 0 | -1 | -1 |
| -1 | 0 | -1 | 0 | -1 | 0 |
| -1 | -1 | -1 | -1 | -1 | -1 |
| 1 | 1 | 1 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 1 | 0 |
| 0 | 1 | 1 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 1 | 1 | 1 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | -1 | 0 | -1 | -1 | -1 |
G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[0,0,-1,0,1,0,0,0,-1,0,0,1,-1,-1,-1,0,1,1,1,0,-1,0,0,0,0,0,-1,1,0,0,0,0,-2,-1,1,1],[-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,-1,1,0,0,0,1,0,0,-1,0,0,0,0,0,-1,0,0,0,0,1,-1],[1,1,-1,-1,0,0,1,1,1,0,-1,-1,1,1,0,0,-1,0,1,1,0,0,-1,-1,2,1,0,0,-1,-1,1,2,0,0,-1,-1],[0,0,-1,-1,1,0,-1,-1,0,-1,1,1,0,-1,-1,-1,1,1,0,0,0,-1,0,0,-1,-1,-1,-1,1,1,0,-1,0,-1,1,0],[0,0,1,0,0,0,1,1,0,0,0,-1,1,0,0,0,0,0,0,0,1,1,0,-1,1,1,1,1,-1,-1,1,0,1,0,0,-1] >;
C2×He3⋊D4 in GAP, Magma, Sage, TeX
C_2\times {\rm He}_3\rtimes D_4 % in TeX
G:=Group("C2xHe3:D4"); // GroupNames label
G:=SmallGroup(432,530);
// by ID
G=gap.SmallGroup(432,530);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,141,1124,851,165,348,530,537,14118,7069]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f=b*c=c*b,d*b*d^-1=e*d*e^-1=b*c^-1,e*b*e^-1=c*d^-1,c*d=d*c,c*e=e*c,f*c*f=c^-1,f*d*f=d^-1,f*e*f=e^-1>;
// generators/relations
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