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## G = C3×F8order 168 = 23·3·7

### Direct product of C3 and F8

Aliases: C3×F8, C23⋊C21, (C22×C6)⋊C7, SmallGroup(168,44)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C3×F8
 Chief series C1 — C23 — F8 — C3×F8
 Lower central C23 — C3×F8
 Upper central C1 — C3

Generators and relations for C3×F8
G = < a,b,c,d,e | a3=b2=c2=d2=e7=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=dc=cd, ece-1=b, ede-1=c >

Character table of C3×F8

 class 1 2 3A 3B 6A 6B 7A 7B 7C 7D 7E 7F 21A 21B 21C 21D 21E 21F 21G 21H 21I 21J 21K 21L size 1 7 1 1 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 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 trivial ρ2 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 linear of order 3 ρ3 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 linear of order 3 ρ4 1 1 1 1 1 1 ζ76 ζ73 ζ74 ζ7 ζ75 ζ72 ζ73 ζ72 ζ76 ζ73 ζ74 ζ7 ζ74 ζ7 ζ75 ζ72 ζ76 ζ75 linear of order 7 ρ5 1 1 1 1 1 1 ζ74 ζ72 ζ75 ζ73 ζ7 ζ76 ζ72 ζ76 ζ74 ζ72 ζ75 ζ73 ζ75 ζ73 ζ7 ζ76 ζ74 ζ7 linear of order 7 ρ6 1 1 1 1 1 1 ζ73 ζ75 ζ72 ζ74 ζ76 ζ7 ζ75 ζ7 ζ73 ζ75 ζ72 ζ74 ζ72 ζ74 ζ76 ζ7 ζ73 ζ76 linear of order 7 ρ7 1 1 1 1 1 1 ζ72 ζ7 ζ76 ζ75 ζ74 ζ73 ζ7 ζ73 ζ72 ζ7 ζ76 ζ75 ζ76 ζ75 ζ74 ζ73 ζ72 ζ74 linear of order 7 ρ8 1 1 1 1 1 1 ζ7 ζ74 ζ73 ζ76 ζ72 ζ75 ζ74 ζ75 ζ7 ζ74 ζ73 ζ76 ζ73 ζ76 ζ72 ζ75 ζ7 ζ72 linear of order 7 ρ9 1 1 1 1 1 1 ζ75 ζ76 ζ7 ζ72 ζ73 ζ74 ζ76 ζ74 ζ75 ζ76 ζ7 ζ72 ζ7 ζ72 ζ73 ζ74 ζ75 ζ73 linear of order 7 ρ10 1 1 ζ3 ζ32 ζ32 ζ3 ζ74 ζ72 ζ75 ζ73 ζ7 ζ76 ζ32ζ72 ζ3ζ76 ζ3ζ74 ζ3ζ72 ζ3ζ75 ζ3ζ73 ζ32ζ75 ζ32ζ73 ζ32ζ7 ζ32ζ76 ζ32ζ74 ζ3ζ7 linear of order 21 ρ11 1 1 ζ3 ζ32 ζ32 ζ3 ζ75 ζ76 ζ7 ζ72 ζ73 ζ74 ζ32ζ76 ζ3ζ74 ζ3ζ75 ζ3ζ76 ζ3ζ7 ζ3ζ72 ζ32ζ7 ζ32ζ72 ζ32ζ73 ζ32ζ74 ζ32ζ75 ζ3ζ73 linear of order 21 ρ12 1 1 ζ32 ζ3 ζ3 ζ32 ζ73 ζ75 ζ72 ζ74 ζ76 ζ7 ζ3ζ75 ζ32ζ7 ζ32ζ73 ζ32ζ75 ζ32ζ72 ζ32ζ74 ζ3ζ72 ζ3ζ74 ζ3ζ76 ζ3ζ7 ζ3ζ73 ζ32ζ76 linear of order 21 ρ13 1 1 ζ3 ζ32 ζ32 