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## G = C22×C7⋊C3order 84 = 22·3·7

### Direct product of C22 and C7⋊C3

Aliases: C22×C7⋊C3, C142C6, C72(C2×C6), (C2×C14)⋊3C3, SmallGroup(84,9)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C22×C7⋊C3
G = < a,b,c,d | a2=b2=c7=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Character table of C22×C7⋊C3

 class 1 2A 2B 2C 3A 3B 6A 6B 6C 6D 6E 6F 7A 7B 14A 14B 14C 14D 14E 14F size 1 1 1 1 7 7 7 7 7 7 7 7 3 3 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 -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 linear of order 2 ρ5 1 -1 -1 1 ζ3 ζ32 ζ3 ζ6 ζ65 ζ6 ζ32 ζ65 1 1 -1 -1 1 1 -1 -1 linear of order 6 ρ6 1 -1 1 -1 ζ32 ζ3 ζ6 ζ65 ζ6 ζ3 ζ65 ζ32 1 1 -1 1 -1 -1 1 -1 linear of order 6 ρ7 1 1 -1 -1 ζ3 ζ32 ζ65 ζ32 ζ3 ζ6 ζ6 ζ65 1 1 1 -1 -1 -1 -1 1 linear of order 6 ρ8 1 -1 -1 1 ζ32 ζ3 ζ32 ζ65 ζ6 ζ65 ζ3 ζ6 1 1 -1 -1 1 1 -1 -1 linear of order 6 ρ9 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 1 1 linear of order 3 ρ10 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 1 1 linear of order 3 ρ11 1 1 -1 -1 ζ32 ζ3 ζ6 ζ3 ζ32 ζ65 ζ65 ζ6 1 1 1 -1 -1 -1 -1 1 linear of order 6 ρ12 1 -1 1 -1 ζ3 ζ32 ζ65 ζ6 ζ65 ζ32 ζ6 ζ3 1 1 -1 1 -1 -1 1 -1 linear of order 6 ρ13 3 -3 -3 3 0 0 0 0 0 0 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 ρ14 3 -3 3 -3 0 0 0 0 0 0 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 ρ15 3 3 3 3 0 0 0 0 0 0 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 ρ16 3 3 -3 -3 0 0 0 0 0 0 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 ρ17 3 3 3 3 0 0 0 0 0 0 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 ρ18 3 -3 -3 3 0 0 0 0 0 0 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 0 0 0 0 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 0 0 0 0 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

Permutation representations of C22×C7⋊C3
On 28 points - transitive group 28T14
Generators in S28
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(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)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(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)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(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),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(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(28,14);

C22×C7⋊C3 is a maximal subgroup of   Dic7⋊C6

Matrix representation of C22×C7⋊C3 in GL4(𝔽43) generated by

 42 0 0 0 0 42 0 0 0 0 42 0 0 0 0 42
,
 1 0 0 0 0 42 0 0 0 0 42 0 0 0 0 42
,
 1 0 0 0 0 24 25 1 0 1 0 0 0 0 1 0
,
 6 0 0 0 0 1 0 0 0 18 42 42 0 0 1 0
G:=sub<GL(4,GF(43))| [42,0,0,0,0,42,0,0,0,0,42,0,0,0,0,42],[1,0,0,0,0,42,0,0,0,0,42,0,0,0,0,42],[1,0,0,0,0,24,1,0,0,25,0,1,0,1,0,0],[6,0,0,0,0,1,18,0,0,0,42,1,0,0,42,0] >;

C22×C7⋊C3 in GAP, Magma, Sage, TeX

C_2^2\times C_7\rtimes C_3
% in TeX

G:=Group("C2^2xC7:C3");
// GroupNames label

G:=SmallGroup(84,9);
// by ID

G=gap.SmallGroup(84,9);
# by ID

G:=PCGroup([4,-2,-2,-3,-7,107]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^7=d^3=1,a*b=b*a,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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