Copied to
clipboard

G = C72⋊C4order 196 = 22·72

The semidirect product of C72 and C4 acting faithfully

metabelian, soluble, monomial, A-group

Aliases: C72⋊C4, C7⋊D7.C2, SmallGroup(196,8)

Series: Derived Chief Lower central Upper central

C1C72 — C72⋊C4
C1C72C7⋊D7 — C72⋊C4
C72 — C72⋊C4
C1

Generators and relations for C72⋊C4
 G = < a,b,c | a7=b7=c4=1, ab=ba, cac-1=a3b-1, cbc-1=a3b4 >

49C2
2C7
2C7
2C7
2C7
49C4
14D7
14D7
14D7
14D7

Character table of C72⋊C4

 class 124A4B7A7B7C7D7E7F7G7H7I7J7K7L
 size 1494949444444444444
ρ11111111111111111    trivial
ρ211-1-1111111111111    linear of order 2
ρ31-1-ii111111111111    linear of order 4
ρ41-1i-i111111111111    linear of order 4
ρ540007473-17473-1767-175+2ζ72767-174+2ζ737572-17572-176+2ζ7ζ7572+2ζ7473+2ζ767+2    orthogonal faithful
ρ6400074+2ζ73ζ7572+2ζ7473+27473-176+2ζ7767-1ζ767+275+2ζ727572-1767-17572-17473-1    orthogonal faithful
ρ74000767-1767-17572-1ζ7572+27572-1ζ7473+27473-17473-1ζ767+276+2ζ775+2ζ7274+2ζ73    orthogonal faithful
ρ840007473-17473-1767-1ζ767+2767-1ζ7572+27572-17572-1ζ7473+274+2ζ7376+2ζ775+2ζ72    orthogonal faithful
ρ9400076+2ζ7ζ7473+2ζ767+2767-175+2ζ727572-1ζ7572+274+2ζ737473-17572-17473-1767-1    orthogonal faithful
ρ104000ζ767+275+2ζ7274+2ζ737572-1ζ7572+27473-176+2ζ7ζ7473+2767-17473-1767-17572-1    orthogonal faithful
ρ114000767-1767-17572-174+2ζ737572-176+2ζ77473-17473-175+2ζ72ζ7473+2ζ767+2ζ7572+2    orthogonal faithful
ρ124000ζ7473+276+2ζ775+2ζ72767-1ζ767+27572-174+2ζ73ζ7572+27473-17572-17473-1767-1    orthogonal faithful
ρ13400075+2ζ72ζ767+2ζ7572+27572-174+2ζ737473-1ζ7473+276+2ζ7767-17473-1767-17572-1    orthogonal faithful
ρ144000ζ7572+274+2ζ7376+2ζ77473-1ζ7473+2767-175+2ζ72ζ767+27572-1767-17572-17473-1    orthogonal faithful
ρ1540007572-17572-17473-176+2ζ77473-175+2ζ72767-1767-174+2ζ73ζ767+2ζ7572+2ζ7473+2    orthogonal faithful
ρ1640007572-17572-17473-1ζ7473+27473-1ζ767+2767-1767-1ζ7572+275+2ζ7274+2ζ7376+2ζ7    orthogonal faithful

Permutation representations of C72⋊C4
On 14 points - transitive group 14T12
Generators in S14
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)
(1 6 4 2 7 5 3)(8 14 13 12 11 10 9)
(1 8)(2 11 7 12)(3 14 6 9)(4 10 5 13)

G:=sub<Sym(14)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,6,4,2,7,5,3)(8,14,13,12,11,10,9), (1,8)(2,11,7,12)(3,14,6,9)(4,10,5,13)>;

G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,6,4,2,7,5,3)(8,14,13,12,11,10,9), (1,8)(2,11,7,12)(3,14,6,9)(4,10,5,13) );

G=PermutationGroup([(1,2,3,4,5,6,7),(8,9,10,11,12,13,14)], [(1,6,4,2,7,5,3),(8,14,13,12,11,10,9)], [(1,8),(2,11,7,12),(3,14,6,9),(4,10,5,13)])

G:=TransitiveGroup(14,12);

On 28 points - transitive group 28T35
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 6 4 2 7 5 3)(8 13 11 9 14 12 10)(15 21 20 19 18 17 16)(22 28 27 26 25 24 23)
(1 27 12 15)(2 23 11 19)(3 26 10 16)(4 22 9 20)(5 25 8 17)(6 28 14 21)(7 24 13 18)

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

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

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

G:=TransitiveGroup(28,35);

C72⋊C4 is a maximal subgroup of   C72⋊C8  D7≀C2  C72⋊Q8
C72⋊C4 is a maximal quotient of   C722C8

Polynomial with Galois group C72⋊C4 over ℚ
actionf(x)Disc(f)
14T12x14+28x12-189x11+756x10-4004x9+15953x8-48856x7+129262x6-251559x5+330764x4-272986x3-123305x2+739662x-57791622·518·738·192·1732·7012·21112·171592·9818430376572

Matrix representation of C72⋊C4 in GL4(𝔽29) generated by

10100
272600
2027410
1411428
,
1000
0100
2127410
411428
,
00101
412325
258280
1726100
G:=sub<GL(4,GF(29))| [10,27,20,14,1,26,27,1,0,0,4,14,0,0,10,28],[1,0,21,4,0,1,27,1,0,0,4,14,0,0,10,28],[0,4,25,17,0,1,8,26,10,23,28,10,1,25,0,0] >;

C72⋊C4 in GAP, Magma, Sage, TeX

C_7^2\rtimes C_4
% in TeX

G:=Group("C7^2:C4");
// GroupNames label

G:=SmallGroup(196,8);
// by ID

G=gap.SmallGroup(196,8);
# by ID

G:=PCGroup([4,-2,-2,-7,7,8,530,150,1603,1351]);
// Polycyclic

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

Export

Subgroup lattice of C72⋊C4 in TeX
Character table of C72⋊C4 in TeX

׿
×
𝔽