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G = C7⋊C32  order 441 = 32·72

Direct product of C7⋊C3 and C7⋊C3

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

Aliases: C7⋊C32, C72⋊C32, C72⋊3C3⋊C3, C72⋊C3⋊C3, (C7×C7⋊C3)⋊C3, C7⋊1(C3×C7⋊C3), SmallGroup(441,9)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C72 — C7⋊C32
Chief series
C1 — C7 — C72 — C7×C7⋊C3 — C7⋊C32
Lower central
C72 — C7⋊C32
Upper central
C1

Generators and relations for C7⋊C32
 G = < a,b,c,d | a7=b3=c7=d3=1, bab-1=a4, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

C1
7C3
7C3
49C3
49C3
C7
C7
3C7
3C7
49C32
C7⋊C3
C7⋊C3
7C21
7C7⋊C3
7C7⋊C3
7C7⋊C3
7C21
7C7⋊C3
21C7⋊C3
21C7⋊C3
C72
7C3×C7⋊C3
7C3×C7⋊C3
C7×C7⋊C3
C7×C7⋊C3
C72⋊C3
C72⋊3C3
C7⋊C32

Character table of C7⋊C32

 class 13A3B3C3D3E3F3G3H7A7B7C7D7E7F7G7H21A21B21C21D21E21F21G21H
 size 1777749494949333399992121212121212121
ρ11111111111111111111111111    trivial
ρ21ζ32ζ3ζ32ζ311ζ3ζ3211111111ζ3ζ32ζ32ζ32ζ3ζ3ζ3ζ32    linear of order 3
ρ3111ζ3ζ32ζ32ζ3ζ3ζ3211111111111ζ3ζ32ζ321ζ3    linear of order 3
ρ4111ζ32ζ3ζ3ζ32ζ32ζ311111111111ζ32ζ3ζ31ζ32    linear of order 3
ρ51ζ32ζ311ζ32ζ3ζ32ζ311111111ζ3ζ32ζ32111ζ31    linear of order 3
ρ61ζ3ζ32ζ3ζ3211ζ32ζ311111111ζ32ζ3ζ3ζ3ζ32ζ32ζ32ζ3    linear of order 3
ρ71ζ3ζ3211ζ3ζ32ζ3ζ3211111111ζ32ζ3ζ3111ζ321    linear of order 3
ρ81ζ3ζ32ζ32ζ3ζ32ζ31111111111ζ32ζ3ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ91ζ32ζ3ζ3ζ32ζ3ζ321111111111ζ3ζ32ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ10300330000-1+√-7/233-1-√-7/2-1-√-7/2-1+√-7/2-1-√-7/2-1+√-7/2000-1+√-7/2-1+√-7/2-1-√-7/20-1-√-7/2    complex lifted from C7⋊C3
ρ113330000003-1-√-7/2-1+√-7/23-1+√-7/2-1+√-7/2-1-√-7/2-1-√-7/2-1-√-7/2-1+√-7/2-1-√-7/2000-1+√-7/20    complex lifted from C7⋊C3
ρ12300330000-1-√-7/233-1+√-7/2-1+√-7/2-1-√-7/2-1+√-7/2-1-√-7/2000-1-√-7/2-1-√-7/2-1+√-7/20-1+√-7/2    complex lifted from C7⋊C3
ρ133330000003-1+√-7/2-1-√-7/23-1-√-7/2-1-√-7/2-1+√-7/2-1+√-7/2-1+√-7/2-1-√-7/2-1+√-7/2000-1-√-7/20    complex lifted from C7⋊C3
ρ143-3+3√-3/2-3-3√-3/20000003-1-√-7/2-1+√-7/23-1+√-7/2-1+√-7/2-1-√-7/2-1-√-7/2ζ32ζ76+ζ32ζ75+ζ32ζ73ζ3ζ74+ζ3ζ72+ζ3ζ7ζ3ζ76+ζ3ζ75+ζ3ζ73000ζ32ζ74+ζ32ζ72+ζ32ζ70    complex lifted from C3×C7⋊C3
ρ153-3-3√-3/2-3+3√-3/20000003-1+√-7/2-1-√-7/23-1-√-7/2-1-√-7/2-1+√-7/2-1+√-7/2ζ3ζ74+ζ3ζ72+ζ3ζ7ζ32ζ76+ζ32ζ75+ζ32ζ73ζ32ζ74+ζ32ζ72+ζ32ζ7000ζ3ζ76+ζ3ζ75+ζ3ζ730    complex lifted from C3×C7⋊C3
ρ16300-3-3√-3/2-3+3√-3/20000-1+√-7/233-1-√-7/2-1-√-7/2-1+√-7/2-1-√-7/2-1+√-7/2000ζ32ζ74+ζ32ζ72+ζ32ζ7ζ3ζ74+ζ3ζ72+ζ3ζ7ζ3ζ76+ζ3ζ75+ζ3ζ730ζ32ζ76+ζ32ζ75+ζ32ζ73    complex lifted from C3×C7⋊C3
ρ17300-3+3√-3/2-3-3√-3/20000-1-√-7/233-1+√-7/2-1+√-7/2-1-√-7/2-1+√-7/2-1-√-7/2000ζ3ζ76+ζ3ζ75+ζ3ζ73ζ32ζ76+ζ32ζ75+ζ32ζ73ζ32ζ74+ζ32ζ72+ζ32ζ70ζ3ζ74+ζ3ζ72+ζ3ζ7    complex lifted from C3×C7⋊C3
ρ18300-3-3√-3/2-3+3√-3/20000-1-√-7/233-1+√-7/2-1+√-7/2-1-√-7/2-1+√-7/2-1-√-7/2000ζ32ζ76+ζ32ζ75+ζ32ζ73ζ3ζ76+ζ3ζ75+ζ3ζ73ζ3ζ74+ζ3ζ72+ζ3ζ70ζ32ζ74+ζ32ζ72+ζ32ζ7    complex lifted from C3×C7⋊C3
ρ193-3-3√-3/2-3+3√-3/20000003-1-√-7/2-1+√-7/23-1+√-7/2-1+√-7/2-1-√-7/2-1-√-7/2ζ3ζ76+ζ3ζ75+ζ3ζ73ζ32ζ74+ζ32ζ72+ζ32ζ7ζ32ζ76+ζ32ζ75+ζ32ζ73000ζ3ζ74+ζ3ζ72+ζ3ζ70    complex lifted from C3×C7⋊C3
ρ20300-3+3√-3/2-3-3√-3/20000-1+√-7/233-1-√-7/2-1-√-7/2-1+√-7/2-1-√-7/2-1+√-7/2000ζ3ζ74+ζ3ζ72+ζ3ζ7ζ32ζ74+ζ32ζ72+ζ32ζ7ζ32ζ76+ζ32ζ75+ζ32ζ730ζ3ζ76+ζ3ζ75+ζ3ζ73    complex lifted from C3×C7⋊C3
ρ213-3+3√-3/2-3-3√-3/20000003-1+√-7/2-1-√-7/23-1-√-7/2-1-√-7/2-1+√-7/2-1+√-7/2ζ32ζ74+ζ32ζ72+ζ32ζ7ζ3ζ76+ζ3ζ75+ζ3ζ73ζ3ζ74+ζ3ζ72+ζ3ζ7000ζ32ζ76+ζ32ζ75+ζ32ζ730    complex lifted from C3×C7⋊C3
ρ22900000000-3+3√-7/2-3-3√-7/2-3+3√-7/2-3-3√-7/22-3-√-7/2-3+√-7/2200000000    complex faithful
ρ23900000000-3-3√-7/2-3+3√-7/2-3-3√-7/2-3+3√-7/22-3+√-7/2-3-√-7/2200000000    complex faithful
ρ24900000000-3+3√-7/2-3+3√-7/2-3-3√-7/2-3-3√-7/2-3+√-7/222-3-√-7/200000000    complex faithful
ρ25900000000-3-3√-7/2-3-3√-7/2-3+3√-7/2-3+3√-7/2-3-√-7/222-3+√-7/200000000    complex faithful

