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G = C3×Dic7  order 84 = 22·3·7

Direct product of C3 and Dic7

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

Aliases: C3×Dic7, C21⋊2C4, C7⋊3C12, C6.2D7, C42.2C2, C14.3C6, C2.(C3×D7), SmallGroup(84,4)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C7 — C3×Dic7
Chief series
C1 — C7 — C14 — C42 — C3×Dic7
Lower central
C7 — C3×Dic7
Upper central
C1 — C6

Generators and relations for C3×Dic7
 G = < a,b,c | a3=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >

C1
C2
C3
C7
7C4
C6
C14
C21
7C12
Dic7
C42
C3×Dic7

Character table of C3×Dic7

 class 123A3B4A4B6A6B7A7B7C12A12B12C12D14A14B14C21A21B21C21D21E21F42A42B42C42D42E42F
 size 111177112227777222222222222222
ρ1111111111111111111111111111111    trivial
ρ21111-1-111111-1-1-1-1111111111111111    linear of order 2
ρ311ζ32ζ311ζ3ζ32111ζ3ζ32ζ3ζ32111ζ32ζ3ζ3ζ32ζ32ζ3ζ32ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ411ζ3ζ3211ζ32ζ3111ζ32ζ3ζ32ζ3111ζ3ζ32ζ32ζ3ζ3ζ32ζ3ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ511ζ32ζ3-1-1ζ3ζ32111ζ65ζ6ζ65ζ6111ζ32ζ3ζ3ζ32ζ32ζ3ζ32ζ3ζ3ζ3ζ32ζ32    linear of order 6
ρ611ζ3ζ32-1-1ζ32ζ3111ζ6ζ65ζ6ζ65111ζ3ζ32ζ32ζ3ζ3ζ32ζ3ζ32ζ32ζ32ζ3ζ3    linear of order 6
ρ71-111i-i-1-1111ii-i-i-1-1-1111111-1-1-1-1-1-1    linear of order 4
ρ81-111-ii-1-1111-i-iii-1-1-1111111-1-1-1-1-1-1    linear of order 4
ρ91-1ζ32ζ3i-iζ65ζ6111ζ4ζ3ζ4ζ32ζ43ζ3ζ43ζ32-1-1-1ζ32ζ3ζ3ζ32ζ32ζ3ζ6ζ65ζ65ζ65ζ6ζ6    linear of order 12
ρ101-1ζ3ζ32i-iζ6ζ65111ζ4ζ32ζ4ζ3ζ43ζ32ζ43ζ3-1-1-1ζ3ζ32ζ32ζ3ζ3ζ32ζ65ζ6ζ6ζ6ζ65ζ65    linear of order 12
ρ111-1ζ32ζ3-iiζ65ζ6111ζ43ζ3ζ43ζ32ζ4ζ3ζ4ζ32-1-1-1ζ32ζ3ζ3ζ32ζ32ζ3ζ6ζ65ζ65ζ65ζ6ζ6    linear of order 12
ρ121-1ζ3ζ32-iiζ6ζ65111ζ43ζ32ζ43ζ3ζ4ζ32ζ4ζ3-1-1-1ζ3ζ32ζ32ζ3ζ3ζ32ζ65ζ6ζ6ζ6ζ65ζ65    linear of order 12
ρ1322220022ζ74+ζ73ζ76+ζ7ζ75+ζ720000ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ74+ζ73ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ74+ζ73ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ75+ζ72    orthogonal lifted from D7
ρ1422220022ζ75+ζ72ζ74+ζ73ζ76+ζ70000ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ75+ζ72ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ75+ζ72ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ76+ζ7    orthogonal lifted from D7
