Copied to
clipboard

G = Dic21order 84 = 22·3·7

Dicyclic group

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

Aliases: Dic21, C6.D7, C3⋊Dic7, C7⋊Dic3, C211C4, C14.S3, C2.D21, C42.1C2, SmallGroup(84,5)

Series: Derived Chief Lower central Upper central

C1C21 — Dic21
C1C7C21C42 — Dic21
C21 — Dic21
C1C2

Generators and relations for Dic21
 G = < a,b | a42=1, b2=a21, bab-1=a-1 >

21C4
7Dic3
3Dic7

Character table of Dic21

 class 1234A4B67A7B7C14A14B14C21A21B21C21D21E21F42A42B42C42D42E42F
 size 11221212222222222222222222
ρ1111111111111111111111111    trivial
ρ2111-1-11111111111111111111    linear of order 2
ρ31-11-ii-1111-1-1-1111111-1-1-1-1-1-1    linear of order 4
ρ41-11i-i-1111-1-1-1111111-1-1-1-1-1-1    linear of order 4
ρ522-100-1222222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ622-100-1ζ7572ζ767ζ7473ζ7473ζ7572ζ76732ζ7432ζ7374ζ32ζ7432ζ7373ζ3ζ763ζ773ζ753ζ72753ζ763ζ77632ζ7532ζ727532ζ7432ζ7374ζ32ζ7432ζ73733ζ753ζ72753ζ763ζ776ζ3ζ763ζ7732ζ7532ζ7275    orthogonal lifted from D21
ρ7222002ζ767ζ7473ζ7572ζ7572ζ767ζ7473ζ7572ζ7572ζ7473ζ767ζ7473ζ767ζ7572ζ7572ζ767ζ7473ζ7473ζ767    orthogonal lifted from D7
ρ8222002ζ7572ζ767ζ7473ζ7473ζ7572ζ767ζ7473ζ7473ζ767ζ7572ζ767ζ7572ζ7473ζ7473ζ7572ζ767ζ767ζ7572    orthogonal lifted from D7
ρ9222002ζ7473ζ7572ζ767ζ767ζ7473ζ7572ζ767ζ767ζ7572ζ7473ζ7572ζ7473ζ767ζ767ζ7473ζ7572ζ7572ζ7473    orthogonal lifted from D7
ρ1022-100-1ζ7473ζ7572ζ767ζ767ζ7473ζ75723ζ763ζ776ζ3ζ763ζ7732ζ7532ζ727532ζ7432ζ73743ζ753ζ7275ζ32ζ7432ζ73733ζ763ζ776ζ3ζ763ζ7732ζ7432ζ73743ζ753ζ727532ζ7532ζ7275ζ32ζ7432ζ7373    orthogonal lifted from D21
ρ1122-100-1ζ7473ζ7572ζ767ζ767ζ7473ζ7572ζ3ζ763ζ773ζ763ζ7763ζ753ζ7275ζ32ζ7432ζ737332ζ7532ζ727532ζ7432ζ7374ζ3ζ763ζ773ζ763ζ776ζ32ζ7432ζ737332ζ7532ζ72753ζ753ζ727532ζ7432ζ7374    orthogonal lifted from D21
ρ1222-100-1ζ767ζ7473ζ7572ζ7572ζ767ζ74733ζ753ζ727532ζ7532ζ7275ζ32ζ7432ζ73733ζ763ζ77632ζ7432ζ7374ζ3ζ763ζ773ζ753ζ727532ζ7532ζ72753ζ763ζ77632ζ7432ζ7374ζ32ζ7432ζ7373ζ3ζ763ζ77    orthogonal lifted from D21
ρ1322-100-1ζ7572ζ767ζ7473ζ7473ζ7572ζ767ζ32ζ7432ζ737332ζ7432ζ73743ζ763ζ77632ζ7532ζ7275ζ3ζ763ζ773ζ753ζ7275ζ32ζ7432ζ737332ζ7432ζ737432ζ7532ζ7275ζ3ζ763ζ773ζ763ζ7763ζ753ζ7275    orthogonal lifted from D21
ρ1422-100-1ζ767ζ7473ζ7572ζ7572ζ767ζ747332ζ7532ζ72753ζ753ζ727532ζ7432ζ7374ζ3ζ763ζ77ζ32ζ7432ζ73733ζ763ζ77632ζ7532ζ72753ζ753ζ7275ζ3ζ763ζ77ζ32ζ7432ζ737332ζ7432ζ73743ζ763ζ776    orthogonal lifted from D21
ρ152-2-1001222-2-2-2-1-1-1-1-1-1111111    symplectic lifted from Dic3, Schur index 2
ρ162-2200-2ζ767ζ7473ζ757275727677473ζ7572ζ7572ζ7473ζ767ζ7473ζ7677572757276774737473767    symplectic lifted from Dic7, Schur index 2
ρ172-2200-2ζ7473ζ7572ζ76776774737572ζ767ζ767ζ7572ζ7473ζ7572ζ74737677677473757275727473    symplectic lifted from Dic7, Schur index 2
ρ182-2-1001ζ767ζ7473ζ7572757276774733ζ753ζ727532ζ7532ζ7275ζ32ζ7432ζ73733ζ763ζ77632ζ7432ζ7374ζ3ζ763ζ77ζ3ζ753ζ72753ζ753ζ727232ζ7632ζ77ζ32ζ7432ζ7374ζ3ζ743ζ7374ζ32ζ7632ζ776    symplectic faithful, Schur index 2
ρ192-2-1001ζ7572ζ767ζ747374737572767ζ32ζ7432ζ737332ζ7432ζ73743ζ763ζ77632ζ7532ζ7275ζ3ζ763ζ773ζ753ζ7275ζ3ζ743ζ7374ζ32ζ7432ζ73743ζ753ζ7272ζ32ζ7632ζ77632ζ7632ζ77ζ3ζ753ζ7275    symplectic faithful, Schur index 2
ρ202-2-1001ζ7473ζ7572ζ76776774737572ζ3ζ763ζ773ζ763ζ7763ζ753ζ7275ζ32ζ7432ζ737332ζ7532ζ727532ζ7432ζ7374ζ32ζ7632ζ77632ζ7632ζ77ζ3ζ743ζ73743ζ753ζ7272ζ3ζ753ζ7275ζ32ζ7432ζ7374    symplectic faithful, Schur index 2
ρ212-2-1001ζ7572ζ767ζ74737473757276732ζ7432ζ7374ζ32ζ7432ζ7373ζ3ζ763ζ773ζ753ζ72753ζ763ζ77632ζ7532ζ7275ζ32ζ7432ζ7374ζ3ζ743ζ7374ζ3ζ753ζ727532ζ7632ζ77ζ32ζ7632ζ7763ζ753ζ7272    symplectic faithful, Schur index 2
ρ222-2-1001ζ7473ζ7572ζ767767747375723ζ763ζ776ζ3ζ763ζ7732ζ7532ζ727532ζ7432ζ73743ζ753ζ7275ζ32ζ7432ζ737332ζ7632ζ77ζ32ζ7632ζ776ζ32ζ7432ζ7374ζ3ζ753ζ72753ζ753ζ7272ζ3ζ743ζ7374    symplectic faithful, Schur index 2
ρ232-2200-2ζ7572ζ767ζ747374737572767ζ7473ζ7473ζ767ζ7572ζ767ζ75727473747375727677677572    symplectic lifted from Dic7, Schur index 2
ρ242-2-1001ζ767ζ7473ζ75727572767747332ζ7532ζ72753ζ753ζ727532ζ7432ζ7374ζ3ζ763ζ77ζ32ζ7432ζ73733ζ763ζ7763ζ753ζ7272ζ3ζ753ζ7275ζ32ζ7632ζ776ζ3ζ743ζ7374ζ32ζ7432ζ737432ζ7632ζ77    symplectic faithful, Schur index 2

