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G = C7×A4  order 84 = 22·3·7

Direct product of C7 and A4

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

Aliases: C7×A4, C22⋊C21, (C2×C14)⋊1C3, SmallGroup(84,10)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C7×A4
Chief series
C1 — C22 — C2×C14 — C7×A4
Lower central
C22 — C7×A4
Upper central
C1 — C7

Generators and relations for C7×A4
 G = < a,b,c,d | a7=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

C1
3C2
4C3
C7
C22
3C14
4C21
A4
C2×C14
C7×A4

Character table of C7×A4

 class 123A3B7A7B7C7D7E7F14A14B14C14D14E14F21A21B21C21D21E21F21G21H21I21J21K21L
 size 1344111111333333444444444444
ρ11111111111111111111111111111    trivial
ρ211ζ3ζ32111111111111ζ32ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ32ζ3    linear of order 3
ρ311ζ32ζ3111111111111ζ3ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ3ζ32    linear of order 3
ρ41111ζ7ζ75ζ72ζ76ζ73ζ74ζ75ζ76ζ7ζ73ζ72ζ74ζ73ζ75ζ72ζ76ζ73ζ74ζ74ζ7ζ75ζ72ζ76ζ7    linear of order 7
ρ51111ζ76ζ72ζ75ζ7ζ74ζ73ζ72ζ7ζ76ζ74ζ75ζ73ζ74ζ72ζ75ζ7ζ74ζ73ζ73ζ76ζ72ζ75ζ7ζ76    linear of order 7
ρ61111ζ73ζ7ζ76ζ74ζ72ζ75ζ7ζ74ζ73ζ72ζ76ζ75ζ72ζ7ζ76ζ74ζ72ζ75ζ75ζ73ζ7ζ76ζ74ζ73    linear of order 7
ρ71111ζ74ζ76ζ7ζ73ζ75ζ72ζ76ζ73ζ74ζ75ζ7ζ72ζ75ζ76ζ7ζ73ζ75ζ72ζ72ζ74ζ76ζ7ζ73ζ74    linear of order 7
ρ81111ζ72ζ73ζ74ζ75ζ76ζ7ζ73ζ75ζ72ζ76ζ74ζ7ζ76ζ73ζ74ζ75ζ76ζ7ζ7ζ72ζ73ζ74ζ75ζ72    linear of order 7
ρ91111ζ75ζ74ζ73ζ72ζ7ζ76ζ74ζ72ζ75ζ7ζ73ζ76ζ7ζ74ζ73ζ72ζ7ζ76ζ76ζ75ζ74ζ73ζ72ζ75    linear of order 7
ρ1011ζ3ζ32ζ72ζ73ζ74ζ75ζ76ζ7ζ73ζ75ζ72ζ76ζ74ζ7ζ32ζ76ζ3ζ73ζ3ζ74ζ3ζ75ζ3ζ76ζ3ζ7ζ32ζ7ζ32ζ72ζ32ζ73ζ32ζ74ζ32ζ75ζ3ζ72    linear of order 21
ρ1111ζ3ζ32ζ75ζ74ζ73ζ72ζ7ζ76ζ74ζ72ζ75ζ7ζ73ζ76ζ32ζ7ζ3ζ74ζ3ζ73ζ3ζ72ζ3ζ7ζ3ζ76ζ32ζ76ζ32ζ75ζ32ζ74ζ32ζ73ζ32ζ72ζ3ζ75    linear of order 21
ρ1211ζ3ζ32ζ74ζ76ζ7ζ73ζ75ζ72ζ76ζ73ζ74ζ75ζ7ζ72ζ32ζ75ζ3ζ76ζ3ζ7ζ3ζ73ζ3ζ75ζ3ζ72ζ32ζ72ζ32ζ74ζ32ζ76ζ32ζ7ζ32ζ73ζ3ζ74    linear of order 21
