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G = S3×Dic7order 168 = 23·3·7

Direct product of S3 and Dic7

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

Aliases: S3×Dic7, D6.D7, C14.2D6, C6.2D14, Dic213C2, C42.2C22, (S3×C7)⋊C4, C73(C4×S3), C212(C2×C4), (S3×C14).C2, C2.2(S3×D7), C31(C2×Dic7), (C3×Dic7)⋊1C2, SmallGroup(168,13)

Series: Derived Chief Lower central Upper central

C1C21 — S3×Dic7
C1C7C21C42C3×Dic7 — S3×Dic7
C21 — S3×Dic7
C1C2

Generators and relations for S3×Dic7
 G = < a,b,c,d | a3=b2=c14=1, d2=c7, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

3C2
3C2
3C22
7C4
21C4
3C14
3C14
21C2×C4
7C12
7Dic3
3Dic7
3C2×C14
7C4×S3
3C2×Dic7

Character table of S3×Dic7

 class 12A2B2C34A4B4C4D67A7B7C12A12B14A14B14C14D14E14F14G14H14I21A21B21C42A42B42C
 size 1133277212122221414222666666444444
ρ1111111111111111111111111111111    trivial
ρ211111-1-1-1-11111-1-1111111111111111    linear of order 2
ρ311-1-11-1-1111111-1-1111-1-1-1-1-1-1111111    linear of order 2
ρ411-1-1111-1-1111111111-1-1-1-1-1-1111111    linear of order 2
ρ51-11-11-iii-i-1111-ii-1-1-111-1-1-11111-1-1-1    linear of order 4
ρ61-11-11i-i-ii-1111i-i-1-1-111-1-1-11111-1-1-1    linear of order 4
ρ71-1-111i-ii-i-1111i-i-1-1-1-1-1111-1111-1-1-1    linear of order 4
ρ81-1-111-ii-ii-1111-ii-1-1-1-1-1111-1111-1-1-1    linear of order 4
ρ92200-12200-1222-1-1222000000-1-1-1-1-1-1    orthogonal lifted from S3
ρ102200-1-2-200-122211222000000-1-1-1-1-1-1    orthogonal lifted from D6
ρ1122-2-2200002ζ7572ζ7473ζ76700ζ767ζ7572ζ74737572747376775727473767ζ767ζ7572ζ7473ζ7572ζ7473ζ767    orthogonal lifted from D14
ρ1222-2-2200002ζ767ζ7572ζ747300ζ7473ζ767ζ75727677572747376775727473ζ7473ζ767ζ7572ζ767ζ7572ζ7473    orthogonal lifted from D14
ρ1322-2-2200002ζ7473ζ767ζ757200ζ7572ζ7473ζ7677473767757274737677572ζ7572ζ7473ζ767ζ7473ζ767ζ7572    orthogonal lifted from D14
ρ142222200002ζ7473ζ767ζ757200ζ7572ζ7473ζ767ζ7473ζ767ζ7572ζ7473ζ767ζ7572ζ7572ζ7473ζ767ζ7473ζ767ζ7572    orthogonal lifted from D7
ρ152222200002ζ7572ζ7473ζ76700ζ767ζ7572ζ7473ζ7572ζ7473ζ767ζ7572ζ7473ζ767ζ767ζ7572ζ7473ζ7572ζ7473ζ767    orthogonal lifted from D7
ρ162222200002ζ767ζ7572ζ747300ζ7473ζ767ζ7572ζ767ζ7572ζ7473ζ767ζ7572ζ7473ζ7473ζ767ζ7572ζ767ζ7572ζ7473    orthogonal lifted from D7
ρ172-22-220000-2ζ767ζ7572ζ74730074737677572ζ767ζ757274737677572ζ7473ζ7473ζ767ζ757276775727473    symplectic lifted from Dic7, Schur index 2
ρ182-22-220000-2ζ7572ζ7473ζ7670076775727473ζ7572ζ747376775727473ζ767ζ767ζ7572ζ747375727473767    symplectic lifted from Dic7, Schur index 2
ρ192-2-2220000-2ζ767ζ7572ζ747300747376775727677572ζ7473ζ767ζ75727473ζ7473ζ767ζ757276775727473    symplectic lifted from Dic7, Schur index 2
ρ202-2-2220000-2ζ7473ζ767ζ757200757274737677473767ζ7572ζ7473ζ7677572ζ7572ζ7473ζ76774737677572    symplectic lifted from Dic7, Schur index 2
ρ212-22-220000-2ζ7473ζ767ζ75720075727473767ζ7473ζ76775727473767ζ7572ζ7572ζ7473ζ76774737677572    symplectic lifted from Dic7, Schur index 2
ρ222-2-2220000-2ζ7572ζ7473ζ767007677572747375727473ζ767ζ7572ζ7473767ζ767ζ7572ζ747375727473767    symplectic lifted from Dic7, Schur index 2
ρ232-200-1-2i2i001222i-i-2-2-2000000-1-1-1111    complex lifted from C4×S3
ρ242-200-12i-2i001222-ii-2-2-2000000-1-1-1111    complex lifted from C4×S3
ρ254400-20000-276+2ζ775+2ζ7274+2ζ730074+2ζ7376+2ζ775+2ζ720000007473767757276775727473    orthogonal lifted from S3×D7
ρ264400-20000-274+2ζ7376+2ζ775+2ζ720075+2ζ7274+2ζ7376+2ζ70000007572747376774737677572    orthogonal lifted from S3×D7
ρ274400-20000-275+2ζ7274+2ζ7376+2ζ70076+2ζ775+2ζ7274+2ζ730000007677572747375727473767    orthogonal lifted from S3×D7
ρ284-400-20000274+2ζ7376+2ζ775+2ζ7200-2ζ75-2ζ72-2ζ74-2ζ73-2ζ76-2ζ700000075727473767ζ7473ζ767ζ7572    symplectic faithful, Schur index 2
ρ294-400-20000276+2ζ775+2ζ7274+2ζ7300-2ζ74-2ζ73-2ζ76-2ζ7-2ζ75-2ζ7200000074737677572ζ767ζ7572ζ7473    symplectic faithful, Schur index 2
ρ304-400-20000275+2ζ7274+2ζ7376+2ζ700-2ζ76-2ζ7-2ζ75-2ζ72-2ζ74-2ζ7300000076775727473ζ7572ζ7473ζ767    symplectic faithful, Schur index 2