ζ3 ζ72 ζ7 ζ76 ζ75 ζ74 ζ73 ζ32ζ7 ζ3ζ73 ζ3ζ72 ζ3ζ7 ζ3ζ76 ζ3ζ75 ζ32ζ76 ζ32ζ75 ζ32ζ74 ζ32ζ73 ζ32ζ72 ζ3ζ74 linear of order 21 ρ14 1 1 ζ32 ζ3 ζ3 ζ32 ζ76 ζ73 ζ74 ζ7 ζ75 ζ72 ζ3ζ73 ζ32ζ72 ζ32ζ76 ζ32ζ73 ζ32ζ74 ζ32ζ7 ζ3ζ74 ζ3ζ7 ζ3ζ75 ζ3ζ72 ζ3ζ76 ζ32ζ75 linear of order 21 ρ15 1 1 ζ32 ζ3 ζ3 ζ32 ζ7 ζ74 ζ73 ζ76 ζ72 ζ75 ζ3ζ74 ζ32ζ75 ζ32ζ7 ζ32ζ74 ζ32ζ73 ζ32ζ76 ζ3ζ73 ζ3ζ76 ζ3ζ72 ζ3ζ75 ζ3ζ7 ζ32ζ72 linear of order 21 ρ16 1 1 ζ32 ζ3 ζ3 ζ32 ζ74 ζ72 ζ75 ζ73 ζ7 ζ76 ζ3ζ72 ζ32ζ76 ζ32ζ74 ζ32ζ72 ζ32ζ75 ζ32ζ73 ζ3ζ75 ζ3ζ73 ζ3ζ7 ζ3ζ76 ζ3ζ74 ζ32ζ7 linear of order 21 ρ17 1 1 ζ3 ζ32 ζ32 ζ3 ζ76 ζ73 ζ74 ζ7 ζ75 ζ72 ζ32ζ73 ζ3ζ72 ζ3ζ76 ζ3ζ73 ζ3ζ74 ζ3ζ7 ζ32ζ74 ζ32ζ7 ζ32ζ75 ζ32ζ72 ζ32ζ76 ζ3ζ75 linear of order 21 ρ18 1 1 ζ32 ζ3 ζ3 ζ32 ζ75 ζ76 ζ7 ζ72 ζ73 ζ74 ζ3ζ76 ζ32ζ74 ζ32ζ75 ζ32ζ76 ζ32ζ7 ζ32ζ72 ζ3ζ7 ζ3ζ72 ζ3ζ73 ζ3ζ74 ζ3ζ75 ζ32ζ73 linear of order 21 ρ19 1 1 ζ3 ζ32 ζ32 ζ3 ζ73 ζ75 ζ72 ζ74 ζ76 ζ7 ζ32ζ75 ζ3ζ7 ζ3ζ73 ζ3ζ75 ζ3ζ72 ζ3ζ74 ζ32ζ72 ζ32ζ74 ζ32ζ76 ζ32ζ7 ζ32ζ73 ζ3ζ76 linear of order 21 ρ20 1 1 ζ32 ζ3 ζ3 ζ32 ζ72 ζ7 ζ76 ζ75 ζ74 ζ73 ζ3ζ7 ζ32ζ73 ζ32ζ72 ζ32ζ7 ζ32ζ76 ζ32ζ75 ζ3ζ76 ζ3ζ75 ζ3ζ74 ζ3ζ73 ζ3ζ72 ζ32ζ74 linear of order 21 ρ21 1 1 ζ3 ζ32 ζ32 ζ3 ζ7 ζ74 ζ73 ζ76 ζ72 ζ75 ζ32ζ74 ζ3ζ75 ζ3ζ7 ζ3ζ74 ζ3ζ73 ζ3ζ76 ζ32ζ73 ζ32ζ76 ζ32ζ72 ζ32ζ75 ζ32ζ7 ζ3ζ72 linear of order 21 ρ22 7 -1 7 7 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from F8 ρ23 7 -1 -7-7√-3/2 -7+7√-3/2 ζ65 ζ6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ24 7 -1 -7+7√-3/2 -7-7√-3/2 ζ6 ζ65 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C3×F8
On 24 points - transitive group 24T282
Generators in S24
(1 2 3)(4 15 23)(5 16 24)(6 17 18)(7 11 19)(8 12 20)(9 13 21)(10 14 22)
(1 15)(2 23)(3 4)(5 9)(6 7)(8 10)(11 17)(12 14)(13 16)(18 19)(20 22)(21 24)
(1 16)(2 24)(3 5)(4 9)(6 10)(7 8)(11 12)(13 15)(14 17)(18 22)(19 20)(21 23)
(1 17)(2 18)(3 6)(4 7)(5 10)(8 9)(11 15)(12 13)(14 16)(19 23)(20 21)(22 24)
(4 5 6 7 8 9 10)(11 12 13 14 15 16 17)(18 19 20 21 22 23 24)

