direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×Dic9, C18⋊C4, C6.9D6, C22.D9, C2.2D18, C6.2Dic3, C18.4C22, C9⋊2(C2×C4), (C2×C18).C2, (C2×C6).2S3, C3.(C2×Dic3), SmallGroup(72,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C2×Dic9 |
Generators and relations for C2×Dic9
G = < a,b,c | a2=b18=1, c2=b9, ab=ba, ac=ca, cbc-1=b-1 >
Character table of C2×Dic9
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 9A | 9B | 9C | 18A | 18B | 18C | 18D | 18E | 18F | 18G | 18H | 18I | |
| size | 1 | 1 | 1 | 1 | 2 | 9 | 9 | 9 | 9 | 2 | 2 | 2 | 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 | -i | i | i | -i | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | -i | -i | i | i | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | i | i | -i | -i | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | 1 | i | -i | -i | i | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D18 |
| ρ10 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ13 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ14 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D18 |
| ρ15 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D18 |
| ρ16 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ17 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ18 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ19 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | ζ95+ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | symplectic lifted from Dic9, Schur index 2 |
| ρ20 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | ζ97+ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | symplectic lifted from Dic9, Schur index 2 |
| ρ21 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | ζ98+ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | symplectic lifted from Dic9, Schur index 2 |
| ρ22 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | -ζ95-ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | symplectic lifted from Dic9, Schur index 2 |
| ρ23 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | -ζ98-ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | symplectic lifted from Dic9, Schur index 2 |
| ρ24 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | -ζ97-ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | symplectic lifted from Dic9, Schur index 2 |
►Regular action on 72 points
Generators in S72
(1 33)(2 34)(3 35)(4 36)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 31)(18 32)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)
(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 40 10 49)(2 39 11 48)(3 38 12 47)(4 37 13 46)(5 54 14 45)(6 53 15 44)(7 52 16 43)(8 51 17 42)(9 50 18 41)(19 66 28 57)(20 65 29 56)(21 64 30 55)(22 63 31 72)(23 62 32 71)(24 61 33 70)(25 60 34 69)(26 59 35 68)(27 58 36 67)
G:=sub<Sym(72)| (1,33)(2,34)(3,35)(4,36)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66), (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), (1,40,10,49)(2,39,11,48)(3,38,12,47)(4,37,13,46)(5,54,14,45)(6,53,15,44)(7,52,16,43)(8,51,17,42)(9,50,18,41)(19,66,28,57)(20,65,29,56)(21,64,30,55)(22,63,31,72)(23,62,32,71)(24,61,33,70)(25,60,34,69)(26,59,35,68)(27,58,36,67)>;
G:=Group( (1,33)(2,34)(3,35)(4,36)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66), (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), (1,40,10,49)(2,39,11,48)(3,38,12,47)(4,37,13,46)(5,54,14,45)(6,53,15,44)(7,52,16,43)(8,51,17,42)(9,50,18,41)(19,66,28,57)(20,65,29,56)(21,64,30,55)(22,63,31,72)(23,62,32,71)(24,61,33,70)(25,60,34,69)(26,59,35,68)(27,58,36,67) );
G=PermutationGroup([[(1,33),(2,34),(3,35),(4,36),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30),(17,31),(18,32),(37,67),(38,68),(39,69),(40,70),(41,71),(42,72),(43,55),(44,56),(45,57),(46,58),(47,59),(48,60),(49,61),(50,62),(51,63),(52,64),(53,65),(54,66)], [(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)], [(1,40,10,49),(2,39,11,48),(3,38,12,47),(4,37,13,46),(5,54,14,45),(6,53,15,44),(7,52,16,43),(8,51,17,42),(9,50,18,41),(19,66,28,57),(20,65,29,56),(21,64,30,55),(22,63,31,72),(23,62,32,71),(24,61,33,70),(25,60,34,69),(26,59,35,68),(27,58,36,67)]])
C2×Dic9 is a maximal subgroup of
Dic9⋊C4 C4⋊Dic9 D18⋊C4 C18.D4 C2×C4×D9 D4⋊2D9 Q8⋊Dic9
C2×Dic9 is a maximal quotient of C4.Dic9 C4⋊Dic9 C18.D4
Matrix representation of C2×Dic9 ►in GL3(𝔽37) generated by
| 1 | 0 | 0 |
| 0 | 36 | 0 |
| 0 | 0 | 36 |
| 36 | 0 | 0 |
| 0 | 11 | 20 |
| 0 | 17 | 31 |
| 6 | 0 | 0 |
| 0 | 26 | 17 |
| 0 | 6 | 11 |
G:=sub<GL(3,GF(37))| [1,0,0,0,36,0,0,0,36],[36,0,0,0,11,17,0,20,31],[6,0,0,0,26,6,0,17,11] >;
C2×Dic9 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_9 % in TeX
G:=Group("C2xDic9"); // GroupNames label
G:=SmallGroup(72,7);
// by ID
G=gap.SmallGroup(72,7);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-3,20,803,138,1204]);
// Polycyclic
G:=Group<a,b,c|a^2=b^18=1,c^2=b^9,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C2×Dic9 in TeX
Character table of C2×Dic9 in TeX