ρ1311ζ32ζ3ζ75ζ74ζ73ζ72ζ7ζ76ζ74ζ72ζ75ζ7ζ73ζ76ζ3ζ7ζ32ζ74ζ32ζ73ζ32ζ72ζ32ζ7ζ32ζ76ζ3ζ76ζ3ζ75ζ3ζ74ζ3ζ73ζ3ζ72ζ32ζ75    linear of order 21
ρ1411ζ3ζ32ζ76ζ72ζ75ζ7ζ74ζ73ζ72ζ7ζ76ζ74ζ75ζ73ζ32ζ74ζ3ζ72ζ3ζ75ζ3ζ7ζ3ζ74ζ3ζ73ζ32ζ73ζ32ζ76ζ32ζ72ζ32ζ75ζ32ζ7ζ3ζ76    linear of order 21
ρ1511ζ32ζ3ζ76ζ72ζ75ζ7ζ74ζ73ζ72ζ7ζ76ζ74ζ75ζ73ζ3ζ74ζ32ζ72ζ32ζ75ζ32ζ7ζ32ζ74ζ32ζ73ζ3ζ73ζ3ζ76ζ3ζ72ζ3ζ75ζ3ζ7ζ32ζ76    linear of order 21
ρ1611ζ32ζ3ζ72ζ73ζ74ζ75ζ76ζ7ζ73ζ75ζ72ζ76ζ74ζ7ζ3ζ76ζ32ζ73ζ32ζ74ζ32ζ75ζ32ζ76ζ32ζ7ζ3ζ7ζ3ζ72ζ3ζ73ζ3ζ74ζ3ζ75ζ32ζ72    linear of order 21
ρ1711ζ32ζ3ζ7ζ75ζ72ζ76ζ73ζ74ζ75ζ76ζ7ζ73ζ72ζ74ζ3ζ73ζ32ζ75ζ32ζ72ζ32ζ76ζ32ζ73ζ32ζ74ζ3ζ74ζ3ζ7ζ3ζ75ζ3ζ72ζ3ζ76ζ32ζ7    linear of order 21
ρ1811ζ32ζ3ζ74ζ76ζ7ζ73ζ75ζ72ζ76ζ73ζ74ζ75ζ7ζ72ζ3ζ75ζ32ζ76ζ32ζ7ζ32ζ73ζ32ζ75ζ32ζ72ζ3ζ72ζ3ζ74ζ3ζ76ζ3ζ7ζ3ζ73ζ32ζ74    linear of order 21
ρ1911ζ3ζ32ζ7ζ75ζ72ζ76ζ73ζ74ζ75ζ76ζ7ζ73ζ72ζ74ζ32ζ73ζ3ζ75ζ3ζ72ζ3ζ76ζ3ζ73ζ3ζ74ζ32ζ74ζ32ζ7ζ32ζ75ζ32ζ72ζ32ζ76ζ3ζ7    linear of order 21
ρ2011ζ32ζ3ζ73ζ7ζ76ζ74ζ72ζ75ζ7ζ74ζ73ζ72ζ76ζ75ζ3ζ72ζ32ζ7ζ32ζ76ζ32ζ74ζ32ζ72ζ32ζ75ζ3ζ75ζ3ζ73ζ3ζ7ζ3ζ76ζ3ζ74ζ32ζ73    linear of order 21
ρ2111ζ3ζ32ζ73ζ7ζ76ζ74ζ72ζ75ζ7ζ74ζ73ζ72ζ76ζ75ζ32ζ72ζ3ζ7ζ3ζ76ζ3ζ74ζ3ζ72ζ3ζ75ζ32ζ75ζ32ζ73ζ32ζ7ζ32ζ76ζ32ζ74ζ3ζ73    linear of order 21
ρ223-100333333-1-1-1-1-1-1000000000000    orthogonal lifted from A4
ρ233-1003ζ73ζ753ζ723ζ763ζ733ζ74-ζ75-ζ76-ζ7-ζ73-ζ72-ζ74000000000000    complex faithful
ρ243-1003ζ763ζ723ζ753ζ73ζ743ζ73-ζ72-ζ7-ζ76-ζ74-ζ75-ζ73000000000000    complex faithful
ρ253-1003ζ723ζ733ζ743ζ753ζ763ζ7-ζ73-ζ75-ζ72-ζ76-ζ74-ζ7000000000000    complex faithful
ρ263-1003ζ733ζ73ζ763ζ743ζ723ζ75-ζ7-ζ74-ζ73-ζ72-ζ76-ζ75000000000000    complex faithful
ρ273-1003ζ753ζ743ζ733ζ723ζ73ζ76-ζ74-ζ72-ζ75-ζ7-ζ73-ζ76000000000000    complex faithful
ρ283-1003ζ743ζ763ζ73ζ733ζ753ζ72-ζ76-ζ73-ζ74-ζ75-ζ7-ζ72000000000000    complex faithful

