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## G = C2×S3×D7order 168 = 23·3·7

### Direct product of C2, S3 and D7

Aliases: C2×S3×D7, C21⋊C23, C141D6, C61D14, C42⋊C22, D425C2, D21⋊C22, (C6×D7)⋊3C2, (S3×C7)⋊C22, C71(C22×S3), (C3×D7)⋊C22, (S3×C14)⋊3C2, C31(C22×D7), SmallGroup(168,50)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C2×S3×D7
 Chief series C1 — C7 — C21 — C3×D7 — S3×D7 — C2×S3×D7
 Lower central C21 — C2×S3×D7
 Upper central C1 — C2

Generators and relations for C2×S3×D7
G = < a,b,c,d,e | a2=b3=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 336 in 64 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2, C3, C22, S3, S3, C6, C6, C7, C23, D6, D6, C2×C6, D7, D7, C14, C14, C21, C22×S3, D14, D14, C2×C14, S3×C7, C3×D7, D21, C42, C22×D7, S3×D7, C6×D7, S3×C14, D42, C2×S3×D7
Quotients: C1, C2, C22, S3, C23, D6, D7, C22×S3, D14, C22×D7, S3×D7, C2×S3×D7

Character table of C2×S3×D7

 class 1 2A 2B 2C 2D 2E 2F 2G 3 6A 6B 6C 7A 7B 7C 14A 14B 14C 14D 14E 14F 14G 14H 14I 21A 21B 21C 42A 42B 42C size 1 1 3 3 7 7 21 21 2 2 14 14 2 2 2 2 2 2 6 6 6 6 6 6 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 -1 1 -1 1 1 -1 1 -1 1 -1 1 1 1 -1 -1 -1 1 -1 -1 1 1 -1 1 1 1 -1 -1 -1 linear of order 2 ρ3 1 1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ4 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ6 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 1 -1 -1 1 1 -1 1 1 1 -1 -1 -1 linear of order 2 ρ7 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ8 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 linear of order 2 ρ9 2 -2 0 0 2 -2 0 0 -1 1 1 -1 2 2 2 -2 -2 -2 0 0 0 0 0 0 -1 -1 -1 1 1 1 orthogonal lifted from D6 ρ10 2 2 0 0 2 2 0 0 -1 -1 -1 -1 2 2 2 2 2 2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 0 0 -2 -2 0 0 -1 -1 1 1 2 2 2 2 2 2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from D6 ρ12 2 -2 0 0 -2 2 0 0 -1 1 -1 1 2 2 2 -2 -2 -2 0 0 0 0 0 0 -1 -1 -1 1 1 1 orthogonal lifted from D6 ρ13 2 -2 -2 2 0 0 0 0 2 -2 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 orthogonal lifted from D14 ρ14 2 2 -2 -2 0 0 0 0 2 2 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ75-ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ74-ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 orthogonal lifted from D14 ρ15 2 -2 2 -2 0 0 0 0 2 -2 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ76-ζ7 ζ74+ζ73 ζ76+ζ7 -ζ75-ζ72 -ζ74-ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 orthogonal lifted from D14 ρ16 2 2 2 2 0 0 0 0 2 2 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 orthogonal lifted from D7 ρ17 2 -2 -2 2 0 0 0 0 2 -2 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 orthogonal lifted from D14 ρ18 2 2 2 2 0 0 0 0 2 2 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 orthogonal lifted from D7 ρ19 2 -2 2 -2 0 0 0 0 2 -2 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ75-ζ72 ζ76+ζ7 ζ75+ζ72 -ζ74-ζ73 -ζ76-ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 orthogonal lifted from D14 ρ20 2 2 -2 -2 0 0 0 0 2 2 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ76-ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ75-ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 orthogonal lifted from D14 ρ21 2 2 -2 -2 0 0 0 0 2 2 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ74-ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ76-ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 orthogonal lifted from D14 ρ22 2 -2 2 -2 0 0 0 0 2 -2 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ74-ζ73 ζ75+ζ72 ζ74+ζ73 -ζ76-ζ7 -ζ75-ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 orthogonal lifted from D14 ρ23 2 2 2 2 0 0 0 0 2 2 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 orthogonal lifted from D7 ρ24 2 -2 -2 2 0 0 0 0 2 -2 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 orthogonal lifted from D14 ρ25 4 4 0 0 0 0 0 0 -2 -2 0 0 2ζ76+2ζ7 2ζ74+2ζ73 2ζ75+2ζ72 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 0 0 0 0 0 0 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 orthogonal lifted from S3×D7 ρ26 4 -4 0 0 0 0 0 0 -2 2 0 0 2ζ75+2ζ72 2ζ76+2ζ7 2ζ74+2ζ73 -2ζ74-2ζ73 -2ζ76-2ζ7 -2ζ75-2ζ72 0 0 0 0 0 0 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 orthogonal faithful ρ27 4 -4 0 0 0 0 0 0 -2 2 0 0 2ζ76+2ζ7 2ζ74+2ζ73 2ζ75+2ζ72 -2ζ75-2ζ72 -2ζ74-2ζ73 -2ζ76-2ζ7 0 0 0 0 0 0 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 orthogonal faithful ρ28 4 4 0 0 0 0 0 0 -2 -2 0 0 2ζ75+2ζ72 2ζ76+2ζ7 2ζ74+2ζ73 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 0 0 0 0 0 0 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 orthogonal lifted from S3×D7 ρ29 4 4 0 0 0 0 0 0 -2 -2 0 0 2ζ74+2ζ73 2ζ75+2ζ72 2ζ76+2ζ7 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 0 0 0 0 0 0 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 orthogonal lifted from S3×D7 ρ30 4 -4 0 0 0 0 0 0 -2 2 0 0 2ζ74+2ζ73 2ζ75+2ζ72 2ζ76+2ζ7 -2ζ76-2ζ7 -2ζ75-2ζ72 -2ζ74-2ζ73 0 0 0 0 0 0 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 orthogonal faithful