Smallest permutation representation of S3×Dic7
On 84 points
Generators in S84
(1 17 53)(2 18 54)(3 19 55)(4 20 56)(5 21 43)(6 22 44)(7 23 45)(8 24 46)(9 25 47)(10 26 48)(11 27 49)(12 28 50)(13 15 51)(14 16 52)(29 71 70)(30 72 57)(31 73 58)(32 74 59)(33 75 60)(34 76 61)(35 77 62)(36 78 63)(37 79 64)(38 80 65)(39 81 66)(40 82 67)(41 83 68)(42 84 69)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 50)(22 51)(23 52)(24 53)(25 54)(26 55)(27 56)(28 43)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 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 69 8 62)(2 68 9 61)(3 67 10 60)(4 66 11 59)(5 65 12 58)(6 64 13 57)(7 63 14 70)(15 30 22 37)(16 29 23 36)(17 42 24 35)(18 41 25 34)(19 40 26 33)(20 39 27 32)(21 38 28 31)(43 80 50 73)(44 79 51 72)(45 78 52 71)(46 77 53 84)(47 76 54 83)(48 75 55 82)(49 74 56 81)

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

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

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

S3×Dic7 is a maximal subgroup of   D12⋊D7  D125D7  C4×S3×D7  C42.C23  Dic3.D14
S3×Dic7 is a maximal quotient of   D6.Dic7  D6⋊Dic7  C14.Dic6

Matrix representation of S3×Dic7 in GL5(𝔽337)

10000
033633600
01000
00010
00001
,
10000
01000
033633600
00010
00001
,
3360000
01000
00100
0001091
00033534
,
1480000
01000
00100
000310263
00019227

G:=sub<GL(5,GF(337))| [1,0,0,0,0,0,336,1,0,0,0,336,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,336,0,0,0,0,336,0,0,0,0,0,1,0,0,0,0,0,1],[336,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,109,335,0,0,0,1,34],[148,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,310,192,0,0,0,263,27] >;

S3×Dic7 in GAP, Magma, Sage, TeX

S_3\times {\rm Dic}_7
% in TeX

G:=Group("S3xDic7");
// GroupNames label

G:=SmallGroup(168,13);
// by ID

G=gap.SmallGroup(168,13);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-7,20,168,3604]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^14=1,d^2=c^7,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of S3×Dic7 in TeX
Character table of S3×Dic7 in TeX

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