Copied to
clipboard

G = Dic27order 108 = 22·33

Dicyclic group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: Dic27, C27⋊C4, C54.C2, C2.D27, C6.1D9, C9.Dic3, C3.Dic9, C18.1S3, SmallGroup(108,1)

Series: Derived Chief Lower central Upper central

C1C27 — Dic27
C1C3C9C27C54 — Dic27
C27 — Dic27
C1C2

Generators and relations for Dic27
 G = < a,b | a54=1, b2=a27, bab-1=a-1 >

27C4
9Dic3
3Dic9

Character table of Dic27

 class 1234A4B69A9B9C18A18B18C27A27B27C27D27E27F27G27H27I54A54B54C54D54E54F54G54H54I
 size 11227272222222222222222222222222
ρ1111111111111111111111111111111    trivial
ρ2111-1-11111111111111111111111111    linear of order 2
ρ31-11i-i-1111-1-1-1111111111-1-1-1-1-1-1-1-1-1    linear of order 4
ρ41-11-ii-1111-1-1-1111111111-1-1-1-1-1-1-1-1-1    linear of order 4
ρ5222002222222-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ6222002-1-1-1-1-1-1ζ9594ζ9594ζ989ζ989ζ9792ζ989ζ9792ζ9792ζ9594ζ989ζ9792ζ9792ζ9792ζ9594ζ9594ζ9594ζ989ζ989    orthogonal lifted from D9
ρ722-100-1ζ2721276ζ2724273ζ27152712ζ27152712ζ2721276ζ2724273ζ272627ζ27172710ζ27162711ζ2725272ζ27142713ζ2720277ζ2722275ζ2723274ζ2719278ζ2725272ζ27142713ζ2722275ζ2723274ζ2719278ζ272627ζ27172710ζ2720277ζ27162711    orthogonal lifted from D27
ρ8222002-1-1-1-1-1-1ζ9792ζ9792ζ9594ζ9594ζ989ζ9594ζ989ζ989ζ9792ζ9594ζ989ζ989ζ989ζ9792ζ9792ζ9792ζ9594ζ9594    orthogonal lifted from D9
ρ9222002-1-1-1-1-1-1ζ989ζ989ζ9792ζ9792ζ9594ζ9792ζ9594ζ9594ζ989ζ9792ζ9594ζ9594ζ9594ζ989ζ989ζ989ζ9792ζ9792    orthogonal lifted from D9
ρ1022-100-1ζ27152712ζ2721276ζ2724273ζ2724273ζ27152712ζ2721276ζ2720277ζ27162711ζ2723274ζ27142713ζ27172710ζ2722275ζ2719278ζ272627ζ2725272ζ27142713ζ27172710ζ2719278ζ272627ζ2725272ζ2720277ζ27162711ζ2722275ζ2723274    orthogonal lifted from D27
ρ1122-100-1ζ27152712ζ2721276ζ2724273ζ2724273ζ27152712ζ2721276ζ27162711ζ2725272ζ27142713ζ2722275ζ2719278ζ2723274ζ272627ζ27172710ζ2720277ζ2722275ζ2719278ζ272627ζ27172710ζ2720277ζ27162711ζ2725272ζ2723274ζ27142713    orthogonal lifted from D27
ρ1222-100-1ζ2721276ζ2724273ζ27152712ζ27152712ζ2721276ζ2724273ζ2719278ζ272627ζ2720277ζ27162711ζ2723274ζ2725272ζ27142713ζ2722275ζ27172710ζ27162711ζ2723274ζ27142713ζ2722275ζ27172710ζ2719278ζ272627ζ2725272ζ2720277    orthogonal lifted from D27
ρ1322-100-1ζ2721276ζ2724273ζ27152712ζ27152712ζ2721276ζ2724273ζ27172710ζ2719278ζ2725272ζ2720277ζ2722275ζ27162711ζ2723274ζ27142713ζ272627ζ2720277ζ2722275ζ2723274ζ27142713ζ272627ζ27172710ζ2719278ζ27162711ζ2725272    orthogonal lifted from D27
ρ1422-100-1ζ2724273ζ27152712ζ2721276ζ2721276ζ2724273ζ27152712ζ27142713ζ2722275ζ2719278ζ272627ζ2720277ζ27172710ζ27162711ζ2725272ζ2723274ζ272627ζ2720277ζ27162711ζ2725272ζ2723274ζ27142713ζ2722275ζ27172710ζ2719278    orthogonal lifted from D27
ρ1522-100-1ζ2724273ζ27152712ζ2721276ζ2721276ζ2724273ζ27152712ζ2722275ζ2723274ζ272627ζ27172710ζ27162711ζ2719278ζ2725272ζ2720277ζ27142713ζ27172710ζ27162711ζ2725272ζ2720277ζ27142713ζ2722275ζ2723274ζ2719278ζ272627    orthogonal lifted from D27
