non-abelian, supersoluble, monomial
Aliases: He3⋊2Q8, C32⋊Dic6, C6.16S32, (C3×C6).1D6, C3⋊Dic3.S3, C32⋊C12.1C2, He3⋊3C4.2C2, C2.3(C32⋊D6), C3.2(C32⋊2Q8), (C2×He3).1C22, SmallGroup(216,33)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for He3⋊2Q8
G = < a,b,c,d,e | a3=b3=c3=d4=1, e2=d2, ab=ba, cac-1=ab-1, ad=da, eae-1=a-1, bc=cb, dbd-1=ebe-1=b-1, dcd-1=c-1, ce=ec, ede-1=d-1 >
3C3
3C3
6C3
9C4
9C4
9C4
3C6
3C6
6C6
2C32
27Q8
3Dic3
3Dic3
3Dic3
3Dic3
6Dic3
9Dic3
9C12
9C12
9C12
9Dic3
9Dic6
9Dic6
9Dic6
Character table of He3⋊2Q8
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 2 | 6 | 6 | 12 | 18 | 18 | 18 | 2 | 6 | 6 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | -1 | 2 | -1 | 0 | 0 | -2 | 2 | 2 | -1 | -1 | 0 | 1 | 1 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 2 | -1 | 2 | -1 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | 0 | -1 | -1 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | 2 | 2 | -1 | -1 | 0 | -2 | 0 | 2 | -1 | 2 | -1 | 1 | 0 | 0 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | 2 | 2 | -1 | -1 | 0 | 2 | 0 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ9 | 2 | -2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ10 | 2 | -2 | 2 | -1 | 2 | -1 | 0 | 0 | 0 | -2 | -2 | 1 | 1 | 0 | √3 | -√3 | 0 | 0 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ11 | 2 | -2 | 2 | -1 | 2 | -1 | 0 | 0 | 0 | -2 | -2 | 1 | 1 | 0 | -√3 | √3 | 0 | 0 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ12 | 2 | -2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | -2 | 1 | -2 | 1 | -√3 | 0 | 0 | √3 | 0 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ13 | 2 | -2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | -2 | 1 | -2 | 1 | √3 | 0 | 0 | -√3 | 0 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ14 | 4 | 4 | 4 | -2 | -2 | 1 | 0 | 0 | 0 | 4 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ15 | 4 | -4 | 4 | -2 | -2 | 1 | 0 | 0 | 0 | -4 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C32⋊2Q8, Schur index 2 |
| ρ16 | 6 | 6 | -3 | 0 | 0 | 0 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from C32⋊D6 |
| ρ17 | 6 | 6 | -3 | 0 | 0 | 0 | 2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from C32⋊D6 |
| ρ18 | 6 | -6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√3 | √3 | symplectic faithful, Schur index 2 |
| ρ19 | 6 | -6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √3 | -√3 | symplectic faithful, Schur index 2 |
►On 72 points
Generators in S72
(1 40 7)(2 37 8)(3 38 5)(4 39 6)(9 66 46)(10 67 47)(11 68 48)(12 65 45)(13 51 31)(14 52 32)(15 49 29)(16 50 30)(17 42 62)(18 43 63)(19 44 64)(20 41 61)(21 58 33)(22 59 34)(23 60 35)(24 57 36)(25 56 71)(26 53 72)(27 54 69)(28 55 70)
