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G = C7⋊D4  order 56 = 23·7

The semidirect product of C7 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C7⋊2D4, C22⋊D7, Dic7⋊C2, D14⋊2C2, C2.5D14, C14.5C22, (C2×C14)⋊2C2, SmallGroup(56,7)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — C7⋊D4
Chief series
C1 — C7 — C14 — D14 — C7⋊D4
Lower central
C7 — C14 — C7⋊D4
Upper central
C1 — C2 — C22

Generators and relations for C7⋊D4
 G = < a,b,c | a7=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

C1
C2
2C2
14C2
C7
C22
7C4
7C22
C14
2D7
2C14
7D4
D14
Dic7
C2×C14
C7⋊D4

Character table of C7⋊D4

 class 12A2B2C47A7B7C14A14B14C14D14E14F14G14H14I
 size 1121414222222222222
ρ111111111111111111    trivial
ρ211-11-11111-1-1-1-1-1-111    linear of order 2
ρ311-1-111111-1-1-1-1-1-111    linear of order 2
ρ4111-1-1111111111111    linear of order 2
ρ52-2000222-2000000-2-2    orthogonal lifted from D4
ρ622200ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ76+ζ7ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ76+ζ7ζ75+ζ72ζ74+ζ73    orthogonal lifted from D7
ρ722200ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ75+ζ72ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ75+ζ72ζ74+ζ73ζ76+ζ7    orthogonal lifted from D7
ρ822200ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ74+ζ73ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ74+ζ73ζ76+ζ7ζ75+ζ72    orthogonal lifted from D7
ρ922-200ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ74+ζ73-ζ74-ζ73-ζ76-ζ7-ζ75-ζ72-ζ75-ζ72-ζ76-ζ7-ζ74-ζ73ζ76+ζ7ζ75+ζ72    orthogonal lifted from D14
ρ1022-200ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ75+ζ72-ζ75-ζ72-ζ74-ζ73-ζ76-ζ7-ζ76-ζ7-ζ74-ζ73-ζ75-ζ72ζ74+ζ73ζ76+ζ7    orthogonal lifted from D14
ρ1122-200ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ76+ζ7-ζ76-ζ7-ζ75-ζ72-ζ74-ζ73-ζ74-ζ73-ζ75-ζ72-ζ76-ζ7ζ75+ζ72ζ74+ζ73    orthogonal lifted from D14
ρ122-2000ζ74+ζ73ζ76+ζ7ζ75+ζ72-ζ76-ζ7-ζ76+ζ7-ζ75+ζ72-ζ74+ζ73ζ74-ζ73ζ75-ζ72ζ76-ζ7-ζ75-ζ72-ζ74-ζ73    complex faithful
ρ132-2000ζ76+ζ7ζ75+ζ72ζ74+ζ73-ζ75-ζ72ζ75-ζ72-ζ74+ζ73-ζ76+ζ7ζ76-ζ7ζ74-ζ73-ζ75+ζ72-ζ74-ζ73-ζ76-ζ7    complex faithful
ρ142-2000ζ75+ζ72ζ74+ζ73ζ76+ζ7-ζ74-ζ73ζ74-ζ73-ζ76+ζ7ζ75-ζ72-ζ75+ζ72ζ76-ζ7-ζ74+ζ73-ζ76-ζ7-ζ75-ζ72    complex faithful
ρ152-2000ζ74+ζ73ζ76+ζ7ζ75+ζ72-ζ76-ζ7ζ76-ζ7ζ75-ζ72ζ74-ζ73-ζ74+ζ73-ζ75+ζ72-ζ76+ζ7-ζ75-ζ72-ζ74-ζ73    complex faithful
ρ162-2000ζ76+ζ7ζ75+ζ72ζ74+ζ73-ζ75-ζ72-ζ75+ζ72ζ74-ζ73ζ76-ζ7-ζ76+ζ7-ζ74+ζ73ζ75-ζ72-ζ74-ζ73-ζ76-ζ7    complex faithful
ρ172-2000ζ75+ζ72ζ74+ζ73ζ76+ζ7-ζ74-ζ73-ζ74+ζ73ζ76-ζ7-ζ75+ζ72ζ75-ζ72-ζ76+ζ7ζ74-ζ73-ζ76-ζ7-ζ75-ζ72    complex faithful

Permutation representations of C7⋊D4
►On 28 points - transitive group 28T6
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 13 15)(2 28 14 21)(3 27 8 20)(4 26 9 19)(5 25 10 18)(6 24 11 17)(7 23 12 16)

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

►On 28 points - transitive group 28T7
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 20 13 27)(2 19 14 26)(3 18 8 25)(4 17 9 24)(5 16 10 23)(6 15 11 22)(7 21 12 28)

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

C7⋊D4 is a maximal subgroup of
 C4○D28  D4×D7  D4⋊2D7  Dic7⋊C6  C21⋊D4  C7⋊D12  C21⋊7D4  C7⋊S4  C35⋊D4  C7⋊D20  C35⋊7D4  C49⋊D4  C72⋊2D4  C7⋊D28  C72⋊7D4
C7⋊D4 is a maximal quotient of
 Dic7⋊C4  D14⋊C4  D4⋊D7  D4.D7  Q8⋊D7  C7⋊Q16  C23.D7  C21⋊D4  C7⋊D12  C21⋊7D4  C35⋊D4  C7⋊D20  C35⋊7D4  C49⋊D4  C72⋊2D4  C7⋊D28  C72⋊7D4

Matrix representation of C7⋊D4 ►in GL2(𝔽29) generated by

71
280
,
919
1420
,
17
028
G:=sub<GL(2,GF(29))| [7,28,1,0],[9,14,19,20],[1,0,7,28] >;
 

C7⋊D4 in GAP, Magma, Sage, TeX 

C_7\rtimes D_4
 
% in TeX
 
G:=Group("C7:D4");
 
// GroupNames label
 
G:=SmallGroup(56,7);
 
// by ID
 
G=gap.SmallGroup(56,7);
 
# by ID
 
G:=PCGroup([4,-2,-2,-2,-7,49,771]);
 
// Polycyclic
 
G:=Group<a,b,c|a^7=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C7⋊D4 in TeX
Character table of C7⋊D4 in TeX

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