ρ1622-100-1ζ27152712ζ2721276ζ2724273ζ2724273ζ27152712ζ2721276ζ2725272ζ2720277ζ2722275ζ2723274ζ272627ζ27142713ζ27172710ζ2719278ζ27162711ζ2723274ζ272627ζ27172710ζ2719278ζ27162711ζ2725272ζ2720277ζ27142713ζ2722275    orthogonal lifted from D27
ρ1722-100-1ζ2724273ζ27152712ζ2721276ζ2721276ζ2724273ζ27152712ζ2723274ζ27142713ζ27172710ζ2719278ζ2725272ζ272627ζ2720277ζ27162711ζ2722275ζ2719278ζ2725272ζ2720277ζ27162711ζ2722275ζ2723274ζ27142713ζ272627ζ27172710    orthogonal lifted from D27
ρ182-2200-2-1-1-1111ζ989ζ989ζ9792ζ9792ζ9594ζ9792ζ9594ζ9594ζ989979295949594959498998998997929792    symplectic lifted from Dic9, Schur index 2
ρ192-2200-2-1-1-1111ζ9594ζ9594ζ989ζ989ζ9792ζ989ζ9792ζ9792ζ9594989979297929792959495949594989989    symplectic lifted from Dic9, Schur index 2
ρ202-2-1001ζ2724273ζ27152712ζ27212762721276272427327152712ζ27142713ζ2722275ζ2719278ζ272627ζ2720277ζ27172710ζ27162711ζ2725272ζ272327427262727202772716271127252722723274271427132722275271727102719278    symplectic faithful, Schur index 2
ρ212-2-1001ζ27152712ζ2721276ζ27242732724273271527122721276ζ27162711ζ2725272ζ27142713ζ2722275ζ2719278ζ2723274ζ272627ζ27172710ζ272027727222752719278272627271727102720277271627112725272272327427142713    symplectic faithful, Schur index 2
ρ222-2200-2222-2-2-2-1-1-1-1-1-1-1-1-1111111111    symplectic lifted from Dic3, Schur index 2
ρ232-2200-2-1-1-1111ζ9792ζ9792ζ9594ζ9594ζ989ζ9594ζ989ζ989ζ9792959498998998997929792979295949594    symplectic lifted from Dic9, Schur index 2
ρ242-2-1001ζ2721276ζ2724273ζ271527122715271227212762724273ζ272627ζ27172710ζ27162711ζ2725272ζ27142713ζ2720277ζ2722275ζ2723274ζ271927827252722714271327222752723274271927827262727172710272027727162711    symplectic faithful, Schur index 2
ρ252-2-1001ζ27152712ζ2721276ζ27242732724273271527122721276ζ2720277ζ27162711ζ2723274ζ27142713ζ27172710ζ2722275ζ2719278ζ272627ζ272527227142713271727102719278272627272527227202772716271127222752723274    symplectic faithful, Schur index 2
ρ262-2-1001ζ2724273ζ27152712ζ27212762721276272427327152712ζ2722275ζ2723274ζ272627ζ27172710ζ27162711ζ2719278ζ2725272ζ2720277ζ2714271327172710271627112725272272027727142713272227527232742719278272627    symplectic faithful, Schur index 2
ρ272-2-1001ζ27152712ζ2721276ζ27242732724273271527122721276ζ2725272ζ2720277ζ2722275ζ2723274ζ272627ζ27142713ζ27172710ζ2719278ζ2716271127232742726272717271027192782716271127252722720277271427132722275    symplectic faithful, Schur index 2
ρ282-2-1001ζ2721276ζ2724273ζ271527122715271227212762724273ζ2719278ζ272627ζ2720277ζ27162711ζ2723274ζ2725272ζ27142713ζ2722275ζ2717271027162711272327427142713272227527172710271927827262727252722720277    symplectic faithful, Schur index 2
ρ292-2-1001ζ2724273ζ27152712ζ27212762721276272427327152712ζ2723274ζ27142713ζ27172710ζ2719278ζ2725272ζ272627ζ2720277ζ27162711ζ272227527192782725272272027727162711272227527232742714271327262727172710    symplectic faithful, Schur index 2
ρ302-2-1001ζ2721276ζ2724273ζ271527122715271227212762724273ζ27172710ζ2719278ζ2725272ζ2720277ζ2722275ζ27162711ζ2723274ζ27142713ζ27262727202772722275272327427142713272627271727102719278271627112725272    symplectic faithful, Schur index 2