Permutation representations of C7×A4
►On 28 points - transitive group 28T17
Generators in S28
(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)

(1 12)(2 13)(3 14)(4 8)(5 9)(6 10)(7 11)(15 26)(16 27)(17 28)(18 22)(19 23)(20 24)(21 25)

(1 24)(2 25)(3 26)(4 27)(5 28)(6 22)(7 23)(8 16)(9 17)(10 18)(11 19)(12 20)(13 21)(14 15)

(8 27 16)(9 28 17)(10 22 18)(11 23 19)(12 24 20)(13 25 21)(14 26 15)
 
G:=sub<Sym(28)| (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), (1,12)(2,13)(3,14)(4,8)(5,9)(6,10)(7,11)(15,26)(16,27)(17,28)(18,22)(19,23)(20,24)(21,25), (1,24)(2,25)(3,26)(4,27)(5,28)(6,22)(7,23)(8,16)(9,17)(10,18)(11,19)(12,20)(13,21)(14,15), (8,27,16)(9,28,17)(10,22,18)(11,23,19)(12,24,20)(13,25,21)(14,26,15)>;
 
G:=Group( (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), (1,12)(2,13)(3,14)(4,8)(5,9)(6,10)(7,11)(15,26)(16,27)(17,28)(18,22)(19,23)(20,24)(21,25), (1,24)(2,25)(3,26)(4,27)(5,28)(6,22)(7,23)(8,16)(9,17)(10,18)(11,19)(12,20)(13,21)(14,15), (8,27,16)(9,28,17)(10,22,18)(11,23,19)(12,24,20)(13,25,21)(14,26,15) );
 
G=PermutationGroup([[(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)], [(1,12),(2,13),(3,14),(4,8),(5,9),(6,10),(7,11),(15,26),(16,27),(17,28),(18,22),(19,23),(20,24),(21,25)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,22),(7,23),(8,16),(9,17),(10,18),(11,19),(12,20),(13,21),(14,15)], [(8,27,16),(9,28,17),(10,22,18),(11,23,19),(12,24,20),(13,25,21),(14,26,15)]])
 
G:=TransitiveGroup(28,17);
 

C7×A4 is a maximal subgroup of   C7⋊S4

Matrix representation of C7×A4 ►in GL3(𝔽43) generated by

1100
0110
0011
,
424242
001
010
,
010
100
424242
,
010
001
100
G:=sub<GL(3,GF(43))| [11,0,0,0,11,0,0,0,11],[42,0,0,42,0,1,42,1,0],[0,1,42,1,0,42,0,0,42],[0,0,1,1,0,0,0,1,0] >;
 

C7×A4 in GAP, Magma, Sage, TeX 

C_7\times A_4
 
% in TeX
 
G:=Group("C7xA4");
 
// GroupNames label
 
G:=SmallGroup(84,10);
 
// by ID
 
G=gap.SmallGroup(84,10);
 
# by ID
 
G:=PCGroup([4,-3,-7,-2,2,506,1011]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^7=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
 
// generators/relations
 

Export

Subgroup lattice of C7×A4 in TeX
Character table of C7×A4 in TeX

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