(1 21 29)(2 30 22)(3 23 31)(4 32 24)(5 35 51)(6 52 36)(7 33 49)(8 50 34)(9 69 43)(10 44 70)(11 71 41)(12 42 72)(13 38 60)(14 57 39)(15 40 58)(16 59 37)(17 53 45)(18 46 54)(19 55 47)(20 48 56)(25 61 68)(26 65 62)(27 63 66)(28 67 64)
(1 7 15)(2 16 8)(3 5 13)(4 14 6)(9 27 46)(10 47 28)(11 25 48)(12 45 26)(17 65 42)(18 43 66)(19 67 44)(20 41 68)(21 33 40)(22 37 34)(23 35 38)(24 39 36)(29 49 58)(30 59 50)(31 51 60)(32 57 52)(53 62 72)(54 69 63)(55 64 70)(56 71 61)
(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)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)(65 66 67 68)(69 70 71 72)
(1 18 3 20)(2 17 4 19)(5 41 7 43)(6 44 8 42)(9 51 11 49)(10 50 12 52)(13 68 15 66)(14 67 16 65)(21 54 23 56)(22 53 24 55)(25 58 27 60)(26 57 28 59)(29 46 31 48)(30 45 32 47)(33 69 35 71)(34 72 36 70)(37 62 39 64)(38 61 40 63)
G:=sub<Sym(72)| (1,40,7)(2,37,8)(3,38,5)(4,39,6)(9,66,46)(10,67,47)(11,68,48)(12,65,45)(13,51,31)(14,52,32)(15,49,29)(16,50,30)(17,42,62)(18,43,63)(19,44,64)(20,41,61)(21,58,33)(22,59,34)(23,60,35)(24,57,36)(25,56,71)(26,53,72)(27,54,69)(28,55,70), (1,21,29)(2,30,22)(3,23,31)(4,32,24)(5,35,51)(6,52,36)(7,33,49)(8,50,34)(9,69,43)(10,44,70)(11,71,41)(12,42,72)(13,38,60)(14,57,39)(15,40,58)(16,59,37)(17,53,45)(18,46,54)(19,55,47)(20,48,56)(25,61,68)(26,65,62)(27,63,66)(28,67,64), (1,7,15)(2,16,8)(3,5,13)(4,14,6)(9,27,46)(10,47,28)(11,25,48)(12,45,26)(17,65,42)(18,43,66)(19,67,44)(20,41,68)(21,33,40)(22,37,34)(23,35,38)(24,39,36)(29,49,58)(30,59,50)(31,51,60)(32,57,52)(53,62,72)(54,69,63)(55,64,70)(56,71,61), (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)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72), (1,18,3,20)(2,17,4,19)(5,41,7,43)(6,44,8,42)(9,51,11,49)(10,50,12,52)(13,68,15,66)(14,67,16,65)(21,54,23,56)(22,53,24,55)(25,58,27,60)(26,57,28,59)(29,46,31,48)(30,45,32,47)(33,69,35,71)(34,72,36,70)(37,62,39,64)(38,61,40,63)>;
G:=Group( (1,40,7)(2,37,8)(3,38,5)(4,39,6)(9,66,46)(10,67,47)(11,68,48)(12,65,45)(13,51,31)(14,52,32)(15,49,29)(16,50,30)(17,42,62)(18,43,63)(19,44,64)(20,41,61)(21,58,33)(22,59,34)(23,60,35)(24,57,36)(25,56,71)(26,53,72)(27,54,69)(28,55,70), (1,21,29)(2,30,22)(3,23,31)(4,32,24)(5,35,51)(6,52,36)(7,33,49)(8,50,34)(9,69,43)(10,44,70)(11,71,41)(12,42,72)(13,38,60)(14,57,39)(15,40,58)(16,59,37)(17,53,45)(18,46,54)(19,55,47)(20,48,56)(25,61,68)(26,65,62)(27,63,66)(28,67,64), (1,7,15)(2,16,8)(3,5,13)(4,14,6)(9,27,46)(10,47,28)(11,25,48)(12,45,26)(17,65,42)(18,43,66)(19,67,44)(20,41,68)(21,33,40)(22,37,34)(23,35,38)(24,39,36)(29,49,58)(30,59,50)(31,51,60)(32,57,52)(53,62,72)(54,69,63)(55,64,70)(56,71,61), (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)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72), (1,18,3,20)(2,17,4,19)(5,41,7,43)(6,44,8,42)(9,51,11,49)(10,50,12,52)(13,68,15,66)(14,67,16,65)(21,54,23,56)(22,53,24,55)(25,58,27,60)(26,57,28,59)(29,46,31,48)(30,45,32,47)(33,69,35,71)(34,72,36,70)(37,62,39,64)(38,61,40,63) );
