direct product, non-abelian, soluble
Aliases: C2×He3⋊C4, (C2×He3)⋊C4, He3⋊1(C2×C4), He3⋊C2⋊1C4, C6.3(C32⋊C4), He3⋊C2.3C22, C3.(C2×C32⋊C4), (C2×He3⋊C2).2C2, SmallGroup(216,100)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — C2×He3⋊C4 |
Generators and relations for C2×He3⋊C4
G = < a,b,c,d,e | a2=b3=c3=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe-1=bcd, cd=dc, ce=ec, ede-1=bd-1 >
Character table of C2×He3⋊C4
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 9 | 9 | 1 | 1 | 12 | 12 | 9 | 9 | 9 | 9 | 1 | 1 | 9 | 9 | 9 | 9 | 12 | 12 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | i | -i | i | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | i | -i | -i | i | i | i | -i | -i | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -i | -i | i | i | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -i | i | i | i | i | -i | -i | -i | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | i | i | -i | -i | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | i | -i | -i | -i | -i | i | i | i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | i | -i | i | -i | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | i | i | -i | -i | -i | i | i | linear of order 4 |
| ρ9 | 3 | 3 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | -1 | -1 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | ζ65 | ζ65 | ζ6 | ζ6 | 0 | 0 | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | ζ65 | ζ65 | ζ6 | complex lifted from He3⋊C4 |
| ρ10 | 3 | -3 | 1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -1 | 1 | 1 | -1 | 3-3√-3/2 | 3+3√-3/2 | ζ32 | ζ6 | ζ3 | ζ65 | 0 | 0 | ζ3 | ζ3 | ζ32 | ζ6 | ζ65 | ζ32 | ζ6 | ζ65 | complex faithful |
| ρ11 | 3 | -3 | 1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 1 | -1 | -1 | 1 | 3+3√-3/2 | 3-3√-3/2 | ζ3 | ζ65 | ζ32 | ζ6 | 0 | 0 | ζ6 | ζ6 | ζ65 | ζ3 | ζ32 | ζ65 | ζ3 | ζ32 | complex faithful |
| ρ12 | 3 | 3 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 1 | 1 | 1 | 1 | -3+3√-3/2 | -3-3√-3/2 | ζ6 | ζ6 | ζ65 | ζ65 | 0 | 0 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | complex lifted from He3⋊C4 |
| ρ13 | 3 | 3 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 1 | 1 | 1 | 1 | -3-3√-3/2 | -3+3√-3/2 | ζ65 | ζ65 | ζ6 | ζ6 | 0 | 0 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | complex lifted from He3⋊C4 |
| ρ14 | 3 | 3 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -1 | -1 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | ζ6 | ζ6 | ζ65 | ζ65 | 0 | 0 | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | ζ6 | ζ6 | ζ65 | complex lifted from He3⋊C4 |
| ρ15 | 3 | -3 | 1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 1 | -1 | -1 | 1 | 3-3√-3/2 | 3+3√-3/2 | ζ32 | ζ6 | ζ3 | ζ65 | 0 | 0 | ζ65 | ζ65 | ζ6 | ζ32 | ζ3 | ζ6 | ζ32 | ζ3 | complex faithful |