Smallest permutation representation of Dic21
Regular action on 84 points
Generators in S84
(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)
(1 81 22 60)(2 80 23 59)(3 79 24 58)(4 78 25 57)(5 77 26 56)(6 76 27 55)(7 75 28 54)(8 74 29 53)(9 73 30 52)(10 72 31 51)(11 71 32 50)(12 70 33 49)(13 69 34 48)(14 68 35 47)(15 67 36 46)(16 66 37 45)(17 65 38 44)(18 64 39 43)(19 63 40 84)(20 62 41 83)(21 61 42 82)

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

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), (1,81,22,60)(2,80,23,59)(3,79,24,58)(4,78,25,57)(5,77,26,56)(6,76,27,55)(7,75,28,54)(8,74,29,53)(9,73,30,52)(10,72,31,51)(11,71,32,50)(12,70,33,49)(13,69,34,48)(14,68,35,47)(15,67,36,46)(16,66,37,45)(17,65,38,44)(18,64,39,43)(19,63,40,84)(20,62,41,83)(21,61,42,82) );

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)], [(1,81,22,60),(2,80,23,59),(3,79,24,58),(4,78,25,57),(5,77,26,56),(6,76,27,55),(7,75,28,54),(8,74,29,53),(9,73,30,52),(10,72,31,51),(11,71,32,50),(12,70,33,49),(13,69,34,48),(14,68,35,47),(15,67,36,46),(16,66,37,45),(17,65,38,44),(18,64,39,43),(19,63,40,84),(20,62,41,83),(21,61,42,82)])

Dic21 is a maximal subgroup of
Dic3×D7  S3×Dic7  C21⋊D4  C21⋊Q8  Dic42  C4×D21  C217D4  Dic63  C6.F7  C3⋊Dic21  Q8.D21  A4⋊Dic7  Dic105  C5⋊Dic21
Dic21 is a maximal quotient of
C21⋊C8  Dic63  C3⋊Dic21  A4⋊Dic7  Dic105  C5⋊Dic21

Matrix representation of Dic21 in GL2(𝔽41) generated by

262
227
,
925
032
G:=sub<GL(2,GF(41))| [26,2,2,27],[9,0,25,32] >;

Dic21 in GAP, Magma, Sage, TeX

{\rm Dic}_{21}
% in TeX

G:=Group("Dic21");
// GroupNames label

G:=SmallGroup(84,5);
// by ID

G=gap.SmallGroup(84,5);
# by ID

G:=PCGroup([4,-2,-2,-3,-7,8,98,1155]);
// Polycyclic

G:=Group<a,b|a^42=1,b^2=a^21,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic21 in TeX
Character table of Dic21 in TeX

׿
×
𝔽