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G = C3×Dic7order 84 = 22·3·7

Direct product of C3 and Dic7

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

Aliases: C3×Dic7, C212C4, C73C12, C6.2D7, C42.2C2, C14.3C6, C2.(C3×D7), SmallGroup(84,4)

Series: Derived Chief Lower central Upper central

C1C7 — C3×Dic7
C1C7C14C42 — C3×Dic7
C7 — C3×Dic7
C1C6

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

7C4
7C12

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
ρ1322220022ζ7473ζ767ζ75720000ζ7572ζ7473ζ767ζ7473ζ7473ζ767ζ767ζ7572ζ7572ζ7473ζ7572ζ7473ζ767ζ767ζ7572    orthogonal lifted from D7
ρ1422220022ζ7572ζ7473ζ7670000ζ767ζ7572ζ7473ζ7572ζ7572ζ7473ζ7473ζ767ζ767ζ7572ζ767ζ7572ζ7473ζ7473ζ767    orthogonal lifted from D7
ρ1522220022ζ767ζ7572ζ74730000ζ7473ζ767ζ7572ζ767ζ767ζ7572ζ7572ζ7473ζ7473ζ767ζ7473ζ767ζ7572ζ7572ζ7473    orthogonal lifted from D7
ρ162-22200-2-2ζ767ζ7572ζ7473000074737677572ζ767ζ767ζ7572ζ7572ζ7473ζ74737677473767757275727473    symplectic lifted from Dic7, Schur index 2
ρ172-22200-2-2ζ7473ζ767ζ7572000075727473767ζ7473ζ7473ζ767ζ767ζ7572ζ75727473757274737677677572    symplectic lifted from Dic7, Schur index 2
ρ182-22200-2-2ζ7572ζ7473ζ767000076775727473ζ7572ζ7572ζ7473ζ7473ζ767ζ7677572767757274737473767    symplectic lifted from Dic7, Schur index 2
ρ1922-1--3-1+-300-1+-3-1--3ζ7473ζ767ζ75720000ζ7572ζ7473ζ767ζ32ζ7432ζ73ζ3ζ743ζ73ζ3ζ763ζ7ζ32ζ7632ζ7ζ32ζ7532ζ72ζ3ζ753ζ72ζ32ζ7432ζ73ζ3ζ753ζ72ζ3ζ743ζ73ζ3ζ763ζ7ζ32ζ7632ζ7ζ32ζ7532ζ72    complex lifted from C3×D7
ρ202-2-1+-3-1--3001+-31--3ζ7473ζ767ζ7572000075727473767ζ3ζ743ζ73ζ32ζ7432ζ73ζ32ζ7632ζ7ζ3ζ763ζ7ζ3ζ753ζ72ζ32ζ7532ζ723ζ743ζ7332ζ7532ζ7232ζ7432ζ7332ζ7632ζ73ζ763ζ73ζ753ζ72    complex faithful, Schur index 2
ρ2122-1+-3-1--300-1--3-1+-3ζ767ζ7572ζ74730000ζ7473ζ767ζ7572ζ3ζ763ζ7ζ32ζ7632ζ7ζ32ζ7532ζ72ζ3ζ753ζ72ζ3ζ743ζ73ζ32ζ7432ζ73ζ3ζ763ζ7ζ32ζ7432ζ73ζ32ζ7632ζ7ζ32ζ7532ζ72ζ3ζ753ζ72ζ3ζ743ζ73    complex lifted from C3×D7
ρ2222-1+-3-1--300-1--3-1+-3ζ7572ζ7473ζ7670000ζ767ζ7572ζ7473ζ3ζ753ζ72ζ32ζ7532ζ72ζ32ζ7432ζ73ζ3ζ743ζ73ζ3ζ763ζ7ζ32ζ7632ζ7ζ3ζ753ζ72ζ32ζ7632ζ7ζ32ζ7532ζ72ζ32ζ7432ζ73ζ3ζ743ζ73ζ3ζ763ζ7    complex lifted from C3×D7