| ρ16 | 3 | -3 | 1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | -1 | 1 | 1 | -1 | 3+3√-3/2 | 3-3√-3/2 | ζ3 | ζ65 | ζ32 | ζ6 | 0 | 0 | ζ32 | ζ32 | ζ3 | ζ65 | ζ6 | ζ3 | ζ65 | ζ6 | complex faithful |
| ρ17 | 3 | 3 | 1 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | i | -i | i | -i | -3-3√-3/2 | -3+3√-3/2 | ζ3 | ζ3 | ζ32 | ζ32 | 0 | 0 | ζ43ζ32 | ζ4ζ32 | ζ4ζ3 | ζ43ζ3 | ζ43ζ32 | ζ43ζ3 | ζ4ζ3 | ζ4ζ32 | complex lifted from He3⋊C4 |
| ρ18 | 3 | -3 | -1 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | i | i | -i | -i | 3+3√-3/2 | 3-3√-3/2 | ζ65 | ζ3 | ζ6 | ζ32 | 0 | 0 | ζ4ζ32 | ζ43ζ32 | ζ43ζ3 | ζ43ζ3 | ζ43ζ32 | ζ4ζ3 | ζ4ζ3 | ζ4ζ32 | complex faithful |
| ρ19 | 3 | 3 | 1 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -i | i | -i | i | -3+3√-3/2 | -3-3√-3/2 | ζ32 | ζ32 | ζ3 | ζ3 | 0 | 0 | ζ4ζ3 | ζ43ζ3 | ζ43ζ32 | ζ4ζ32 | ζ4ζ3 | ζ4ζ32 | ζ43ζ32 | ζ43ζ3 | complex lifted from He3⋊C4 |
| ρ20 | 3 | -3 | -1 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | -i | -i | i | i | 3+3√-3/2 | 3-3√-3/2 | ζ65 | ζ3 | ζ6 | ζ32 | 0 | 0 | ζ43ζ32 | ζ4ζ32 | ζ4ζ3 | ζ4ζ3 | ζ4ζ32 | ζ43ζ3 | ζ43ζ3 | ζ43ζ32 | complex faithful |
| ρ21 | 3 | 3 | 1 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | i | -i | i | -i | -3+3√-3/2 | -3-3√-3/2 | ζ32 | ζ32 | ζ3 | ζ3 | 0 | 0 | ζ43ζ3 | ζ4ζ3 | ζ4ζ32 | ζ43ζ32 | ζ43ζ3 | ζ43ζ32 | ζ4ζ32 | ζ4ζ3 | complex lifted from He3⋊C4 |
| ρ22 | 3 | 3 | 1 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | -i | i | -i | i | -3-3√-3/2 | -3+3√-3/2 | ζ3 | ζ3 | ζ32 | ζ32 | 0 | 0 | ζ4ζ32 | ζ43ζ32 | ζ43ζ3 | ζ4ζ3 | ζ4ζ32 | ζ4ζ3 | ζ43ζ3 | ζ43ζ32 | complex lifted from He3⋊C4 |
| ρ23 | 3 | -3 | -1 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | i | i | -i | -i | 3-3√-3/2 | 3+3√-3/2 | ζ6 | ζ32 | ζ65 | ζ3 | 0 | 0 | ζ4ζ3 | ζ43ζ3 | ζ43ζ32 | ζ43ζ32 | ζ43ζ3 | ζ4ζ32 | ζ4ζ32 | ζ4ζ3 | complex faithful |
| ρ24 | 3 | -3 | -1 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -i | -i | i | i | 3-3√-3/2 | 3+3√-3/2 | ζ6 | ζ32 | ζ65 | ζ3 | 0 | 0 | ζ43ζ3 | ζ4ζ3 | ζ4ζ32 | ζ4ζ32 | ζ4ζ3 | ζ43ζ32 | ζ43ζ32 | ζ43ζ3 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | 4 | 4 | -2 | 1 | 0 | 0 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×C32⋊C4 |
| ρ26 | 4 | 4 | 0 | 0 | 4 | 4 | -2 | 1 | 0 | 0 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
| ρ27 | 4 | -4 | 0 | 0 | 4 | 4 | 1 | -2 | 0 | 0 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×C32⋊C4 |
| ρ28 | 4 | 4 | 0 | 0 | 4 | 4 | 1 | -2 | 0 | 0 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
►On 36 points
Generators in S36
(1 5)(2 6)(3 10)(4 9)(7 12)(8 11)(13 28)(14 25)(15 26)(16 27)(17 31)(18 32)(19 29)(20 30)(21 33)(22 34)(23 35)(24 36)
(1 16 21)(2 24 19)(3 30 25)(4 28 34)(5 27 33)(6 36 29)(7 31 26)(8 35 32)(9 13 22)(10 20 14)(11 23 18)(12 17 15)
(1 10 11)(2 9 12)(3 8 5)(4 7 6)(13 17 24)(14 18 21)(15 19 22)(16 20 23)(25 32 33)(26 29 34)(27 30 35)(28 31 36)
