direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×C3⋊Dic3, C6⋊Dic3, C6.15D6, C62.2C2, (C3×C6)⋊3C4, (C2×C6).5S3, C32⋊7(C2×C4), C22.(C3⋊S3), C3⋊2(C2×Dic3), (C3×C6).14C22, C2.2(C2×C3⋊S3), SmallGroup(72,34)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C2×C3⋊Dic3 |
Generators and relations for C2×C3⋊Dic3
G = < a,b,c,d | a2=b3=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Character table of C2×C3⋊Dic3
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 9 | 9 | 9 | 9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | 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 | 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 | 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 | 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 | -1 | -1 | -1 | 1 | 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 | -1 | -1 | -1 | -1 | 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 | -1 | -1 | -1 | 1 | 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 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | orthogonal lifted from S3 |
| ρ11 | 2 | -2 | -2 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -1 | -2 | 1 | 1 | 1 | -1 | -1 | 2 | -2 | orthogonal lifted from D6 |
| ρ12 | 2 | -2 | -2 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | -1 | 1 | 1 | -2 | 1 | 2 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | -2 | -2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -2 | -1 | 1 | 1 | 1 | -2 | -1 | 2 | -1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ15 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ16 | 2 | -2 | -2 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | -2 | 1 | 1 | 2 | 1 | -2 | 1 | 1 | -1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ17 | 2 | -2 | 2 | -2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -2 | 1 | -2 | 1 | 1 | 1 | 1 | 1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ18 | 2 | 2 | -2 | -2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | -2 | 1 | 1 | -2 | -1 | 2 | -1 | -1 | 1 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ19 | 2 | 2 | -2 | -2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -2 | 1 | -1 | -1 | -1 | 2 | 1 | -2 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ20 | 2 | -2 | 2 | -2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | 1 | 1 | 1 | 1 | -2 | 1 | -2 | 1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ21 | 2 | 2 | -2 | -2 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | 1 | -1 | -1 | 2 | -1 | -2 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ22 | 2 | -2 | 2 | -2 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | -2 | 1 | 1 | 1 | 1 | 1 | -2 | 2 | symplectic lifted from Dic3, Schur index 2 |
| ρ23 | 2 | 2 | -2 | -2 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 2 | -1 | -1 | -1 | 1 | 1 | -2 | -2 | symplectic lifted from Dic3, Schur index 2 |
| ρ24 | 2 | -2 | 2 | -2 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | 1 | 1 | 1 | -2 | 1 | -2 | 1 | 1 | -1 | symplectic lifted from Dic3, Schur index 2 |
►Regular action on 72 points
Generators in S72
(1 47)(2 48)(3 43)(4 44)(5 45)(6 46)(7 63)(8 64)(9 65)(10 66)(11 61)(12 62)(13 57)(14 58)(15 59)(16 60)(17 55)(18 56)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 35)(26 36)(27 31)(28 32)(29 33)(30 34)(49 70)(50 71)(51 72)(52 67)(53 68)(54 69)
(1 23 26)(2 24 27)(3 19 28)(4 20 29)(5 21 30)(6 22 25)(7 13 71)(8 14 72)(9 15 67)(10 16 68)(11 17 69)(12 18 70)(31 48 42)(32 43 37)(33 44 38)(34 45 39)(35 46 40)(36 47 41)(49 62 56)(50 63 57)(51 64 58)(52 65 59)(53 66 60)(54 61 55)
