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## G = C4×C7⋊C3order 84 = 22·3·7

### Direct product of C4 and C7⋊C3

Aliases: C4×C7⋊C3, C28⋊C3, C72C12, C14.2C6, C2.(C2×C7⋊C3), (C2×C7⋊C3).2C2, SmallGroup(84,2)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4×C7⋊C3
G = < a,b,c | a4=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Character table of C4×C7⋊C3

 class 1 2 3A 3B 4A 4B 6A 6B 7A 7B 12A 12B 12C 12D 14A 14B 28A 28B 28C 28D size 1 1 7 7 1 1 7 7 3 3 7 7 7 7 3 3 3 3 3 3 ρ1 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 linear of order 2 ρ3 1 1 ζ32 ζ3 1 1 ζ32 ζ3 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 1 linear of order 3 ρ4 1 1 ζ32 ζ3 -1 -1 ζ32 ζ3 1 1 ζ65 ζ6 ζ65 ζ6 1 1 -1 -1 -1 -1 linear of order 6 ρ5 1 1 ζ3 ζ32 1 1 ζ3 ζ32 1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 1 linear of order 3 ρ6 1 1 ζ3 ζ32 -1 -1 ζ3 ζ32 1 1 ζ6 ζ65 ζ6 ζ65 1 1 -1 -1 -1 -1 linear of order 6 ρ7 1 -1 1 1 i -i -1 -1 1 1 -i -i i i -1 -1 i -i i -i linear of order 4 ρ8 1 -1 1 1 -i i -1 -1 1 1 i i -i -i -1 -1 -i i -i i linear of order 4 ρ9 1 -1 ζ32 ζ3 -i i ζ6 ζ65 1 1 ζ4ζ3 ζ4ζ32 ζ43ζ3 ζ43ζ32 -1 -1 -i i -i i linear of order 12 ρ10 1 -1 ζ3 ζ32 -i i ζ65 ζ6 1 1 ζ4ζ32 ζ4ζ3 ζ43ζ32 ζ43ζ3 -1 -1 -i i -i i linear of order 12 ρ11 1 -1 ζ3 ζ32 i -i ζ65 ζ6 1 1 ζ43ζ32 ζ43ζ3 ζ4ζ32 ζ4ζ3 -1 -1 i -i i -i linear of order 12 ρ12 1 -1 ζ32 ζ3 i -i ζ6 ζ65 1 1 ζ43ζ3 ζ43ζ32 ζ4ζ3 ζ4ζ32 -1 -1 i -i i -i linear of order 12 ρ13 3 3 0 0 3 3 0 0 -1-√-7/2 -1+√-7/2 0 0 0 0 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 complex lifted from C7⋊C3 ρ14 3 3 0 0 3 3 0 0 -1+√-7/2 -1-√-7/2 0 0 0 0 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 complex lifted from C7⋊C3 ρ15 3 3 0 0 -3 -3 0 0 -1+√-7/2 -1-√-7/2 0 0 0 0 -1-√-7/2 -1+√-7/2 1-√-7/2 1-√-7/2 1+√-7/2 1+√-7/2 complex lifted from C2×C7⋊C3 ρ16 3 3 0 0 -3 -3 0 0 -1-√-7/2 -1+√-7/2 0 0 0 0 -1+√-7/2 -1-√-7/2 1+√-7/2 1+√-7/2 1-√-7/2 1-√-7/2 complex lifted from C2×C7⋊C3 ρ17 3 -3 0 0 3i -3i 0 0 -1+√-7/2 -1-√-7/2 0 0 0 0 1+√-7/2 1-√-7/2 ζ4ζ74+ζ4ζ72+ζ4ζ7 ζ43ζ74+ζ43ζ72+ζ43ζ7 ζ4ζ76+ζ4ζ75+ζ4ζ73 ζ43ζ76+ζ43ζ75+ζ43ζ73 complex faithful ρ18 3 -3 0 0 -3i 3i 0 0 -1+√-7/2 -1-√-7/2 0 0 0 0 1+√-7/2 1-√-7/2 ζ43ζ74+ζ43ζ72+ζ43ζ7 ζ4ζ74+ζ4ζ72+ζ4ζ7 ζ43ζ76+ζ43ζ75+ζ43ζ73 ζ4ζ76+ζ4ζ75+ζ4ζ73 complex faithful ρ19 3 -3 0 0 -3i 3i 0 0 -1-√-7/2 -1+√-7/2 0 0 0 0 1-√-7/2 1+√-7/2 ζ43ζ76+ζ43ζ75+ζ43ζ73 ζ4ζ76+ζ4ζ75+ζ4ζ73 ζ43ζ74+ζ43ζ72+ζ43ζ7 ζ4ζ74+ζ4ζ72+ζ4ζ7 complex faithful ρ20 3 -3 0 0 3i -3i 0 0 -1-√-7/2 -1+√-7/2 0 0 0 0 1-√-7/2 1+√-7/2 ζ4ζ76+ζ4ζ75+ζ4ζ73 ζ43ζ76+ζ43ζ75+ζ43ζ73 ζ4ζ74+ζ4ζ72+ζ4ζ7 ζ43ζ74+ζ43ζ72+ζ43ζ7 complex faithful

Permutation representations of C4×C7⋊C3
On 28 points - transitive group 28T13
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 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)(23 24 26)(25 28 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), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27)>;

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

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

G:=TransitiveGroup(28,13);

C4×C7⋊C3 is a maximal subgroup of   C7⋊C24  C4.F7  C4⋊F7  C28.A4

Matrix representation of C4×C7⋊C3 in GL4(𝔽337) generated by

 189 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 336 124 1 0 0 124 1 0 336 125 1
,
 208 0 0 0 0 125 1 213 0 1 0 0 0 1 1 212
G:=sub<GL(4,GF(337))| [189,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,336,0,336,0,124,124,125,0,1,1,1],[208,0,0,0,0,125,1,1,0,1,0,1,0,213,0,212] >;

C4×C7⋊C3 in GAP, Magma, Sage, TeX

C_4\times C_7\rtimes C_3
% in TeX

G:=Group("C4xC7:C3");
// GroupNames label

G:=SmallGroup(84,2);
// by ID

G=gap.SmallGroup(84,2);
# by ID

G:=PCGroup([4,-2,-3,-2,-7,24,199]);
// Polycyclic

G:=Group<a,b,c|a^4=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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