(2 24 22)(4 28 26)(6 36 34)(7 31 29)(9 13 15)(12 17 19)(14 21 18)(16 20 23)(25 33 32)(27 30 35)
(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)
G:=sub<Sym(36)| (1,5)(2,6)(3,10)(4,9)(7,12)(8,11)(13,28)(14,25)(15,26)(16,27)(17,31)(18,32)(19,29)(20,30)(21,33)(22,34)(23,35)(24,36), (1,16,21)(2,24,19)(3,30,25)(4,28,34)(5,27,33)(6,36,29)(7,31,26)(8,35,32)(9,13,22)(10,20,14)(11,23,18)(12,17,15), (1,10,11)(2,9,12)(3,8,5)(4,7,6)(13,17,24)(14,18,21)(15,19,22)(16,20,23)(25,32,33)(26,29,34)(27,30,35)(28,31,36), (2,24,22)(4,28,26)(6,36,34)(7,31,29)(9,13,15)(12,17,19)(14,21,18)(16,20,23)(25,33,32)(27,30,35), (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)>;
G:=Group( (1,5)(2,6)(3,10)(4,9)(7,12)(8,11)(13,28)(14,25)(15,26)(16,27)(17,31)(18,32)(19,29)(20,30)(21,33)(22,34)(23,35)(24,36), (1,16,21)(2,24,19)(3,30,25)(4,28,34)(5,27,33)(6,36,29)(7,31,26)(8,35,32)(9,13,22)(10,20,14)(11,23,18)(12,17,15), (1,10,11)(2,9,12)(3,8,5)(4,7,6)(13,17,24)(14,18,21)(15,19,22)(16,20,23)(25,32,33)(26,29,34)(27,30,35)(28,31,36), (2,24,22)(4,28,26)(6,36,34)(7,31,29)(9,13,15)(12,17,19)(14,21,18)(16,20,23)(25,33,32)(27,30,35), (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) );
G=PermutationGroup([[(1,5),(2,6),(3,10),(4,9),(7,12),(8,11),(13,28),(14,25),(15,26),(16,27),(17,31),(18,32),(19,29),(20,30),(21,33),(22,34),(23,35),(24,36)], [(1,16,21),(2,24,19),(3,30,25),(4,28,34),(5,27,33),(6,36,29),(7,31,26),(8,35,32),(9,13,22),(10,20,14),(11,23,18),(12,17,15)], [(1,10,11),(2,9,12),(3,8,5),(4,7,6),(13,17,24),(14,18,21),(15,19,22),(16,20,23),(25,32,33),(26,29,34),(27,30,35),(28,31,36)], [(2,24,22),(4,28,26),(6,36,34),(7,31,29),(9,13,15),(12,17,19),(14,21,18),(16,20,23),(25,33,32),(27,30,35)], [(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)]])
C2×He3⋊C4 is a maximal subgroup of
C6.S3≀C2 C32⋊D6⋊C4 C2.SU3(𝔽2) C4⋊(He3⋊C4) C22⋊(He3⋊C4)
C2×He3⋊C4 is a maximal quotient of He3⋊2(C2×C8) He3⋊1M4(2) C4⋊(He3⋊C4) He3⋊4M4(2) C22⋊(He3⋊C4)
Matrix representation of C2×He3⋊C4 ►in GL3(𝔽7) generated by
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 2 | 1 | 0 |
| 4 | 5 | 4 |
| 5 | 5 | 0 |
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 1 | 0 | 0 |
| 2 | 2 | 0 |
| 6 | 0 | 4 |
| 0 | 4 | 1 |
| 5 | 1 | 0 |
| 0 | 3 | 0 |
G:=sub<GL(3,GF(7))| [6,0,0,0,6,0,0,0,6],[2,4,5,1,5,5,0,4,0],[4,0,0,0,4,0,0,0,4],[1,2,6,0,2,0,0,0,4],[0,5,0,4,1,3,1,0,0] >;
C2×He3⋊C4 in GAP, Magma, Sage, TeX
C_2\times {\rm He}_3\rtimes C_4 % in TeX
G:=Group("C2xHe3:C4"); // GroupNames label
G:=SmallGroup(216,100);
// by ID
G=gap.SmallGroup(216,100);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,3,-3,24,1347,111,1924,916,382]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^3=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e^-1=b*c*d,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^-1>;
// generators/relations
Export
Subgroup lattice of C2×He3⋊C4 in TeX
Character table of C2×He3⋊C4 in TeX