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G = C3×F8  order 168 = 23·3·7

Direct product of C3 and F8

direct product, metabelian, soluble, monomial, A-group

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 >

C1
7C2
C3
8C7
7C22
7C6
8C21
C23
7C2×C6
C22×C6
F8
C3×F8

Character table of C3×F8

 class 123A3B6A6B7A7B7C7D7E7F21A21B21C21D21E21F21G21H21I21J21K21L
 size 171177888888888888888888
ρ1111111111111111111111111    trivial
ρ211ζ32ζ3ζ3ζ32111111ζ3ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ3ζ32    linear of order 3
ρ311ζ3ζ32ζ32ζ3111111ζ32ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ32ζ3    linear of order 3
ρ4111111ζ76ζ73ζ74ζ7ζ75ζ72ζ73ζ72ζ76ζ73ζ74ζ7ζ74ζ7ζ75ζ72ζ76ζ75    linear of order 7
ρ5111111ζ74ζ72ζ75ζ73ζ7ζ76ζ72ζ76ζ74ζ72ζ75ζ73ζ75ζ73ζ7ζ76ζ74ζ7    linear of order 7
ρ6111111ζ73ζ75ζ72ζ74ζ76ζ7ζ75ζ7ζ73ζ75ζ72ζ74ζ72ζ74ζ76ζ7ζ73ζ76    linear of order 7
ρ7111111ζ72ζ7ζ76ζ75ζ74ζ73ζ7ζ73ζ72ζ7ζ76ζ75ζ76ζ75ζ74ζ73ζ72ζ74    linear of order 7
ρ8111111ζ7ζ74ζ73ζ76ζ72ζ75ζ74ζ75ζ7ζ74ζ73ζ76ζ73ζ76ζ72ζ75ζ7ζ72    linear of order 7
ρ9111111ζ75ζ76ζ7ζ72ζ73ζ74ζ76ζ74ζ75ζ76ζ7ζ72ζ7ζ72ζ73ζ74ζ75ζ73    linear of order 7
ρ1011ζ3ζ32ζ32ζ3ζ74ζ72ζ75ζ73ζ7ζ76ζ32ζ72ζ3ζ76ζ3ζ74ζ3ζ72ζ3ζ75ζ3ζ73ζ32ζ75ζ32ζ73ζ32ζ7ζ32ζ76ζ32ζ74ζ3ζ7    linear of order 21
ρ1111ζ3ζ32ζ32ζ3ζ75ζ76ζ7ζ72ζ73ζ74ζ32ζ76ζ3ζ74ζ3ζ75ζ3ζ76ζ3ζ7ζ3ζ72ζ32ζ7ζ32ζ72ζ32ζ73ζ32ζ74ζ32ζ75ζ3ζ73    linear of order 21
ρ1211ζ32ζ3ζ3ζ32ζ73ζ75ζ72ζ74ζ76ζ7ζ3ζ75ζ32ζ7ζ32ζ73ζ32ζ75ζ32ζ72ζ32ζ74ζ3ζ72ζ3ζ74ζ3ζ76ζ3ζ7ζ3ζ73ζ32ζ76    linear of order 21
ρ1311ζ3ζ32ζ32ζ3ζ72ζ7ζ76ζ75ζ74ζ73ζ32ζ7ζ3ζ73ζ3ζ72ζ3ζ7ζ3ζ76ζ3ζ75ζ32ζ76ζ32ζ75ζ32ζ74ζ32ζ73ζ32ζ72ζ3ζ74    linear of order 21
ρ1411ζ32ζ3ζ3ζ32ζ76ζ73ζ74ζ7ζ75ζ72ζ3ζ73ζ32ζ72ζ32ζ76ζ32ζ73ζ32ζ74ζ32ζ7ζ3ζ74ζ3ζ7ζ3ζ75ζ3ζ72ζ3ζ76ζ32ζ75    linear of order 21
ρ1511ζ32ζ3ζ3ζ32ζ7ζ74ζ73ζ76ζ72ζ75ζ3ζ74ζ32ζ75ζ32ζ7ζ32ζ74ζ32ζ73ζ32ζ76ζ3ζ73ζ3ζ76ζ3ζ72ζ3ζ75ζ3ζ7ζ32ζ72    linear of order 21
ρ1611ζ32ζ3ζ3ζ32ζ74ζ72ζ75ζ73ζ7ζ76ζ3ζ72ζ32ζ76ζ32ζ74ζ32ζ72ζ32ζ75ζ32ζ73ζ3ζ75ζ3ζ73ζ3ζ7ζ3ζ76ζ3ζ74ζ32ζ7    linear of order 21
ρ1711ζ3ζ32ζ32ζ3ζ76ζ73ζ74ζ7ζ75ζ72ζ32ζ73ζ3ζ72ζ3ζ76ζ3ζ73ζ3ζ74ζ3ζ7ζ32ζ74ζ32ζ7ζ32ζ75ζ32ζ72ζ32ζ76ζ3ζ75    linear of order 21
ρ1811ζ32ζ3ζ3ζ32ζ75ζ76ζ7ζ72ζ73ζ74ζ3ζ76ζ32ζ74ζ32ζ75ζ32ζ76ζ32ζ7ζ32ζ72ζ3ζ7ζ3ζ72ζ3ζ73ζ3ζ74ζ3ζ75ζ32ζ73    linear of order 21
ρ1911ζ3ζ32ζ32ζ3ζ73ζ75ζ72ζ74ζ76ζ7ζ32ζ75ζ3ζ7ζ3ζ73ζ3ζ75ζ3ζ72ζ3ζ74ζ32ζ72ζ32ζ74ζ32ζ76ζ32ζ7ζ32ζ73ζ3ζ76    linear of order 21
ρ2011ζ32ζ3ζ3ζ32ζ72ζ7ζ76ζ75ζ74ζ73ζ3ζ7ζ32ζ73ζ32ζ72ζ32ζ7ζ32ζ76ζ32ζ75ζ3ζ76ζ3ζ75ζ3ζ74ζ3ζ73ζ3ζ72ζ32ζ74    linear of order 21
ρ2111ζ3ζ32ζ32ζ3ζ7ζ74ζ73ζ76ζ72ζ75ζ32ζ74ζ3ζ75ζ3ζ7ζ3ζ74ζ3ζ73ζ3ζ76ζ32ζ73ζ32ζ76ζ32ζ72ζ32ζ75ζ32ζ7ζ3ζ72    linear of order 21
ρ227-177-1-1000000000000000000    orthogonal lifted from F8
ρ237-1-7-7√-3/2-7+7√-3/2ζ65ζ6000000000000000000    complex faithful
ρ247-1-7+7√-3/2-7-7√-3/2ζ6ζ65000000000000000000    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)

36000000
03600000
00360000
00036000
00003600
00000360
00000036
,
0000001
42424242424242
0000100
0000010
0010000
0001000
1000000
,
0001000
0010000
0100000
1000000
42424242424242
0000001
0000010
,
0000100
0000010
0000001
42424242424242
1000000
0100000
0010000
,
1000000
0010000
0000100
0000001
0001000
0100000
42424242424242

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
 

Export

Subgroup lattice of C3×F8 in TeX
Character table of C3×F8 in TeX

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