Permutation representations of C7⋊C32
►On 21 points - transitive group 21T21
Generators in S21
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)

(1 8 15)(2 10 19)(3 12 16)(4 14 20)(5 9 17)(6 11 21)(7 13 18)

(1 7 6 5 4 3 2)(8 13 11 9 14 12 10)(15 18 21 17 20 16 19)

(1 15 8)(2 16 9)(3 17 10)(4 18 11)(5 19 12)(6 20 13)(7 21 14)
 
G:=sub<Sym(21)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,8,15)(2,10,19)(3,12,16)(4,14,20)(5,9,17)(6,11,21)(7,13,18), (1,7,6,5,4,3,2)(8,13,11,9,14,12,10)(15,18,21,17,20,16,19), (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14)>;
 
G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,8,15)(2,10,19)(3,12,16)(4,14,20)(5,9,17)(6,11,21)(7,13,18), (1,7,6,5,4,3,2)(8,13,11,9,14,12,10)(15,18,21,17,20,16,19), (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21)], [(1,8,15),(2,10,19),(3,12,16),(4,14,20),(5,9,17),(6,11,21),(7,13,18)], [(1,7,6,5,4,3,2),(8,13,11,9,14,12,10),(15,18,21,17,20,16,19)], [(1,15,8),(2,16,9),(3,17,10),(4,18,11),(5,19,12),(6,20,13),(7,21,14)]])
 
G:=TransitiveGroup(21,21);
 

Matrix representation of C7⋊C32 ►in GL6(𝔽43)

1810000
1901000
100000
000100
000010
000001
,
3604000
007000
0367000
0003600
0000360
0000036
,
100000
010000
001000
0004210
0009934
00083410
,
600000
060000
006000
000006
000600
000060

G:=sub<GL(6,GF(43))| [18,19,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,0,0,0,0,0,0,0,36,0,0,0,4,7,7,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,42,9,8,0,0,0,1,9,34,0,0,0,0,34,10],[6,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,6,0,0] >;
 

C7⋊C32 in GAP, Magma, Sage, TeX 

C_7\rtimes C_3^2
 
% in TeX
 
G:=Group("C7:C3^2");
 
// GroupNames label
 
G:=SmallGroup(441,9);
 
// by ID
 
G=gap.SmallGroup(441,9);
 
# by ID
 
G:=PCGroup([4,-3,-3,-7,-7,78,2019]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^7=b^3=c^7=d^3=1,b*a*b^-1=a^4,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
 
// generators/relations
 

Export

Subgroup lattice of C7⋊C32 in TeX
Character table of C7⋊C32 in TeX

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