Smallest permutation representation of Dic27
Regular action on 108 points
Generators in S108
(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 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108)
(1 65 28 92)(2 64 29 91)(3 63 30 90)(4 62 31 89)(5 61 32 88)(6 60 33 87)(7 59 34 86)(8 58 35 85)(9 57 36 84)(10 56 37 83)(11 55 38 82)(12 108 39 81)(13 107 40 80)(14 106 41 79)(15 105 42 78)(16 104 43 77)(17 103 44 76)(18 102 45 75)(19 101 46 74)(20 100 47 73)(21 99 48 72)(22 98 49 71)(23 97 50 70)(24 96 51 69)(25 95 52 68)(26 94 53 67)(27 93 54 66)

G:=sub<Sym(108)| (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)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108), (1,65,28,92)(2,64,29,91)(3,63,30,90)(4,62,31,89)(5,61,32,88)(6,60,33,87)(7,59,34,86)(8,58,35,85)(9,57,36,84)(10,56,37,83)(11,55,38,82)(12,108,39,81)(13,107,40,80)(14,106,41,79)(15,105,42,78)(16,104,43,77)(17,103,44,76)(18,102,45,75)(19,101,46,74)(20,100,47,73)(21,99,48,72)(22,98,49,71)(23,97,50,70)(24,96,51,69)(25,95,52,68)(26,94,53,67)(27,93,54,66)>;

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)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108), (1,65,28,92)(2,64,29,91)(3,63,30,90)(4,62,31,89)(5,61,32,88)(6,60,33,87)(7,59,34,86)(8,58,35,85)(9,57,36,84)(10,56,37,83)(11,55,38,82)(12,108,39,81)(13,107,40,80)(14,106,41,79)(15,105,42,78)(16,104,43,77)(17,103,44,76)(18,102,45,75)(19,101,46,74)(20,100,47,73)(21,99,48,72)(22,98,49,71)(23,97,50,70)(24,96,51,69)(25,95,52,68)(26,94,53,67)(27,93,54,66) );

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),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)], [(1,65,28,92),(2,64,29,91),(3,63,30,90),(4,62,31,89),(5,61,32,88),(6,60,33,87),(7,59,34,86),(8,58,35,85),(9,57,36,84),(10,56,37,83),(11,55,38,82),(12,108,39,81),(13,107,40,80),(14,106,41,79),(15,105,42,78),(16,104,43,77),(17,103,44,76),(18,102,45,75),(19,101,46,74),(20,100,47,73),(21,99,48,72),(22,98,49,71),(23,97,50,70),(24,96,51,69),(25,95,52,68),(26,94,53,67),(27,93,54,66)])

Dic27 is a maximal subgroup of
Dic54  C4×D27  C27⋊D4  Dic81  C27⋊C12  C27⋊Dic3  Q8.D27  C18.S4
Dic27 is a maximal quotient of
C27⋊C8  Dic81  C27⋊Dic3  C18.S4

Matrix representation of Dic27 in GL2(𝔽109) generated by

1679
3046
,
6567
244
G:=sub<GL(2,GF(109))| [16,30,79,46],[65,2,67,44] >;

Dic27 in GAP, Magma, Sage, TeX

{\rm Dic}_{27}
% in TeX

G:=Group("Dic27");
// GroupNames label

G:=SmallGroup(108,1);
// by ID

G=gap.SmallGroup(108,1);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-3,10,302,237,1203,138,1804]);
// Polycyclic

G:=Group<a,b|a^54=1,b^2=a^27,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic27 in TeX
Character table of Dic27 in TeX

׿
×
𝔽