(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 54 4 51)(2 53 5 50)(3 52 6 49)(7 31 10 34)(8 36 11 33)(9 35 12 32)(13 42 16 39)(14 41 17 38)(15 40 18 37)(19 59 22 56)(20 58 23 55)(21 57 24 60)(25 62 28 65)(26 61 29 64)(27 66 30 63)(43 67 46 70)(44 72 47 69)(45 71 48 68)
G:=sub<Sym(72)| (1,47)(2,48)(3,43)(4,44)(5,45)(6,46)(7,63)(8,64)(9,65)(10,66)(11,61)(12,62)(13,57)(14,58)(15,59)(16,60)(17,55)(18,56)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,35)(26,36)(27,31)(28,32)(29,33)(30,34)(49,70)(50,71)(51,72)(52,67)(53,68)(54,69), (1,23,26)(2,24,27)(3,19,28)(4,20,29)(5,21,30)(6,22,25)(7,13,71)(8,14,72)(9,15,67)(10,16,68)(11,17,69)(12,18,70)(31,48,42)(32,43,37)(33,44,38)(34,45,39)(35,46,40)(36,47,41)(49,62,56)(50,63,57)(51,64,58)(52,65,59)(53,66,60)(54,61,55), (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), (1,54,4,51)(2,53,5,50)(3,52,6,49)(7,31,10,34)(8,36,11,33)(9,35,12,32)(13,42,16,39)(14,41,17,38)(15,40,18,37)(19,59,22,56)(20,58,23,55)(21,57,24,60)(25,62,28,65)(26,61,29,64)(27,66,30,63)(43,67,46,70)(44,72,47,69)(45,71,48,68)>;
G:=Group( (1,47)(2,48)(3,43)(4,44)(5,45)(6,46)(7,63)(8,64)(9,65)(10,66)(11,61)(12,62)(13,57)(14,58)(15,59)(16,60)(17,55)(18,56)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,35)(26,36)(27,31)(28,32)(29,33)(30,34)(49,70)(50,71)(51,72)(52,67)(53,68)(54,69), (1,23,26)(2,24,27)(3,19,28)(4,20,29)(5,21,30)(6,22,25)(7,13,71)(8,14,72)(9,15,67)(10,16,68)(11,17,69)(12,18,70)(31,48,42)(32,43,37)(33,44,38)(34,45,39)(35,46,40)(36,47,41)(49,62,56)(50,63,57)(51,64,58)(52,65,59)(53,66,60)(54,61,55), (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), (1,54,4,51)(2,53,5,50)(3,52,6,49)(7,31,10,34)(8,36,11,33)(9,35,12,32)(13,42,16,39)(14,41,17,38)(15,40,18,37)(19,59,22,56)(20,58,23,55)(21,57,24,60)(25,62,28,65)(26,61,29,64)(27,66,30,63)(43,67,46,70)(44,72,47,69)(45,71,48,68) );
G=PermutationGroup([[(1,47),(2,48),(3,43),(4,44),(5,45),(6,46),(7,63),(8,64),(9,65),(10,66),(11,61),(12,62),(13,57),(14,58),(15,59),(16,60),(17,55),(18,56),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,35),(26,36),(27,31),(28,32),(29,33),(30,34),(49,70),(50,71),(51,72),(52,67),(53,68),(54,69)], [(1,23,26),(2,24,27),(3,19,28),(4,20,29),(5,21,30),(6,22,25),(7,13,71),(8,14,72),(9,15,67),(10,16,68),(11,17,69),(12,18,70),(31,48,42),(32,43,37),(33,44,38),(34,45,39),(35,46,40),(36,47,41),(49,62,56),(50,63,57),(51,64,58),(52,65,59),(53,66,60),(54,61,55)], [(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)], [(1,54,4,51),(2,53,5,50),(3,52,6,49),(7,31,10,34),(8,36,11,33),(9,35,12,32),(13,42,16,39),(14,41,17,38),(15,40,18,37),(19,59,22,56),(20,58,23,55),(21,57,24,60),(25,62,28,65),(26,61,29,64),(27,66,30,63),(43,67,46,70),(44,72,47,69),(45,71,48,68)]])
C2×C3⋊Dic3 is a maximal subgroup of
Dic32 D6⋊Dic3 Dic3⋊Dic3 C62.C22 C6.Dic6 C12⋊Dic3 C6.11D12 C62⋊5C4 C62.C4 C2×S3×Dic3 D6.4D6 C2×C4×C3⋊S3 C12.D6 C6.GL2(𝔽3)
C2×C3⋊Dic3 is a maximal quotient of
C12.58D6 C12⋊Dic3 C62⋊5C4
Matrix representation of C2×C3⋊Dic3 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 12 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 10 |
| 0 | 0 | 0 | 7 | 10 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,9,0,0,0,0,0,3,0,0,0,0,0,0,12,0,0,0,1,12],[12,0,0,0,0,0,9,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1],[5,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,3,7,0,0,0,10,10] >;
C2×C3⋊Dic3 in GAP, Magma, Sage, TeX
C_2\times C_3\rtimes {\rm Dic}_3 % in TeX
G:=Group("C2xC3:Dic3"); // GroupNames label
G:=SmallGroup(72,34);
// by ID
G=gap.SmallGroup(72,34);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-3,20,323,1204]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^3=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of C2×C3⋊Dic3 in TeX
Character table of C2×C3⋊Dic3 in TeX