ρ1522220022ζ76+ζ7ζ75+ζ72ζ74+ζ730000ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ76+ζ7ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ76+ζ7ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ74+ζ73    orthogonal lifted from D7
ρ162-22200-2-2ζ76+ζ7ζ75+ζ72ζ74+ζ730000-ζ74-ζ73-ζ76-ζ7-ζ75-ζ72ζ76+ζ7ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ74+ζ73ζ74+ζ73-ζ76-ζ7-ζ74-ζ73-ζ76-ζ7-ζ75-ζ72-ζ75-ζ72-ζ74-ζ73    symplectic lifted from Dic7, Schur index 2
ρ172-22200-2-2ζ74+ζ73ζ76+ζ7ζ75+ζ720000-ζ75-ζ72-ζ74-ζ73-ζ76-ζ7ζ74+ζ73ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ75+ζ72ζ75+ζ72-ζ74-ζ73-ζ75-ζ72-ζ74-ζ73-ζ76-ζ7-ζ76-ζ7-ζ75-ζ72    symplectic lifted from Dic7, Schur index 2
ρ182-22200-2-2ζ75+ζ72ζ74+ζ73ζ76+ζ70000-ζ76-ζ7-ζ75-ζ72-ζ74-ζ73ζ75+ζ72ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ76+ζ7ζ76+ζ7-ζ75-ζ72-ζ76-ζ7-ζ75-ζ72-ζ74-ζ73-ζ74-ζ73-ζ76-ζ7    symplectic lifted from Dic7, Schur index 2
ρ1922-1-√-3-1+√-300-1+√-3-1-√-3ζ74+ζ73ζ76+ζ7ζ75+ζ720000ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ32ζ74+ζ32ζ73ζ3ζ74+ζ3ζ73ζ3ζ76+ζ3ζ7ζ32ζ76+ζ32ζ7ζ32ζ75+ζ32ζ72ζ3ζ75+ζ3ζ72ζ32ζ74+ζ32ζ73ζ3ζ75+ζ3ζ72ζ3ζ74+ζ3ζ73ζ3ζ76+ζ3ζ7ζ32ζ76+ζ32ζ7ζ32ζ75+ζ32ζ72    complex lifted from C3×D7
ρ202-2-1+√-3-1-√-3001+√-31-√-3ζ74+ζ73ζ76+ζ7ζ75+ζ720000-ζ75-ζ72-ζ74-ζ73-ζ76-ζ7ζ3ζ74+ζ3ζ73ζ32ζ74+ζ32ζ73ζ32ζ76+ζ32ζ7ζ3ζ76+ζ3ζ7ζ3ζ75+ζ3ζ72ζ32ζ75+ζ32ζ72-ζ3ζ74-ζ3ζ73-ζ32ζ75-ζ32ζ72-ζ32ζ74-ζ32ζ73-ζ32ζ76-ζ32ζ7-ζ3ζ76-ζ3ζ7-ζ3ζ75-ζ3ζ72    complex faithful, Schur index 2
ρ2122-1+√-3-1-√-300-1-√-3-1+√-3ζ76+ζ7ζ75+ζ72ζ74+ζ730000ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ3ζ76+ζ3ζ7ζ32ζ76+ζ32ζ7ζ32ζ75+ζ32ζ72ζ3ζ75+ζ3ζ72ζ3ζ74+ζ3ζ73ζ32ζ74+ζ32ζ73ζ3ζ76+ζ3ζ7ζ32ζ74+ζ32ζ73ζ32ζ76+ζ32ζ7ζ32ζ75+ζ32ζ72ζ3ζ75+ζ3ζ72ζ3ζ74+ζ3ζ73    complex lifted from C3×D7
ρ2222-1+√-3-1-√-300-1-√-3-1+√-3ζ75+ζ72ζ74+ζ73ζ76+ζ70000ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ3ζ75+ζ3ζ72ζ32ζ75+ζ32ζ72ζ32ζ74+ζ32ζ73ζ3ζ74+ζ3ζ73ζ3ζ76+ζ3ζ7ζ32ζ76+ζ32ζ7ζ3ζ75+ζ3ζ72ζ32ζ76+ζ32ζ7ζ32ζ75+ζ32ζ72ζ32ζ74+ζ32ζ73ζ3ζ74+ζ3ζ73ζ3ζ76+ζ3ζ7    complex lifted from C3×D7
