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## G = C22×D7order 56 = 23·7

### Direct product of C22 and D7

Aliases: C22×D7, C7⋊C23, C14⋊C22, (C2×C14)⋊3C2, SmallGroup(56,12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C22×D7
 Chief series C1 — C7 — D7 — D14 — C22×D7
 Lower central C7 — C22×D7
 Upper central C1 — C22

Generators and relations for C22×D7
G = < a,b,c,d | a2=b2=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Character table of C22×D7

 class 1 2A 2B 2C 2D 2E 2F 2G 7A 7B 7C 14A 14B 14C 14D 14E 14F 14G 14H 14I size 1 1 1 1 7 7 7 7 2 2 2 2 2 2 2 2 2 2 2 2 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 1 1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 -1 -1 1 linear of order 2 ρ6 1 -1 -1 1 -1 1 1 -1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ7 1 1 -1 -1 1 -1 1 -1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 linear of order 2 ρ8 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ9 2 2 2 2 0 0 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D7 ρ10 2 2 2 2 0 0 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D7 ρ11 2 -2 2 -2 0 0 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 ζ75+ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 ζ76+ζ7 orthogonal lifted from D14 ρ12 2 -2 2 -2 0 0 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 ζ74+ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 ζ75+ζ72 orthogonal lifted from D14 ρ13 2 2 -2 -2 0 0 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 -ζ76-ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ75-ζ72 ζ76+ζ7 ζ75+ζ72 -ζ74-ζ73 orthogonal lifted from D14 ρ14 2 2 -2 -2 0 0 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 -ζ75-ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ74-ζ73 ζ75+ζ72 ζ74+ζ73 -ζ76-ζ7 orthogonal lifted from D14 ρ15 2 -2 -2 2 0 0 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 orthogonal lifted from D14 ρ16 2 2 -2 -2 0 0 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 -ζ74-ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ76-ζ7 ζ74+ζ73 ζ76+ζ7 -ζ75-ζ72 orthogonal lifted from D14 ρ17 2 -2 -2 2 0 0 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 orthogonal lifted from D14 ρ18 2 -2 -2 2 0 0 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 orthogonal lifted from D14 ρ19 2 2 2 2 0 0 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D7 ρ20 2 -2 2 -2 0 0 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 ζ76+ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 ζ74+ζ73 orthogonal lifted from D14

Permutation representations of C22×D7
On 28 points - transitive group 28T9
Generators in S28
(1 27)(2 28)(3 22)(4 23)(5 24)(6 25)(7 26)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)
(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(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)
(1 19)(2 18)(3 17)(4 16)(5 15)(6 21)(7 20)(8 24)(9 23)(10 22)(11 28)(12 27)(13 26)(14 25)

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

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

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

G:=TransitiveGroup(28,9);

C22×D7 is a maximal subgroup of   D14⋊C4  D7⋊A4
C22×D7 is a maximal quotient of   C4○D28  D42D7  Q82D7

Matrix representation of C22×D7 in GL3(𝔽29) generated by

 28 0 0 0 28 0 0 0 28
,
 28 0 0 0 1 0 0 0 1
,
 1 0 0 0 0 1 0 28 3
,
 1 0 0 0 0 28 0 28 0
G:=sub<GL(3,GF(29))| [28,0,0,0,28,0,0,0,28],[28,0,0,0,1,0,0,0,1],[1,0,0,0,0,28,0,1,3],[1,0,0,0,0,28,0,28,0] >;

C22×D7 in GAP, Magma, Sage, TeX

C_2^2\times D_7
% in TeX

G:=Group("C2^2xD7");
// GroupNames label

G:=SmallGroup(56,12);
// by ID

G=gap.SmallGroup(56,12);
# by ID

G:=PCGroup([4,-2,-2,-2,-7,771]);
// Polycyclic

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

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