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G = Dic3×D7order 168 = 23·3·7

Direct product of Dic3 and D7

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: Dic3×D7, D14.S3, C6.1D14, C14.1D6, Dic212C2, C42.1C22, (C3×D7)⋊C4, C33(C4×D7), C211(C2×C4), (C6×D7).C2, C2.1(S3×D7), C71(C2×Dic3), (C7×Dic3)⋊1C2, SmallGroup(168,12)

Series: Derived Chief Lower central Upper central

C1C21 — Dic3×D7
C1C7C21C42C6×D7 — Dic3×D7
C21 — Dic3×D7
C1C2

Generators and relations for Dic3×D7
 G = < a,b,c,d | a6=c7=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

7C2
7C2
3C4
7C22
21C4
7C6
7C6
21C2×C4
7Dic3
7C2×C6
3C28
3Dic7
7C2×Dic3
3C4×D7

Character table of Dic3×D7

 class 12A2B2C34A4B4C4D6A6B6C7A7B7C14A14B14C21A21B21C28A28B28C28D28E28F42A42B42C
 size 1177233212121414222222444666666444
ρ1111111111111111111111111111111    trivial
ρ211111-1-1-1-1111111111111-1-1-1-1-1-1111    linear of order 2
ρ311-1-1111-1-11-1-1111111111111111111    linear of order 2
ρ411-1-11-1-1111-1-1111111111-1-1-1-1-1-1111    linear of order 2
ρ51-11-11-iii-i-11-1111-1-1-1111i-ii-i-ii-1-1-1    linear of order 4
ρ61-1-111i-ii-i-1-11111-1-1-1111-ii-iii-i-1-1-1    linear of order 4
ρ71-11-11i-i-ii-11-1111-1-1-1111-ii-iii-i-1-1-1    linear of order 4
ρ81-1-111-ii-ii-1-11111-1-1-1111i-ii-i-ii-1-1-1    linear of order 4
ρ9220022200200ζ7572ζ767ζ7473ζ7572ζ7473ζ767ζ767ζ7572ζ7473ζ7473ζ767ζ767ζ7572ζ7473ζ7572ζ7572ζ767ζ7473    orthogonal lifted from D7
ρ1022-2-2-10000-111222222-1-1-1000000-1-1-1    orthogonal lifted from D6
ρ112222-10000-1-1-1222222-1-1-1000000-1-1-1    orthogonal lifted from S3
ρ1222002-2-200200ζ7473ζ7572ζ767ζ7473ζ767ζ7572ζ7572ζ7473ζ7677677572757274737677473ζ7473ζ7572ζ767    orthogonal lifted from D14
ρ1322002-2-200200ζ7572ζ767ζ7473ζ7572ζ7473ζ767ζ767ζ7572ζ74737473767767757274737572ζ7572ζ767ζ7473    orthogonal lifted from D14
ρ14220022200200ζ767ζ7473ζ7572ζ767ζ7572ζ7473ζ7473ζ767ζ7572ζ7572ζ7473ζ7473ζ767ζ7572ζ767ζ767ζ7473ζ7572    orthogonal lifted from D7
ρ1522002-2-200200ζ767ζ7473ζ7572ζ767ζ7572ζ7473ζ7473ζ767ζ75727572747374737677572767ζ767ζ7473ζ7572    orthogonal lifted from D14
ρ16220022200200ζ7473ζ7572ζ767ζ7473ζ767ζ7572ζ7572ζ7473ζ767ζ767ζ7572ζ7572ζ7473ζ767ζ7473ζ7473ζ7572ζ767    orthogonal lifted from D7
ρ172-2-22-1000011-1222-2-2-2-1-1-1000000111    symplectic lifted from Dic3, Schur index 2
ρ182-22-2-100001-11222-2-2-2-1-1-1000000111    symplectic lifted from Dic3, Schur index 2
ρ192-20022i-2i00-200ζ7473ζ7572ζ76774737677572ζ7572ζ7473ζ767ζ43ζ7643ζ7ζ4ζ754ζ72ζ43ζ7543ζ72ζ4ζ744ζ73ζ4ζ764ζ7ζ43ζ7443ζ7374737572767    complex lifted from C4×D7
ρ202-20022i-2i00-200ζ7572ζ767ζ747375727473767ζ767ζ7572ζ7473ζ43ζ7443ζ73ζ4ζ764ζ7ζ43ζ7643ζ7ζ4ζ754ζ72ζ4ζ744ζ73ζ43ζ7543ζ7275727677473    complex lifted from C4×D7
ρ212-20022i-2i00-200ζ767ζ7473ζ757276775727473ζ7473ζ767ζ7572ζ43ζ7543ζ72ζ4ζ744ζ73ζ43ζ7443ζ73ζ4ζ764ζ7ζ4ζ754ζ72ζ43ζ7643ζ776774737572    complex lifted from C4×D7
ρ222-2002-2i2i00-200ζ7473ζ7572ζ76774737677572ζ7572ζ7473ζ767ζ4ζ764ζ7ζ43ζ7543ζ72ζ4ζ754ζ72ζ43ζ7443ζ73ζ43ζ7643ζ7ζ4ζ744ζ7374737572767    complex lifted from C4×D7
ρ232-2002-2i2i00-200ζ767ζ7473ζ757276775727473ζ7473ζ767ζ7572ζ4ζ754ζ72ζ43ζ7443ζ73ζ4ζ744ζ73ζ43ζ7643ζ7ζ43ζ7543ζ72ζ4ζ764ζ776774737572    complex lifted from C4×D7
ρ242-2002-2i2i00-200ζ7572ζ767ζ747375727473767ζ767ζ7572ζ7473ζ4ζ744ζ73ζ43ζ7643ζ7ζ4ζ764ζ7ζ43ζ7543ζ72ζ43ζ7443ζ73ζ4ζ754ζ7275727677473    complex lifted from C4×D7
ρ254400-20000-20074+2ζ7375+2ζ7276+2ζ774+2ζ7376+2ζ775+2ζ727572747376700000074737572767    orthogonal lifted from S3×D7
ρ264400-20000-20075+2ζ7276+2ζ774+2ζ7375+2ζ7274+2ζ7376+2ζ77677572747300000075727677473    orthogonal lifted from S3×D7
ρ274400-20000-20076+2ζ774+2ζ7375+2ζ7276+2ζ775+2ζ7274+2ζ737473767757200000076774737572    orthogonal lifted from S3×D7
ρ284-400-2000020075+2ζ7276+2ζ774+2ζ73-2ζ75-2ζ72-2ζ74-2ζ73-2ζ76-2ζ776775727473000000ζ7572ζ767ζ7473    symplectic faithful, Schur index 2
ρ294-400-2000020076+2ζ774+2ζ7375+2ζ72-2ζ76-2ζ7-2ζ75-2ζ72-2ζ74-2ζ7374737677572000000ζ767ζ7473ζ7572    symplectic faithful, Schur index 2
ρ304-400-2000020074+2ζ7375+2ζ7276+2ζ7-2ζ74-2ζ73-2ζ76-2ζ7-2ζ75-2ζ7275727473767000000ζ7473ζ7572ζ767    symplectic faithful, Schur index 2