ρ232-2-1-√-3-1+√-3001-√-31+√-3ζ74+ζ73ζ76+ζ7ζ75+ζ720000-ζ75-ζ72-ζ74-ζ73-ζ76-ζ7ζ32ζ74+ζ32ζ73ζ3ζ74+ζ3ζ73ζ3ζ76+ζ3ζ7ζ32ζ76+ζ32ζ7ζ32ζ75+ζ32ζ72ζ3ζ75+ζ3ζ72-ζ32ζ74-ζ32ζ73-ζ3ζ75-ζ3ζ72-ζ3ζ74-ζ3ζ73-ζ3ζ76-ζ3ζ7-ζ32ζ76-ζ32ζ7-ζ32ζ75-ζ32ζ72    complex faithful, Schur index 2
ρ2422-1-√-3-1+√-300-1+√-3-1-√-3ζ76+ζ7ζ75+ζ72ζ74+ζ730000ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ32ζ76+ζ32ζ7ζ3ζ76+ζ3ζ7ζ3ζ75+ζ3ζ72ζ32ζ75+ζ32ζ72ζ32ζ74+ζ32ζ73ζ3ζ74+ζ3ζ73ζ32ζ76+ζ32ζ7ζ3ζ74+ζ3ζ73ζ3ζ76+ζ3ζ7ζ3ζ75+ζ3ζ72ζ32ζ75+ζ32ζ72ζ32ζ74+ζ32ζ73    complex lifted from C3×D7
ρ252-2-1-√-3-1+√-3001-√-31+√-3ζ76+ζ7ζ75+ζ72ζ74+ζ730000-ζ74-ζ73-ζ76-ζ7-ζ75-ζ72ζ32ζ76+ζ32ζ7ζ3ζ76+ζ3ζ7ζ3ζ75+ζ3ζ72ζ32ζ75+ζ32ζ72ζ32ζ74+ζ32ζ73ζ3ζ74+ζ3ζ73-ζ32ζ76-ζ32ζ7-ζ3ζ74-ζ3ζ73-ζ3ζ76-ζ3ζ7-ζ3ζ75-ζ3ζ72-ζ32ζ75-ζ32ζ72-ζ32ζ74-ζ32ζ73    complex faithful, Schur index 2
ρ262-2-1+√-3-1-√-3001+√-31-√-3ζ75+ζ72ζ74+ζ73ζ76+ζ70000-ζ76-ζ7-ζ75-ζ72-ζ74-ζ73ζ3ζ75+ζ3ζ72ζ32ζ75+ζ32ζ72ζ32ζ74+ζ32ζ73ζ3ζ74+ζ3ζ73ζ3ζ76+ζ3ζ7ζ32ζ76+ζ32ζ7-ζ3ζ75-ζ3ζ72-ζ32ζ76-ζ32ζ7-ζ32ζ75-ζ32ζ72-ζ32ζ74-ζ32ζ73-ζ3ζ74-ζ3ζ73-ζ3ζ76-ζ3ζ7    complex faithful, Schur index 2
ρ272-2-1-√-3-1+√-3001-√-31+√-3ζ75+ζ72ζ74+ζ73ζ76+ζ70000-ζ76-ζ7-ζ75-ζ72-ζ74-ζ73ζ32ζ75+ζ32ζ72ζ3ζ75+ζ3ζ72ζ3ζ74+ζ3ζ73ζ32ζ74+ζ32ζ73ζ32ζ76+ζ32ζ7ζ3ζ76+ζ3ζ7-ζ32ζ75-ζ32ζ72-ζ3ζ76-ζ3ζ7-ζ3ζ75-ζ3ζ72-ζ3ζ74-ζ3ζ73-ζ32ζ74-ζ32ζ73-ζ32ζ76-ζ32ζ7    complex faithful, Schur index 2
ρ2822-1-√-3-1+√-300-1+√-3-1-√-3ζ75+ζ72ζ74+ζ73ζ76+ζ70000ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ32ζ75+ζ32ζ72ζ3ζ75+ζ3ζ72ζ3ζ74+ζ3ζ73ζ32ζ74+ζ32ζ73ζ32ζ76+ζ32ζ7ζ3ζ76+ζ3ζ7ζ32ζ75+ζ32ζ72ζ3ζ76+ζ3ζ7ζ3ζ75+ζ3ζ72ζ3ζ74+ζ3ζ73ζ32ζ74+ζ32ζ73ζ32ζ76+ζ32ζ7    complex lifted from C3×D7
ρ292-2-1+√-3-1-√-3001+√-31-√-3ζ76+ζ7ζ75+ζ72ζ74+ζ730000-ζ74-ζ73-ζ76-ζ7-ζ75-ζ72ζ3ζ76+ζ3ζ7ζ32ζ76+ζ32ζ7ζ32ζ75+ζ32ζ72ζ3ζ75+ζ3ζ72ζ3ζ74+ζ3ζ73ζ32ζ74+ζ32ζ73-ζ3ζ76-ζ3ζ7-ζ32ζ74-ζ32ζ73-ζ32ζ76-ζ32ζ7-ζ32ζ75-ζ32ζ72-ζ3ζ75-ζ3ζ72-ζ3ζ74-ζ3ζ73    complex faithful, Schur index 2
ρ3022-1+√-3-1-√-300-1-√-3-1+√-3ζ74+ζ73ζ76+ζ7ζ75+ζ720000ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ3ζ74+ζ3ζ73ζ32ζ74+ζ32ζ73ζ32ζ76+ζ32ζ7ζ3ζ76+ζ3ζ7ζ3ζ75+ζ3ζ72ζ32ζ75+ζ32ζ72ζ3ζ74+ζ3ζ73ζ32ζ75+ζ32ζ72ζ32ζ74+ζ32ζ73ζ32ζ76+ζ32ζ7ζ3ζ76+ζ3ζ7ζ3ζ75+ζ3ζ72    complex lifted from C3×D7

