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G = Dic18order 72 = 23·32

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic18, C9⋊Q8, C4.D9, C6.6D6, C3.Dic6, C36.1C2, C12.1S3, C2.3D18, Dic9.C2, C18.1C22, SmallGroup(72,4)

Series: Derived Chief Lower central Upper central

C1C18 — Dic18
C1C3C9C18Dic9 — Dic18
C9C18 — Dic18
C1C2C4

Generators and relations for Dic18
 G = < a,b | a36=1, b2=a18, bab-1=a-1 >

9C4
9C4
9Q8
3Dic3
3Dic3
3Dic6

Character table of Dic18

 class 1234A4B4C69A9B9C12A12B18A18B18C36A36B36C36D36E36F
 size 11221818222222222222222
ρ1111111111111111111111    trivial
ρ2111-11-11111-1-1111-1-1-1-1-1-1    linear of order 2
ρ3111-1-111111-1-1111-1-1-1-1-1-1    linear of order 2
ρ41111-1-1111111111111111    linear of order 2
ρ522-1-200-1ζ989ζ9792ζ959411ζ9792ζ9594ζ9899899594979298997929594    orthogonal lifted from D18
ρ62222002-1-1-122-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ7222-2002-1-1-1-2-2-1-1-1111111    orthogonal lifted from D6
ρ822-1-200-1ζ9594ζ989ζ979211ζ989ζ9792ζ95949594979298995949899792    orthogonal lifted from D18
ρ922-1200-1ζ989ζ9792ζ9594-1-1ζ9792ζ9594ζ989ζ989ζ9594ζ9792ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ1022-1200-1ζ9594ζ989ζ9792-1-1ζ989ζ9792ζ9594ζ9594ζ9792ζ989ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ1122-1200-1ζ9792ζ9594ζ989-1-1ζ9594ζ989ζ9792ζ9792ζ989ζ9594ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ1222-1-200-1ζ9792ζ9594ζ98911ζ9594ζ989ζ97929792989959497929594989    orthogonal lifted from D18
ρ132-22000-222200-2-2-2000000    symplectic lifted from Q8, Schur index 2
ρ142-22000-2-1-1-1001113-3-3-333    symplectic lifted from Dic6, Schur index 2
ρ152-22000-2-1-1-100111-3333-3-3    symplectic lifted from Dic6, Schur index 2
ρ162-2-10001ζ989ζ9792ζ9594-3397929594989ζ43ζ9843ζ9ζ4ζ954ζ944ζ974ζ9243ζ9843ζ9ζ4ζ974ζ924ζ954ζ94    symplectic faithful, Schur index 2
ρ172-2-10001ζ9594ζ989ζ9792-33989979295944ζ954ζ944ζ974ζ9243ζ9843ζ9ζ4ζ954ζ94ζ43ζ9843ζ9ζ4ζ974ζ92    symplectic faithful, Schur index 2
ρ182-2-10001ζ9792ζ9594ζ9893-3959498997924ζ974ζ92ζ43ζ9843ζ94ζ954ζ94ζ4ζ974ζ92ζ4ζ954ζ9443ζ9843ζ9    symplectic faithful, Schur index 2
ρ192-2-10001ζ989ζ9792ζ95943-39792959498943ζ9843ζ94ζ954ζ94ζ4ζ974ζ92ζ43ζ9843ζ94ζ974ζ92ζ4ζ954ζ94    symplectic faithful, Schur index 2
ρ202-2-10001ζ9792ζ9594ζ989-3395949899792ζ4ζ974ζ9243ζ9843ζ9ζ4ζ954ζ944ζ974ζ924ζ954ζ94ζ43ζ9843ζ9    symplectic faithful, Schur index 2
ρ212-2-10001ζ9594ζ989ζ97923-398997929594ζ4ζ954ζ94ζ4ζ974ζ92ζ43ζ9843ζ94ζ954ζ9443ζ9843ζ94ζ974ζ92    symplectic faithful, Schur index 2

Smallest permutation representation of Dic18
Regular action on 72 points
Generators in S72
(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 69 19 51)(2 68 20 50)(3 67 21 49)(4 66 22 48)(5 65 23 47)(6 64 24 46)(7 63 25 45)(8 62 26 44)(9 61 27 43)(10 60 28 42)(11 59 29 41)(12 58 30 40)(13 57 31 39)(14 56 32 38)(15 55 33 37)(16 54 34 72)(17 53 35 71)(18 52 36 70)

G:=sub<Sym(72)| (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), (1,69,19,51)(2,68,20,50)(3,67,21,49)(4,66,22,48)(5,65,23,47)(6,64,24,46)(7,63,25,45)(8,62,26,44)(9,61,27,43)(10,60,28,42)(11,59,29,41)(12,58,30,40)(13,57,31,39)(14,56,32,38)(15,55,33,37)(16,54,34,72)(17,53,35,71)(18,52,36,70)>;

G:=Group( (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), (1,69,19,51)(2,68,20,50)(3,67,21,49)(4,66,22,48)(5,65,23,47)(6,64,24,46)(7,63,25,45)(8,62,26,44)(9,61,27,43)(10,60,28,42)(11,59,29,41)(12,58,30,40)(13,57,31,39)(14,56,32,38)(15,55,33,37)(16,54,34,72)(17,53,35,71)(18,52,36,70) );

G=PermutationGroup([(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)], [(1,69,19,51),(2,68,20,50),(3,67,21,49),(4,66,22,48),(5,65,23,47),(6,64,24,46),(7,63,25,45),(8,62,26,44),(9,61,27,43),(10,60,28,42),(11,59,29,41),(12,58,30,40),(13,57,31,39),(14,56,32,38),(15,55,33,37),(16,54,34,72),(17,53,35,71),(18,52,36,70)])

Dic18 is a maximal subgroup of
Dic36  C72⋊C2  D4.D9  C9⋊Q16  D365C2  D42D9  Q8×D9  Dic54  C9⋊Dic6  C36.C6  C12.D9  C12.1S4  C12.3S4  C45⋊Q8  Dic90
Dic18 is a maximal quotient of
Dic9⋊C4  C4⋊Dic9  Dic54  C9⋊Dic6  C12.D9  C12.1S4  C45⋊Q8  Dic90

Matrix representation of Dic18 in GL2(𝔽37) generated by

176
60
,
1525
2522
G:=sub<GL(2,GF(37))| [17,6,6,0],[15,25,25,22] >;

Dic18 in GAP, Magma, Sage, TeX

{\rm Dic}_{18}
% in TeX

G:=Group("Dic18");
// GroupNames label

G:=SmallGroup(72,4);
// by ID

G=gap.SmallGroup(72,4);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,20,61,26,803,138,1204]);
// Polycyclic

G:=Group<a,b|a^36=1,b^2=a^18,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic18 in TeX
Character table of Dic18 in TeX

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