Smallest permutation representation of Dic3×D7
On 84 points
Generators in S84
(1 41 13 27 20 34)(2 42 14 28 21 35)(3 36 8 22 15 29)(4 37 9 23 16 30)(5 38 10 24 17 31)(6 39 11 25 18 32)(7 40 12 26 19 33)(43 71 57 64 50 78)(44 72 58 65 51 79)(45 73 59 66 52 80)(46 74 60 67 53 81)(47 75 61 68 54 82)(48 76 62 69 55 83)(49 77 63 70 56 84)
(1 69 27 48)(2 70 28 49)(3 64 22 43)(4 65 23 44)(5 66 24 45)(6 67 25 46)(7 68 26 47)(8 71 29 50)(9 72 30 51)(10 73 31 52)(11 74 32 53)(12 75 33 54)(13 76 34 55)(14 77 35 56)(15 78 36 57)(16 79 37 58)(17 80 38 59)(18 81 39 60)(19 82 40 61)(20 83 41 62)(21 84 42 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 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 31)(32 35)(33 34)(36 38)(39 42)(40 41)(43 45)(46 49)(47 48)(50 52)(53 56)(54 55)(57 59)(60 63)(61 62)(64 66)(67 70)(68 69)(71 73)(74 77)(75 76)(78 80)(81 84)(82 83)

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

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

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

Dic3×D7 is a maximal subgroup of   D285S3  D28⋊S3  C4×S3×D7  Dic7.D6  C42.C23
Dic3×D7 is a maximal quotient of   C28.32D6  D14⋊Dic3  C42.Q8

Matrix representation of Dic3×D7 in GL4(𝔽337) generated by

1000
0100
001336
0010
,
336000
033600
00310133
0010627
,
0100
33614300
0010
0001
,
0100
1000
0010
0001
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,336,0],[336,0,0,0,0,336,0,0,0,0,310,106,0,0,133,27],[0,336,0,0,1,143,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

Dic3×D7 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times D_7
% in TeX

G:=Group("Dic3xD7");
// GroupNames label

G:=SmallGroup(168,12);
// by ID

G=gap.SmallGroup(168,12);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-7,26,168,3604]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^7=d^2=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of Dic3×D7 in TeX
Character table of Dic3×D7 in TeX

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