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## G = C4×F7order 168 = 23·3·7

### Direct product of C4 and F7

Aliases: C4×F7, D7⋊C12, C282C6, D14.C6, Dic72C6, (C4×D7)⋊C3, C7⋊C122C2, C71(C2×C12), (C2×F7).C2, C2.1(C2×F7), C14.2(C2×C6), C7⋊C31(C2×C4), (C4×C7⋊C3)⋊2C2, (C2×C7⋊C3).2C22, SmallGroup(168,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C4×F7
 Chief series C1 — C7 — C14 — C2×C7⋊C3 — C2×F7 — C4×F7
 Lower central C7 — C4×F7
 Upper central C1 — C4

Generators and relations for C4×F7
G = < a,b,c | a4=b7=c6=1, ab=ba, ac=ca, cbc-1=b5 >

Character table of C4×F7

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 7 12A 12B 12C 12D 12E 12F 12G 12H 14 28A 28B size 1 1 7 7 7 7 1 1 7 7 7 7 7 7 7 7 6 7 7 7 7 7 7 7 7 6 6 6 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 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 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 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 linear of order 2 ρ5 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 1 1 1 linear of order 3 ρ6 1 1 -1 -1 ζ32 ζ3 1 1 -1 -1 ζ6 ζ6 ζ65 ζ65 ζ3 ζ32 1 ζ65 ζ32 ζ3 ζ3 ζ6 ζ6 ζ65 ζ32 1 1 1 linear of order 6 ρ7 1 1 -1 -1 ζ3 ζ32 1 1 -1 -1 ζ65 ζ65 ζ6 ζ6 ζ32 ζ3 1 ζ6 ζ3 ζ32 ζ32 ζ65 ζ65 ζ6 ζ3 1 1 1 linear of order 6 ρ8 1 1 1 1 ζ32 ζ3 -1 -1 -1 -1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 1 ζ65 ζ6 ζ65 ζ65 ζ6 ζ6 ζ65 ζ6 1 -1 -1 linear of order 6 ρ9 1 1 -1 -1 ζ3 ζ32 -1 -1 1 1 ζ65 ζ65 ζ6 ζ6 ζ32 ζ3 1 ζ32 ζ65 ζ6 ζ6 ζ3 ζ3 ζ32 ζ65 1 -1 -1 linear of order 6 ρ10 1 1 -1 -1 ζ32 ζ3 -1 -1 1 1 ζ6 ζ6 ζ65 ζ65 ζ3 ζ32 1 ζ3 ζ6 ζ65 ζ65 ζ32 ζ32 ζ3 ζ6 1 -1 -1 linear of order 6 ρ11 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 1 1 1 linear of order 3 ρ12 1 1 1 1 ζ3 ζ32 -1 -1 -1 -1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 1 ζ6 ζ65 ζ6 ζ6 ζ65 ζ65 ζ6 ζ65 1 -1 -1 linear of order 6 ρ13 1 -1 -1 1 1 1 -i i -i i -1 1 1 -1 -1 -1 1 -i -i i -i i -i i i -1 -i i linear of order 4 ρ14 1 -1 1 -1 1 1 -i i i -i 1 -1 -1 1 -1 -1 1 i -i i -i -i i -i i -1 -i i linear of order 4 ρ15 1 -1 -1 1 1 1 i -i i -i -1 1 1 -1 -1 -1 1 i i -i i -i i -i -i -1 i -i linear of order 4 ρ16 1 -1 1 -1 1 1 i -i -i i 1 -1 -1 1 -1 -1 1 -i i -i i i -i i -i -1 i -i linear of order 4 ρ17 1 -1 -1 1 ζ32 ζ3 -i i -i i ζ6 ζ32 ζ3 ζ65 ζ65 ζ6 1 ζ43ζ3 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ4ζ32 -1 -i i linear of order 12 ρ18 1 -1 1 -1 ζ32 ζ3 -i i i -i ζ32 ζ6 ζ65 ζ3 ζ65 ζ6 1 ζ4ζ3 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ32 -1 -i i linear of order 12 ρ19 1 -1 -1 1 ζ3 ζ32 -i i -i i ζ65 ζ3 ζ32 ζ6 ζ6 ζ65 1 ζ43ζ32 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ4ζ3 -1 -i i linear of order 12 ρ20 1 -1 -1 1 ζ3 ζ32 i -i i -i ζ65 ζ3 ζ32 ζ6 ζ6 ζ65 1 ζ4ζ32 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ43ζ3 -1 i -i linear of order 12 ρ21 1 -1 1 -1 ζ3 ζ32 i -i -i i ζ3 ζ65 ζ6 ζ32 ζ6 ζ65 1 ζ43ζ32 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ3 -1 i -i linear of order 12 ρ22 1 -1 -1 1 ζ32 ζ3 i -i i -i ζ6 ζ32 ζ3 ζ65 ζ65 ζ6 1 ζ4ζ3 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ43ζ32 -1 i -i linear of order 12 ρ23 1 -1 1 -1 ζ3 ζ32 -i i i -i ζ3 ζ65 ζ6 ζ32 ζ6 ζ65 1 ζ4ζ32 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ3 -1 -i i linear of order 12 ρ24 1 -1 1 -1 ζ32 ζ3 i -i -i i ζ32 ζ6 ζ65 ζ3 ζ65 ζ6 1 ζ43ζ3 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ32 -1 i -i linear of order 12 ρ25 6 6 0 0 0 0 -6 -6 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 1 1 orthogonal lifted from C2×F7 ρ26 6 6 0 0 0 0 6 6 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from F7 ρ27 6 -6 0 0 0 0 6i -6i 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -i i complex faithful ρ28 6 -6 0 0 0 0 -6i 6i 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 i -i complex faithful

Permutation representations of C4×F7
On 28 points - transitive group 28T26
Generators in S28
(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)
(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)
(2 4 3 7 5 6)(9 11 10 14 12 13)(16 18 17 21 19 20)(23 25 24 28 26 27)

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

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

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

G:=TransitiveGroup(28,26);

C4×F7 is a maximal subgroup of   C8⋊F7  D286C6  D42F7  Q83F7
C4×F7 is a maximal quotient of   C8⋊F7  Dic7⋊C12  D14⋊C12

Matrix representation of C4×F7 in GL6(𝔽337)

 189 0 0 0 0 0 0 189 0 0 0 0 0 0 189 0 0 0 0 0 0 189 0 0 0 0 0 0 189 0 0 0 0 0 0 189
,
 336 336 336 336 336 336 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 336 0 0 0 0 0 0 0 0 0 0 336 0 0 0 336 0 0 0 336 0 0 0 0 1 1 1 1 1 1 0 0 0 0 336 0

G:=sub<GL(6,GF(337))| [189,0,0,0,0,0,0,189,0,0,0,0,0,0,189,0,0,0,0,0,0,189,0,0,0,0,0,0,189,0,0,0,0,0,0,189],[336,1,0,0,0,0,336,0,1,0,0,0,336,0,0,1,0,0,336,0,0,0,1,0,336,0,0,0,0,1,336,0,0,0,0,0],[336,0,0,0,1,0,0,0,0,336,1,0,0,0,0,0,1,0,0,0,336,0,1,0,0,0,0,0,1,336,0,336,0,0,1,0] >;

C4×F7 in GAP, Magma, Sage, TeX

C_4\times F_7
% in TeX

G:=Group("C4xF7");
// GroupNames label

G:=SmallGroup(168,8);
// by ID

G=gap.SmallGroup(168,8);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-7,66,3604,614]);
// Polycyclic

G:=Group<a,b,c|a^4=b^7=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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