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## G = D4×C7⋊C3order 168 = 23·3·7

### Direct product of D4 and C7⋊C3

Aliases: D4×C7⋊C3, C283C6, (C7×D4)⋊C3, C73(C3×D4), (C2×C14)⋊5C6, C14.7(C2×C6), C4⋊(C2×C7⋊C3), (C4×C7⋊C3)⋊3C2, C222(C2×C7⋊C3), (C22×C7⋊C3)⋊3C2, C2.2(C22×C7⋊C3), (C2×C7⋊C3).7C22, SmallGroup(168,20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D4×C7⋊C3
 Chief series C1 — C7 — C14 — C2×C7⋊C3 — C22×C7⋊C3 — D4×C7⋊C3
 Lower central C7 — C14 — D4×C7⋊C3
 Upper central C1 — C2 — D4

Generators and relations for D4×C7⋊C3
G = < a,b,c,d | a4=b2=c7=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Character table of D4×C7⋊C3

 class 1 2A 2B 2C 3A 3B 4 6A 6B 6C 6D 6E 6F 7A 7B 12A 12B 14A 14B 14C 14D 14E 14F 28A 28B size 1 1 2 2 7 7 2 7 7 14 14 14 14 3 3 14 14 3 3 6 6 6 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 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 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 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 linear of order 2 ρ5 1 1 1 -1 ζ3 ζ32 -1 ζ32 ζ3 ζ65 ζ6 ζ3 ζ32 1 1 ζ6 ζ65 1 1 1 -1 -1 1 -1 -1 linear of order 6 ρ6 1 1 -1 -1 ζ3 ζ32 1 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 1 1 ζ32 ζ3 1 1 -1 -1 -1 -1 1 1 linear of order 6 ρ7 1 1 1 1 ζ3 ζ32 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 linear of order 3 ρ8 1 1 -1 1 ζ3 ζ32 -1 ζ32 ζ3 ζ3 ζ32 ζ65 ζ6 1 1 ζ6 ζ65 1 1 -1 1 1 -1 -1 -1 linear of order 6 ρ9 1 1 1 -1 ζ32 ζ3 -1 ζ3 ζ32 ζ6 ζ65 ζ32 ζ3 1 1 ζ65 ζ6 1 1 1 -1 -1 1 -1 -1 linear of order 6 ρ10 1 1 -1 1 ζ32 ζ3 -1 ζ3 ζ32 ζ32 ζ3 ζ6 ζ65 1 1 ζ65 ζ6 1 1 -1 1 1 -1 -1 -1 linear of order 6 ρ11 1 1 -1 -1 ζ32 ζ3 1 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 1 1 ζ3 ζ32 1 1 -1 -1 -1 -1 1 1 linear of order 6 ρ12 1 1 1 1 ζ32 ζ3 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 1 1 ζ3 ζ32 1 1 1 1 1 1 1 1 linear of order 3 ρ13 2 -2 0 0 2 2 0 -2 -2 0 0 0 0 2 2 0 0 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 -2 0 0 -1-√-3 -1+√-3 0 1-√-3 1+√-3 0 0 0 0 2 2 0 0 -2 -2 0 0 0 0 0 0 complex lifted from C3×D4 ρ15 2 -2 0 0 -1+√-3 -1-√-3 0 1+√-3 1-√-3 0 0 0 0 2 2 0 0 -2 -2 0 0 0 0 0 0 complex lifted from C3×D4 ρ16 3 3 3 3 0 0 3 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 0 0 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C7⋊C3 ρ17 3 3 -3 -3 0 0 3 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 0 0 -1+√-7/2 -1-√-7/2 1+√-7/2 1+√-7/2 1-√-7/2 1-√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C2×C7⋊C3 ρ18 3 3 3 -3 0 0 -3 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 0 0 -1+√-7/2 -1-√-7/2 -1-√-7/2 1+√-7/2 1-√-7/2 -1+√-7/2 1-√-7/2 1+√-7/2 complex lifted from C2×C7⋊C3 ρ19 3 3 -3 -3 0 0 3 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 0 0 -1-√-7/2 -1+√-7/2 1-√-7/2 1-√-7/2 1+√-7/2 1+√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C2×C7⋊C3 ρ20 3 3 -3 3 0 0 -3 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 0 0 -1-√-7/2 -1+√-7/2 1-√-7/2 -1+√-7/2 -1-√-7/2 1+√-7/2 1+√-7/2 1-√-7/2 complex lifted from C2×C7⋊C3 ρ21 3 3 -3 3 0 0 -3 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 0 0 -1+√-7/2 -1-√-7/2 1+√-7/2 -1-√-7/2 -1+√-7/2 1-√-7/2 1-√-7/2 1+√-7/2 complex lifted from C2×C7⋊C3 ρ22 3 3 3 3 0 0 3 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 0 0 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C7⋊C3 ρ23 3 3 3 -3 0 0 -3 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 0 0 -1-√-7/2 -1+√-7/2 -1+√-7/2 1-√-7/2 1+√-7/2 -1-√-7/2 1+√-7/2 1-√-7/2 complex lifted from C2×C7⋊C3 ρ24 6 -6 0 0 0 0 0 0 0 0 0 0 0 -1-√-7 -1+√-7 0 0 1-√-7 1+√-7 0 0 0 0 0 0 complex faithful ρ25 6 -6 0 0 0 0 0 0 0 0 0 0 0 -1+√-7 -1-√-7 0 0 1+√-7 1-√-7 0 0 0 0 0 0 complex faithful

Permutation representations of D4×C7⋊C3
On 28 points - transitive group 28T22
Generators in S28
(1 15 8 22)(2 16 9 23)(3 17 10 24)(4 18 11 25)(5 19 12 26)(6 20 13 27)(7 21 14 28)
(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 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)
(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)| (1,15,8,22)(2,16,9,23)(3,17,10,24)(4,18,11,25)(5,19,12,26)(6,20,13,27)(7,21,14,28), (15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (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), (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( (1,15,8,22)(2,16,9,23)(3,17,10,24)(4,18,11,25)(5,19,12,26)(6,20,13,27)(7,21,14,28), (15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (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), (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([[(1,15,8,22),(2,16,9,23),(3,17,10,24),(4,18,11,25),(5,19,12,26),(6,20,13,27),(7,21,14,28)], [(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28)], [(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)], [(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(28,22);

D4×C7⋊C3 is a maximal subgroup of   D4⋊F7  D4.F7  D42F7

Matrix representation of D4×C7⋊C3 in GL5(𝔽337)

 336 88 0 0 0 314 1 0 0 0 0 0 336 0 0 0 0 0 336 0 0 0 0 0 336
,
 336 88 0 0 0 0 1 0 0 0 0 0 336 0 0 0 0 0 336 0 0 0 0 0 336
,
 1 0 0 0 0 0 1 0 0 0 0 0 212 213 1 0 0 1 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 124 336 336 0 0 0 1 0

G:=sub<GL(5,GF(337))| [336,314,0,0,0,88,1,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336],[336,0,0,0,0,88,1,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336],[1,0,0,0,0,0,1,0,0,0,0,0,212,1,0,0,0,213,0,1,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,124,0,0,0,0,336,1,0,0,0,336,0] >;

D4×C7⋊C3 in GAP, Magma, Sage, TeX

D_4\times C_7\rtimes C_3
% in TeX

G:=Group("D4xC7:C3");
// GroupNames label

G:=SmallGroup(168,20);
// by ID

G=gap.SmallGroup(168,20);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-7,141,314]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^7=d^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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