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G = D4×C7⋊C3  order 168 = 23·3·7

Direct product of D4 and C7⋊C3

direct product, metacyclic, supersoluble, monomial

Aliases: D4×C7⋊C3, C28⋊3C6, (C7×D4)⋊C3, C7⋊3(C3×D4), (C2×C14)⋊5C6, C14.7(C2×C6), C4⋊(C2×C7⋊C3), (C4×C7⋊C3)⋊3C2, C22⋊2(C2×C7⋊C3), (C22×C7⋊C3)⋊3C2, C2.2(C22×C7⋊C3), (C2×C7⋊C3).7C22, SmallGroup(168,20)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — D4×C7⋊C3
Chief series
C1 — C7 — C14 — C2×C7⋊C3 — C22×C7⋊C3 — D4×C7⋊C3
Lower central
C7 — C14 — D4×C7⋊C3
Upper central
C1 — C2 — D4

Generators and relations for D4×C7⋊C3
 G = < a,b,c,d | a4=b2=c7=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

C1
C2
2C2
2C2
7C3
C7
C22
C22
C4
7C6
14C6
14C6
C14
2C14
2C14
C7⋊C3
D4
7C2×C6
7C2×C6
7C12
C2×C14
C2×C14
C28
C2×C7⋊C3
2C2×C7⋊C3
2C2×C7⋊C3
7C3×D4
C7×D4
C22×C7⋊C3
C4×C7⋊C3
C22×C7⋊C3
D4×C7⋊C3

Character table of D4×C7⋊C3

 class 12A2B2C3A3B46A6B6C6D6E6F7A7B12A12B14A14B14C14D14E14F28A28B
 size 1122772771414141433141433666666
ρ11111111111111111111111111    trivial
ρ211-1111-11111-1-111-1-111-111-1-1-1    linear of order 2
ρ3111-111-111-1-11111-1-1111-1-11-1-1    linear of order 2
ρ411-1-111111-1-1-1-1111111-1-1-1-111    linear of order 2
ρ5111-1ζ3ζ32-1ζ32ζ3ζ65ζ6ζ3ζ3211ζ6ζ65111-1-11-1-1    linear of order 6
ρ611-1-1ζ3ζ321ζ32ζ3ζ65ζ6ζ65ζ611ζ32ζ311-1-1-1-111    linear of order 6
ρ71111ζ3ζ321ζ32ζ3ζ3ζ32ζ3ζ3211ζ32ζ311111111    linear of order 3
ρ811-11ζ3ζ32-1ζ32ζ3ζ3ζ32ζ65ζ611ζ6ζ6511-111-1-1-1    linear of order 6
ρ9111-1ζ32ζ3-1ζ3ζ32ζ6ζ65ζ32ζ311ζ65ζ6111-1-11-1-1    linear of order 6
ρ1011-11ζ32ζ3-1ζ3ζ32ζ32ζ3ζ6ζ6511ζ65ζ611-111-1-1-1    linear of order 6
ρ1111-1-1ζ32ζ31ζ3ζ32ζ6ζ65ζ6ζ6511ζ3ζ3211-1-1-1-111    linear of order 6
ρ121111ζ32ζ31ζ3ζ32ζ32ζ3ζ32ζ311ζ3ζ3211111111    linear of order 3
ρ132-200220-2-200002200-2-2000000    orthogonal lifted from D4
ρ142-200-1-√-3-1+√-301-√-31+√-300002200-2-2000000    complex lifted from C3×D4
ρ152-200-1+√-3-1-√-301+√-31-√-300002200-2-2000000    complex lifted from C3×D4
ρ163333003000000-1-√-7/2-1+√-7/200-1+√-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
ρ1733-3-3003000000-1-√-7/2-1+√-7/200-1+√-7/2-1-√-7/21+√-7/21+√-7/21-√-7/21-√-7/2-1+√-7/2-1-√-7/2    complex lifted from C2×C7⋊C3
ρ18333-300-3000000-1-√-7/2-1+√-7/200-1+√-7/2-1-√-7/2-1-√-7/21+√-7/21-√-7/2-1+√-7/21-√-7/21+√-7/2    complex lifted from C2×C7⋊C3
ρ1933-3-3003000000-1+√-7/2-1-√-7/200-1-√-7/2-1+√-7/21-√-7/21-√-7/21+√-7/21+√-7/2-1-√-7/2-1+√-7/2    complex lifted from C2×C7⋊C3
ρ2033-3300-3000000-1+√-7/2-1-√-7/200-1-√-7/2-1+√-7/21-√-7/2-1+√-7/2-1-√-7/21+√-7/21+√-7/21-√-7/2    complex lifted from C2×C7⋊C3
ρ2133-3300-3000000-1-√-7/2-1+√-7/200-1+√-7/2-1-√-7/21+√-7/2-1-√-7/2-1+√-7/21-√-7/21-√-7/21+√-7/2    complex lifted from C2×C7⋊C3
ρ223333003000000-1+√-7/2-1-√-7/200-1-√-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
ρ23333-300-3000000-1+√-7/2-1-√-7/200-1-√-7/2-1+√-7/2-1+√-7/21-√-7/21+√-7/2-1-√-7/21+√-7/21-√-7/2    complex lifted from C2×C7⋊C3
ρ246-600000000000-1-√-7-1+√-7001-√-71+√-7000000    complex faithful
ρ256-600000000000-1+√-7-1-√-7001+√-71-√-7000000    complex faithful

Permutation representations of D4×C7⋊C3
►On 28 points - transitive group 28T22
Generators in S28
(1 15 8 22)(2 16 9 23)(3 17 10 24)(4 18 11 25)(5 19 12 26)(6 20 13 27)(7 21 14 28)

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

D4×C7⋊C3 is a maximal subgroup of   D4⋊F7  D4.F7  D4⋊2F7

Matrix representation of D4×C7⋊C3 ►in GL5(𝔽337)

33688000
3141000
0033600
0003360
0000336
,
33688000
01000
0033600
0003360
0000336
,
10000
01000
002122131
00100
00010
,
10000
01000
00100
00124336336
00010

G:=sub<GL(5,GF(337))| [336,314,0,0,0,88,1,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336],[336,0,0,0,0,88,1,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336],[1,0,0,0,0,0,1,0,0,0,0,0,212,1,0,0,0,213,0,1,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,124,0,0,0,0,336,1,0,0,0,336,0] >;
 

D4×C7⋊C3 in GAP, Magma, Sage, TeX 

D_4\times C_7\rtimes C_3
 
% in TeX
 
G:=Group("D4xC7:C3");
 
// GroupNames label
 
G:=SmallGroup(168,20);
 
// by ID
 
G=gap.SmallGroup(168,20);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-2,-7,141,314]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^7=d^3=1,b*a*b=a^-1,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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Subgroup lattice of D4×C7⋊C3 in TeX
Character table of D4×C7⋊C3 in TeX

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