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## G = Dic7⋊C6order 168 = 23·3·7

### The semidirect product of Dic7 and C6 acting faithfully

Aliases: Dic7⋊C6, D142C6, C222F7, C7⋊C12⋊C2, C7⋊C32D4, C7⋊D4⋊C3, C72(C3×D4), (C2×C14)⋊3C6, (C2×F7)⋊2C2, C2.5(C2×F7), C14.5(C2×C6), (C22×C7⋊C3)⋊1C2, (C2×C7⋊C3).5C22, SmallGroup(168,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — Dic7⋊C6
 Chief series C1 — C7 — C14 — C2×C7⋊C3 — C2×F7 — Dic7⋊C6
 Lower central C7 — C14 — Dic7⋊C6
 Upper central C1 — C2 — C22

Generators and relations for Dic7⋊C6
G = < a,b,c | a14=c6=1, b2=a7, bab-1=a-1, cac-1=a11, cbc-1=a7b >

Character table of Dic7⋊C6

 class 1 2A 2B 2C 3A 3B 4 6A 6B 6C 6D 6E 6F 7 12A 12B 14A 14B 14C size 1 1 2 14 7 7 14 7 7 14 14 14 14 6 14 14 6 6 6 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 1 1 1 ζ3 ζ32 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 1 ζ3 ζ32 1 1 1 linear of order 3 ρ6 1 1 -1 -1 ζ3 ζ32 1 ζ3 ζ32 ζ65 ζ6 ζ6 ζ65 1 ζ3 ζ32 -1 1 -1 linear of order 6 ρ7 1 1 -1 -1 ζ32 ζ3 1 ζ32 ζ3 ζ6 ζ65 ζ65 ζ6 1 ζ32 ζ3 -1 1 -1 linear of order 6 ρ8 1 1 1 -1 ζ3 ζ32 -1 ζ3 ζ32 ζ3 ζ32 ζ6 ζ65 1 ζ65 ζ6 1 1 1 linear of order 6 ρ9 1 1 -1 1 ζ3 ζ32 -1 ζ3 ζ32 ζ65 ζ6 ζ32 ζ3 1 ζ65 ζ6 -1 1 -1 linear of order 6 ρ10 1 1 -1 1 ζ32 ζ3 -1 ζ32 ζ3 ζ6 ζ65 ζ3 ζ32 1 ζ6 ζ65 -1 1 -1 linear of order 6 ρ11 1 1 1 -1 ζ32 ζ3 -1 ζ32 ζ3 ζ32 ζ3 ζ65 ζ6 1 ζ6 ζ65 1 1 1 linear of order 6 ρ12 1 1 1 1 ζ32 ζ3 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 1 ζ32 ζ3 1 1 1 linear of order 3 ρ13 2 -2 0 0 2 2 0 -2 -2 0 0 0 0 2 0 0 0 -2 0 orthogonal lifted from D4 ρ14 2 -2 0 0 -1+√-3 -1-√-3 0 1-√-3 1+√-3 0 0 0 0 2 0 0 0 -2 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 0 0 0 -2 0 complex lifted from C3×D4 ρ16 6 6 6 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 -1 -1 orthogonal lifted from F7 ρ17 6 6 -6 0 0 0 0 0 0 0 0 0 0 -1 0 0 1 -1 1 orthogonal lifted from C2×F7 ρ18 6 -6 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 -√-7 1 √-7 complex faithful ρ19 6 -6 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 √-7 1 -√-7 complex faithful

Permutation representations of Dic7⋊C6
On 28 points - transitive group 28T21
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 22 8 15)(2 21 9 28)(3 20 10 27)(4 19 11 26)(5 18 12 25)(6 17 13 24)(7 16 14 23)
(2 10 12)(3 5 9)(4 14 6)(7 13 11)(15 22)(16 17 26 23 24 19)(18 21 20 25 28 27)

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

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

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

G:=TransitiveGroup(28,21);

On 28 points - transitive group 28T25
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 23 8 16)(2 22 9 15)(3 21 10 28)(4 20 11 27)(5 19 12 26)(6 18 13 25)(7 17 14 24)
(1 23)(2 18 12 24 10 20)(3 27 9 25 5 17)(4 22 6 26 14 28)(7 21 11 15 13 19)(8 16)

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

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

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

G:=TransitiveGroup(28,25);

Dic7⋊C6 is a maximal subgroup of   D286C6  D4×F7  D42F7
Dic7⋊C6 is a maximal quotient of   Dic7⋊C12  D14⋊C12  D4⋊F7  D4.F7  Q82F7  Q8.2F7  C23.2F7

Matrix representation of Dic7⋊C6 in GL6(𝔽337)

 0 336 0 0 0 0 0 0 336 0 0 0 336 212 213 0 0 0 0 0 0 125 124 336 0 0 0 336 0 0 0 0 0 0 336 0
,
 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 336 0 0 0 0 0 0 336 0 0 0 0 0 0 336 0 0 0
,
 1 0 0 0 0 0 124 336 336 0 0 0 0 1 0 0 0 0 0 0 0 336 0 0 0 0 0 213 1 1 0 0 0 0 336 0

G:=sub<GL(6,GF(337))| [0,0,336,0,0,0,336,0,212,0,0,0,0,336,213,0,0,0,0,0,0,125,336,0,0,0,0,124,0,336,0,0,0,336,0,0],[0,0,0,336,0,0,0,0,0,0,336,0,0,0,0,0,0,336,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[1,124,0,0,0,0,0,336,1,0,0,0,0,336,0,0,0,0,0,0,0,336,213,0,0,0,0,0,1,336,0,0,0,0,1,0] >;

Dic7⋊C6 in GAP, Magma, Sage, TeX

{\rm Dic}_7\rtimes C_6
% in TeX

G:=Group("Dic7:C6");
// GroupNames label

G:=SmallGroup(168,11);
// by ID

G=gap.SmallGroup(168,11);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-7,141,3604,614]);
// Polycyclic

G:=Group<a,b,c|a^14=c^6=1,b^2=a^7,b*a*b^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^7*b>;
// generators/relations

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