G:=sub<Sym(24)| (1,2,3)(4,15,23)(5,16,24)(6,17,18)(7,11,19)(8,12,20)(9,13,21)(10,14,22), (1,15)(2,23)(3,4)(5,9)(6,7)(8,10)(11,17)(12,14)(13,16)(18,19)(20,22)(21,24), (1,16)(2,24)(3,5)(4,9)(6,10)(7,8)(11,12)(13,15)(14,17)(18,22)(19,20)(21,23), (1,17)(2,18)(3,6)(4,7)(5,10)(8,9)(11,15)(12,13)(14,16)(19,23)(20,21)(22,24), (4,5,6,7,8,9,10)(11,12,13,14,15,16,17)(18,19,20,21,22,23,24)>;

G:=Group( (1,2,3)(4,15,23)(5,16,24)(6,17,18)(7,11,19)(8,12,20)(9,13,21)(10,14,22), (1,15)(2,23)(3,4)(5,9)(6,7)(8,10)(11,17)(12,14)(13,16)(18,19)(20,22)(21,24), (1,16)(2,24)(3,5)(4,9)(6,10)(7,8)(11,12)(13,15)(14,17)(18,22)(19,20)(21,23), (1,17)(2,18)(3,6)(4,7)(5,10)(8,9)(11,15)(12,13)(14,16)(19,23)(20,21)(22,24), (4,5,6,7,8,9,10)(11,12,13,14,15,16,17)(18,19,20,21,22,23,24) );

G=PermutationGroup([[(1,2,3),(4,15,23),(5,16,24),(6,17,18),(7,11,19),(8,12,20),(9,13,21),(10,14,22)], [(1,15),(2,23),(3,4),(5,9),(6,7),(8,10),(11,17),(12,14),(13,16),(18,19),(20,22),(21,24)], [(1,16),(2,24),(3,5),(4,9),(6,10),(7,8),(11,12),(13,15),(14,17),(18,22),(19,20),(21,23)], [(1,17),(2,18),(3,6),(4,7),(5,10),(8,9),(11,15),(12,13),(14,16),(19,23),(20,21),(22,24)], [(4,5,6,7,8,9,10),(11,12,13,14,15,16,17),(18,19,20,21,22,23,24)]])

G:=TransitiveGroup(24,282);

Matrix representation of C3×F8 in GL7(𝔽43)

 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36
,
 0 0 0 0 0 0 1 42 42 42 42 42 42 42 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0
,
 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 42 42 42 42 42 42 42 0 0 0 0 0 0 1 0 0 0 0 0 1 0
,
 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 42 42 42 42 42 42 42 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 42 42 42 42 42 42 42

G:=sub<GL(7,GF(43))| [36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36],[0,42,0,0,0,0,1,0,42,0,0,0,0,0,0,42,0,0,1,0,0,0,42,0,0,0,1,0,0,42,1,0,0,0,0,0,42,0,1,0,0,0,1,42,0,0,0,0,0],[0,0,0,1,42,0,0,0,0,1,0,42,0,0,0,1,0,0,42,0,0,1,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,1,0,0,0,0,42,1,0],[0,0,0,42,1,0,0,0,0,0,42,0,1,0,0,0,0,42,0,0,1,0,0,0,42,0,0,0,1,0,0,42,0,0,0,0,1,0,42,0,0,0,0,0,1,42,0,0,0],[1,0,0,0,0,0,42,0,0,0,0,0,1,42,0,1,0,0,0,0,42,0,0,0,0,1,0,42,0,0,1,0,0,0,42,0,0,0,0,0,0,42,0,0,0,1,0,0,42] >;

C3×F8 in GAP, Magma, Sage, TeX

C_3\times F_8
% in TeX

G:=Group("C3xF8");
// GroupNames label

G:=SmallGroup(168,44);
// by ID

G=gap.SmallGroup(168,44);
# by ID

G:=PCGroup([5,-3,-7,-2,2,2,217,568,884]);
// Polycyclic

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

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