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G = C22×D7order 56 = 23·7

Direct product of C22 and D7

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C7 — C22×D7
C1C7D7D14 — C22×D7
C7 — C22×D7
C1C22

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 >

7C2
7C2
7C2
7C2
7C22
7C22
7C22
7C22
7C22
7C22
7C23

Character table of C22×D7

 class 12A2B2C2D2E2F2G7A7B7C14A14B14C14D14E14F14G14H14I
 size 11117777222222222222
ρ111111111111111111111    trivial
ρ21-1-111-1-11111-1-1111-1-1-1-1    linear of order 2
ρ31-11-1-1-111111-11-1-1-11-1-11    linear of order 2
ρ411-1-1-11-111111-1-1-1-1-111-1    linear of order 2
ρ51-11-111-1-1111-11-1-1-11-1-11    linear of order 2
ρ61-1-11-111-1111-1-1111-1-1-1-1    linear of order 2
ρ711-1-11-11-11111-1-1-1-1-111-1    linear of order 2
ρ81111-1-1-1-1111111111111    linear of order 2
ρ922220000ζ7572ζ7473ζ767ζ7473ζ767ζ767ζ7572ζ7473ζ7572ζ767ζ7572ζ7473    orthogonal lifted from D7
ρ1022220000ζ767ζ7572ζ7473ζ7572ζ7473ζ7473ζ767ζ7572ζ767ζ7473ζ767ζ7572    orthogonal lifted from D7
ρ112-22-20000ζ7473ζ767ζ7572767ζ757275727473767ζ747375727473ζ767    orthogonal lifted from D14
ρ122-22-20000ζ767ζ7572ζ74737572ζ747374737677572ζ7677473767ζ7572    orthogonal lifted from D14
ρ1322-2-20000ζ7572ζ7473ζ767ζ7473767767757274737572ζ767ζ75727473    orthogonal lifted from D14
ρ1422-2-20000ζ7473ζ767ζ7572ζ7677572757274737677473ζ7572ζ7473767    orthogonal lifted from D14
ρ152-2-220000ζ7473ζ767ζ75727677572ζ7572ζ7473ζ767747375727473767    orthogonal lifted from D14
ρ1622-2-20000ζ767ζ7572ζ7473ζ7572747374737677572767ζ7473ζ7677572    orthogonal lifted from D14
ρ172-2-220000ζ767ζ7572ζ747375727473ζ7473ζ767ζ757276774737677572    orthogonal lifted from D14
ρ182-2-220000ζ7572ζ7473ζ7677473767ζ767ζ7572ζ7473757276775727473    orthogonal lifted from D14
ρ1922220000ζ7473ζ767ζ7572ζ767ζ7572ζ7572ζ7473ζ767ζ7473ζ7572ζ7473ζ767    orthogonal lifted from D7
ρ202-22-20000ζ7572ζ7473ζ7677473ζ76776775727473ζ75727677572ζ7473    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);

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

2800
0280
0028
,
2800
010
001
,
100
001
0283
,
100
0028
0280
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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