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G = C723C3order 147 = 3·72

3rd semidirect product of C72 and C3 acting faithfully

metabelian, supersoluble, monomial, A-group

Aliases: C723C3, C72(C7⋊C3), SmallGroup(147,5)

Series: Derived Chief Lower central Upper central

C1C72 — C723C3
C1C7C72 — C723C3
C72 — C723C3
C1

Generators and relations for C723C3
 G = < a,b,c | a7=b7=c3=1, ab=ba, cac-1=a4, cbc-1=b2 >

49C3
3C7
3C7
7C7⋊C3
7C7⋊C3

Character table of C723C3

 class 13A3B7A7B7C7D7E7F7G7H7I7J7K7L7M7N7O7P
 size 149493333333333333333
ρ11111111111111111111    trivial
ρ21ζ32ζ31111111111111111    linear of order 3
ρ31ζ3ζ321111111111111111    linear of order 3
ρ4300-1--7/2ζ76+2ζ74ζ73+2ζ72ζ7572+1ζ75+2ζ7ζ7473+1ζ767+1-1+-7/2ζ7572+1ζ767+17574ζ7473+17377672-1--7/2-1+-7/2    complex faithful
ρ5300-1--7/2ζ73+2ζ72ζ75+2ζ7ζ767+1ζ76+2ζ74ζ7572+1ζ7473+1-1+-7/2ζ767+1ζ7473+17672ζ7572+17574737-1--7/2-1+-7/2    complex faithful
ρ6300-1--7/2ζ7473+1ζ7572+1ζ76+2ζ74ζ767+1ζ75+2ζ7ζ73+2ζ72-1+-7/27377574ζ7572+17672ζ7473+1ζ767+1-1+-7/2-1--7/2    complex faithful
ρ7300-1--7/2-1--7/2-1--7/2-1--7/2-1--7/2-1--7/2-1--7/2-1+-7/2-1+-7/2-1+-7/2-1+-7/2-1+-7/2-1+-7/2-1+-7/233    complex lifted from C7⋊C3
ρ8300-1+-7/2ζ767+1ζ7473+17672ζ7572+17574737-1--7/2ζ75+2ζ7ζ76+2ζ74ζ7473+1ζ73+2ζ72ζ767+1ζ7572+1-1--7/2-1+-7/2    complex faithful
ρ9300-1+-7/275747672ζ767+1737ζ7572+1ζ7473+1-1--7/2ζ767+1ζ7473+1ζ75+2ζ7ζ7572+1ζ73+2ζ72ζ76+2ζ74-1+-7/2-1--7/2    complex faithful
ρ10300-1+-7/27377574ζ7572+17672ζ7473+1ζ767+1-1--7/2ζ7572+1ζ767+1ζ73+2ζ72ζ7473+1ζ76+2ζ74ζ75+2ζ7-1+-7/2-1--7/2    complex faithful
ρ11300-1+-7/2ζ7473+1ζ7572+1737ζ767+176727574-1--7/2ζ76+2ζ74ζ73+2ζ72ζ7572+1ζ75+2ζ7ζ7473+1ζ767+1-1--7/2-1+-7/2    complex faithful
ρ12300-1--7/2ζ75+2ζ7ζ76+2ζ74ζ7473+1ζ73+2ζ72ζ767+1ζ7572+1-1+-7/2ζ7473+1ζ7572+1737ζ767+176727574-1--7/2-1+-7/2    complex faithful
ρ13300-1--7/2ζ7572+1ζ767+1ζ73+2ζ72ζ7473+1ζ76+2ζ74ζ75+2ζ7-1+-7/275747672ζ767+1737ζ7572+1ζ7473+1-1+-7/2-1--7/2    complex faithful
ρ143003-1--7/2-1--7/2-1+-7/2-1--7/2-1+-7/2-1+-7/23-1--7/2-1--7/2-1+-7/2-1--7/2-1+-7/2-1+-7/2-1--7/2-1+-7/2    complex lifted from C7⋊C3
ρ15300-1+-7/27672737ζ7473+17574ζ767+1ζ7572+1-1--7/2ζ7473+1ζ7572+1ζ76+2ζ74ζ767+1ζ75+2ζ7ζ73+2ζ72-1+-7/2-1--7/2    complex faithful
ρ16300-1--7/2ζ767+1ζ7473+1ζ75+2ζ7ζ7572+1ζ73+2ζ72ζ76+2ζ74-1+-7/27672737ζ7473+17574ζ767+1ζ7572+1-1+-7/2-1--7/2    complex faithful
ρ17300-1+-7/2ζ7572+1ζ767+17574ζ7473+17377672-1--7/2ζ73+2ζ72ζ75+2ζ7ζ767+1ζ76+2ζ74ζ7572+1ζ7473+1-1--7/2-1+-7/2    complex faithful
ρ183003-1+-7/2-1+-7/2-1--7/2-1+-7/2-1--7/2-1--7/23-1+-7/2-1+-7/2-1--7/2-1+-7/2-1--7/2-1--7/2-1+-7/2-1--7/2    complex lifted from C7⋊C3
ρ19300-1+-7/2-1+-7/2-1+-7/2-1+-7/2-1+-7/2-1+-7/2-1+-7/2-1--7/2-1--7/2-1--7/2-1--7/2-1--7/2-1--7/2-1--7/233    complex lifted from C7⋊C3

Permutation representations of C723C3
On 21 points - transitive group 21T12
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 3 5 7 2 4 6)(8 12 9 13 10 14 11)(15 16 17 18 19 20 21)
(1 17 13)(2 19 10)(3 21 14)(4 16 11)(5 18 8)(6 20 12)(7 15 9)

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

G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,3,5,7,2,4,6)(8,12,9,13,10,14,11)(15,16,17,18,19,20,21), (1,17,13)(2,19,10)(3,21,14)(4,16,11)(5,18,8)(6,20,12)(7,15,9) );

G=PermutationGroup([(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21)], [(1,3,5,7,2,4,6),(8,12,9,13,10,14,11),(15,16,17,18,19,20,21)], [(1,17,13),(2,19,10),(3,21,14),(4,16,11),(5,18,8),(6,20,12),(7,15,9)])

G:=TransitiveGroup(21,12);

C723C3 is a maximal subgroup of   C72⋊S3  C73F7  C72⋊C6  C7⋊C32
C723C3 is a maximal quotient of   C723C9

Matrix representation of C723C3 in GL3(𝔽43) generated by

2100
0110
0035
,
4100
0160
004
,
010
001
100
G:=sub<GL(3,GF(43))| [21,0,0,0,11,0,0,0,35],[41,0,0,0,16,0,0,0,4],[0,0,1,1,0,0,0,1,0] >;

C723C3 in GAP, Magma, Sage, TeX

C_7^2\rtimes_3C_3
% in TeX

G:=Group("C7^2:3C3");
// GroupNames label

G:=SmallGroup(147,5);
// by ID

G=gap.SmallGroup(147,5);
# by ID

G:=PCGroup([3,-3,-7,-7,37,758]);
// Polycyclic

G:=Group<a,b,c|a^7=b^7=c^3=1,a*b=b*a,c*a*c^-1=a^4,c*b*c^-1=b^2>;
// generators/relations

Export

Subgroup lattice of C723C3 in TeX
Character table of C723C3 in TeX

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