G=PermutationGroup([[(1,40,7),(2,37,8),(3,38,5),(4,39,6),(9,66,46),(10,67,47),(11,68,48),(12,65,45),(13,51,31),(14,52,32),(15,49,29),(16,50,30),(17,42,62),(18,43,63),(19,44,64),(20,41,61),(21,58,33),(22,59,34),(23,60,35),(24,57,36),(25,56,71),(26,53,72),(27,54,69),(28,55,70)], [(1,21,29),(2,30,22),(3,23,31),(4,32,24),(5,35,51),(6,52,36),(7,33,49),(8,50,34),(9,69,43),(10,44,70),(11,71,41),(12,42,72),(13,38,60),(14,57,39),(15,40,58),(16,59,37),(17,53,45),(18,46,54),(19,55,47),(20,48,56),(25,61,68),(26,65,62),(27,63,66),(28,67,64)], [(1,7,15),(2,16,8),(3,5,13),(4,14,6),(9,27,46),(10,47,28),(11,25,48),(12,45,26),(17,65,42),(18,43,66),(19,67,44),(20,41,68),(21,33,40),(22,37,34),(23,35,38),(24,39,36),(29,49,58),(30,59,50),(31,51,60),(32,57,52),(53,62,72),(54,69,63),(55,64,70),(56,71,61)], [(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),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64),(65,66,67,68),(69,70,71,72)], [(1,18,3,20),(2,17,4,19),(5,41,7,43),(6,44,8,42),(9,51,11,49),(10,50,12,52),(13,68,15,66),(14,67,16,65),(21,54,23,56),(22,53,24,55),(25,58,27,60),(26,57,28,59),(29,46,31,48),(30,45,32,47),(33,69,35,71),(34,72,36,70),(37,62,39,64),(38,61,40,63)]])
He3⋊2Q8 is a maximal subgroup of
He3⋊2SD16 He3⋊Q16 C3⋊S3⋊Dic6 C12.91S32 C12.85S32 C62.8D6 C62.9D6
He3⋊2Q8 is a maximal quotient of C62.D6 C62.3D6
Matrix representation of He3⋊2Q8 ►in GL6(𝔽13)
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 2 | 0 | 0 | 2 | 0 | 2 |
| 11 | 11 | 2 | 0 | 2 | 0 |
| 0 | 2 | 2 | 0 | 0 | 2 |
| 2 | 0 | 11 | 11 | 2 | 0 |
| 0 | 2 | 0 | 2 | 2 | 0 |
| 2 | 0 | 2 | 0 | 11 | 11 |
| 12 | 6 | 7 | 1 | 12 | 6 |
| 7 | 1 | 7 | 6 | 7 | 1 |
| 7 | 1 | 12 | 6 | 12 | 6 |
| 7 | 6 | 7 | 1 | 7 | 1 |
| 12 | 6 | 12 | 6 | 7 | 1 |
| 7 | 1 | 7 | 1 | 7 | 6 |
G:=sub<GL(6,GF(13))| [0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,12,0,0,0,0,1,12,0,0,0,0],[2,11,0,2,0,2,0,11,2,0,2,0,0,2,2,11,0,2,2,0,0,11,2,0,0,2,0,2,2,11,2,0,2,0,0,11],[12,7,7,7,12,7,6,1,1,6,6,1,7,7,12,7,12,7,1,6,6,1,6,1,12,7,12,7,7,7,6,1,6,1,1,6] >;
He3⋊2Q8 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_2Q_8 % in TeX
G:=Group("He3:2Q8"); // GroupNames label
G:=SmallGroup(216,33);
// by ID
G=gap.SmallGroup(216,33);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,73,31,201,1444,382,5189,2603]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,e*a*e^-1=a^-1,b*c=c*b,d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
Export
Subgroup lattice of He3⋊2Q8 in TeX
Character table of He3⋊2Q8 in TeX