ρ232-2-1--3-1+-3001--31+-3ζ7473ζ767ζ7572000075727473767ζ32ζ7432ζ73ζ3ζ743ζ73ζ3ζ763ζ7ζ32ζ7632ζ7ζ32ζ7532ζ72ζ3ζ753ζ7232ζ7432ζ733ζ753ζ723ζ743ζ733ζ763ζ732ζ7632ζ732ζ7532ζ72    complex faithful, Schur index 2
ρ2422-1--3-1+-300-1+-3-1--3ζ767ζ7572ζ74730000ζ7473ζ767ζ7572ζ32ζ7632ζ7ζ3ζ763ζ7ζ3ζ753ζ72ζ32ζ7532ζ72ζ32ζ7432ζ73ζ3ζ743ζ73ζ32ζ7632ζ7ζ3ζ743ζ73ζ3ζ763ζ7ζ3ζ753ζ72ζ32ζ7532ζ72ζ32ζ7432ζ73    complex lifted from C3×D7
ρ252-2-1--3-1+-3001--31+-3ζ767ζ7572ζ7473000074737677572ζ32ζ7632ζ7ζ3ζ763ζ7ζ3ζ753ζ72ζ32ζ7532ζ72ζ32ζ7432ζ73ζ3ζ743ζ7332ζ7632ζ73ζ743ζ733ζ763ζ73ζ753ζ7232ζ7532ζ7232ζ7432ζ73    complex faithful, Schur index 2
ρ262-2-1+-3-1--3001+-31--3ζ7572ζ7473ζ767000076775727473ζ3ζ753ζ72ζ32ζ7532ζ72ζ32ζ7432ζ73ζ3ζ743ζ73ζ3ζ763ζ7ζ32ζ7632ζ73ζ753ζ7232ζ7632ζ732ζ7532ζ7232ζ7432ζ733ζ743ζ733ζ763ζ7    complex faithful, Schur index 2
ρ272-2-1--3-1+-3001--31+-3ζ7572ζ7473ζ767000076775727473ζ32ζ7532ζ72ζ3ζ753ζ72ζ3ζ743ζ73ζ32ζ7432ζ73ζ32ζ7632ζ7ζ3ζ763ζ732ζ7532ζ723ζ763ζ73ζ753ζ723ζ743ζ7332ζ7432ζ7332ζ7632ζ7    complex faithful, Schur index 2
ρ2822-1--3-1+-300-1+-3-1--3ζ7572ζ7473ζ7670000ζ767ζ7572ζ7473ζ32ζ7532ζ72ζ3ζ753ζ72ζ3ζ743ζ73ζ32ζ7432ζ73ζ32ζ7632ζ7ζ3ζ763ζ7ζ32ζ7532ζ72ζ3ζ763ζ7ζ3ζ753ζ72ζ3ζ743ζ73ζ32ζ7432ζ73ζ32ζ7632ζ7    complex lifted from C3×D7
ρ292-2-1+-3-1--3001+-31--3ζ767ζ7572ζ7473000074737677572ζ3ζ763ζ7ζ32ζ7632ζ7ζ32ζ7532ζ72ζ3ζ753ζ72ζ3ζ743ζ73ζ32ζ7432ζ733ζ763ζ732ζ7432ζ7332ζ7632ζ732ζ7532ζ723ζ753ζ723ζ743ζ73    complex faithful, Schur index 2
ρ3022-1+-3-1--300-1--3-1+-3ζ7473ζ767ζ75720000ζ7572ζ7473ζ767ζ3ζ743ζ73ζ32ζ7432ζ73ζ32ζ7632ζ7ζ3ζ763ζ7ζ3ζ753ζ72ζ32ζ7532ζ72ζ3ζ743ζ73ζ32ζ7532ζ72ζ32ζ7432ζ73ζ32ζ7632ζ7ζ3ζ763ζ7ζ3ζ753ζ72    complex lifted from C3×D7

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

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

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

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

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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