Smallest permutation representation of C2×S3×D7
On 42 points
Generators in S42
(1 27)(2 28)(3 22)(4 23)(5 24)(6 25)(7 26)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)
(1 13 20)(2 14 21)(3 8 15)(4 9 16)(5 10 17)(6 11 18)(7 12 19)(22 29 36)(23 30 37)(24 31 38)(25 32 39)(26 33 40)(27 34 41)(28 35 42)
(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)
(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)
(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)

G:=sub<Sym(42)| (1,27)(2,28)(3,22)(4,23)(5,24)(6,25)(7,26)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42), (1,13,20)(2,14,21)(3,8,15)(4,9,16)(5,10,17)(6,11,18)(7,12,19)(22,29,36)(23,30,37)(24,31,38)(25,32,39)(26,33,40)(27,34,41)(28,35,42), (8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (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), (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)>;

G:=Group( (1,27)(2,28)(3,22)(4,23)(5,24)(6,25)(7,26)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42), (1,13,20)(2,14,21)(3,8,15)(4,9,16)(5,10,17)(6,11,18)(7,12,19)(22,29,36)(23,30,37)(24,31,38)(25,32,39)(26,33,40)(27,34,41)(28,35,42), (8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (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), (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) );

G=PermutationGroup([[(1,27),(2,28),(3,22),(4,23),(5,24),(6,25),(7,26),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42)], [(1,13,20),(2,14,21),(3,8,15),(4,9,16),(5,10,17),(6,11,18),(7,12,19),(22,29,36),(23,30,37),(24,31,38),(25,32,39),(26,33,40),(27,34,41),(28,35,42)], [(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42)], [(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)], [(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)]])

C2×S3×D7 is a maximal subgroup of
C28⋊D6  D6⋊D14
C2×S3×D7 is a maximal quotient of
D285S3  D28⋊S3  D12⋊D7  D84⋊C2  D21⋊Q8  D6.D14  D125D7  D14.D6  C28⋊D6  Dic7.D6  C42.C23  Dic3.D14  D6⋊D14

Matrix representation of C2×S3×D7 in GL5(𝔽43)

 42 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 42 42 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 42 0 0 0 0 0 1 0 0 0 0 42 42 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 42 1 0 0 0 22 20
,
 42 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 42 0 0 0 0 22 1

G:=sub<GL(5,GF(43))| [42,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,42,1,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,1],[42,0,0,0,0,0,1,42,0,0,0,0,42,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,42,22,0,0,0,1,20],[42,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,42,22,0,0,0,0,1] >;

C2×S3×D7 in GAP, Magma, Sage, TeX

C_2\times S_3\times D_7
% in TeX

G:=Group("C2xS3xD7");
// GroupNames label

G:=SmallGroup(168,50);
// by ID

G=gap.SmallGroup(168,50);
# by ID

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

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

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