Smallest permutation representation of C3×Dic7
►Regular action on 84 points
Generators in S84
(1 32 27)(2 33 28)(3 34 15)(4 35 16)(5 36 17)(6 37 18)(7 38 19)(8 39 20)(9 40 21)(10 41 22)(11 42 23)(12 29 24)(13 30 25)(14 31 26)(43 71 57)(44 72 58)(45 73 59)(46 74 60)(47 75 61)(48 76 62)(49 77 63)(50 78 64)(51 79 65)(52 80 66)(53 81 67)(54 82 68)(55 83 69)(56 84 70)

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

C3×Dic7 is a maximal subgroup of   D21⋊C4  C7⋊D12  C21⋊Q8  C12×D7  C7⋊C36  Dic7.2A4

Matrix representation of C3×Dic7 ►in GL2(𝔽13) generated by

90
09
,
15
114
,
81
05
G:=sub<GL(2,GF(13))| [9,0,0,9],[1,11,5,4],[8,0,1,5] >;
 

C3×Dic7 in GAP, Magma, Sage, TeX 

C_3\times {\rm Dic}_7
 
% in TeX
 
G:=Group("C3xDic7");
 
// GroupNames label
 
G:=SmallGroup(84,4);
 
// by ID
 
G=gap.SmallGroup(84,4);
 
# by ID
 
G:=PCGroup([4,-2,-3,-2,-7,24,1155]);
 
// Polycyclic
 
G:=Group<a,b,c|a^3=b^14=1,c^2=b^7,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C3×Dic7 in TeX
Character table of C3×Dic7 in TeX

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