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G = C72⋊S3order 294 = 2·3·72

The semidirect product of C72 and S3 acting faithfully

non-abelian, soluble, monomial, A-group

Aliases: C72⋊S3, C723C31C2, SmallGroup(294,7)

Series: Derived Chief Lower central Upper central

C1C72C723C3 — C72⋊S3
C1C72C723C3 — C72⋊S3
C723C3 — C72⋊S3
C1

Generators and relations for C72⋊S3
 G = < a,b,c,d | a7=b7=c3=d2=1, ab=ba, cac-1=a2, dad=b, cbc-1=b4, dbd=a, dcd=c-1 >

21C2
49C3
2C7
3C7
3C7
49S3
3D7
21C14
14C7⋊C3
3C7×D7

Character table of C72⋊S3

 class 1237A7B7C7D7E7F7G7H7I7J7K14A14B14C14D14E14F
 size 1219833333366666212121212121
ρ111111111111111111111    trivial
ρ21-1111111111111-1-1-1-1-1-1    linear of order 2
ρ320-122222222222000000    orthogonal lifted from S3
ρ43107377574ζ76+2ζ747672ζ73+2ζ72ζ75+2ζ7-1+-7/2ζ7572+1-1--7/2ζ7473+1ζ767+1ζ74ζ76ζ72ζ75ζ7ζ73    complex faithful
ρ53-10ζ76+2ζ74ζ73+2ζ72737ζ75+2ζ775747672-1--7/2ζ7572+1-1+-7/2ζ7473+1ζ767+173775727674    complex faithful
ρ63107672737ζ75+2ζ77574ζ76+2ζ74ζ73+2ζ72-1+-7/2ζ7473+1-1--7/2ζ767+1ζ7572+1ζ7ζ75ζ74ζ73ζ72ζ76    complex faithful
ρ731075747672ζ73+2ζ72737ζ75+2ζ7ζ76+2ζ74-1+-7/2ζ767+1-1--7/2ζ7572+1ζ7473+1ζ72ζ73ζ7ζ76ζ74ζ75    complex faithful
ρ83-107377574ζ76+2ζ747672ζ73+2ζ72ζ75+2ζ7-1+-7/2ζ7572+1-1--7/2ζ7473+1ζ767+174767275773    complex faithful
ρ93-1075747672ζ73+2ζ72737ζ75+2ζ7ζ76+2ζ74-1+-7/2ζ767+1-1--7/2ζ7572+1ζ7473+172737767475    complex faithful
ρ10310ζ73+2ζ72ζ75+2ζ77574ζ76+2ζ747672737-1--7/2ζ767+1-1+-7/2ζ7572+1ζ7473+1ζ75ζ74ζ76ζ7ζ73ζ72    complex faithful
ρ113-107672737ζ75+2ζ77574ζ76+2ζ74ζ73+2ζ72-1+-7/2ζ7473+1-1--7/2ζ767+1ζ7572+177574737276    complex faithful
ρ123-10ζ75+2ζ7ζ76+2ζ747672ζ73+2ζ727377574-1--7/2ζ7473+1-1+-7/2ζ767+1ζ7572+176727374757    complex faithful
ρ13310ζ76+2ζ74ζ73+2ζ72737ζ75+2ζ775747672-1--7/2ζ7572+1-1+-7/2ζ7473+1ζ767+1ζ73ζ7ζ75ζ72ζ76ζ74    complex faithful
ρ143-10ζ73+2ζ72ζ75+2ζ77574ζ76+2ζ747672737-1--7/2ζ767+1-1+-7/2ζ7572+1ζ7473+175747677372    complex faithful
ρ15310ζ75+2ζ7ζ76+2ζ747672ζ73+2ζ727377574-1--7/2ζ7473+1-1+-7/2ζ767+1ζ7572+1ζ76ζ72ζ73ζ74ζ75ζ7    complex faithful
ρ1660076+2ζ7+274+2ζ73+276+2ζ7+275+2ζ72+274+2ζ73+275+2ζ72+2-1767572+2ζ7-1757473+2ζ72ζ76+2ζ74+2ζ737000000    orthogonal faithful
ρ1760075+2ζ72+276+2ζ7+275+2ζ72+274+2ζ73+276+2ζ7+274+2ζ73+2-1757473+2ζ72-1ζ76+2ζ74+2ζ737767572+2ζ7000000    orthogonal faithful
ρ1860074+2ζ73+275+2ζ72+274+2ζ73+276+2ζ7+275+2ζ72+276+2ζ7+2-1ζ76+2ζ74+2ζ737-1767572+2ζ7757473+2ζ72000000    orthogonal faithful
ρ19600-1+-7-1+-7-1--7-1+-7-1--7-1--75+-7/2-15--7/2-1-1000000    complex faithful
ρ20600-1--7-1--7-1+-7-1--7-1+-7-1+-75--7/2-15+-7/2-1-1000000    complex faithful

Permutation representations of C72⋊S3
On 14 points - transitive group 14T15
Generators in S14
(8 9 10 11 12 13 14)
(1 7 6 5 4 3 2)
(2 3 5)(4 7 6)(9 12 10)(11 13 14)
(1 8)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)

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

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

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

G:=TransitiveGroup(14,15);

On 21 points - transitive group 21T17
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 7 6 5 4 3 2)(8 13 11 9 14 12 10)(15 18 21 17 20 16 19)
(1 11 16)(2 8 18)(3 12 20)(4 9 15)(5 13 17)(6 10 19)(7 14 21)
(2 7)(3 6)(4 5)(8 21)(9 17)(10 20)(11 16)(12 19)(13 15)(14 18)

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

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

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

G:=TransitiveGroup(21,17);

On 21 points - transitive group 21T18
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 2 3 4 5 6 7)(8 10 12 14 9 11 13)(15 19 16 20 17 21 18)
(1 10 19)(2 14 21)(3 11 16)(4 8 18)(5 12 20)(6 9 15)(7 13 17)
(8 18)(9 15)(10 19)(11 16)(12 20)(13 17)(14 21)

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

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

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

G:=TransitiveGroup(21,18);

Polynomial with Galois group C72⋊S3 over ℚ
actionf(x)Disc(f)
14T15x14-105x12-147x11+5271x10+19838x9-94150x8-607634x7+570164x6+12260920x5+42847770x4+95169270x3+197804880x2+348280352x+336238208-224·724·317·1312·3172·47377272·49178134212·10963085927262635337784632

Matrix representation of C72⋊S3 in GL3(𝔽43) generated by

20276
15266
34721
,
42042
242425
111
,
7288
15830
331628
,
421116
02530
0518
G:=sub<GL(3,GF(43))| [20,15,34,27,26,7,6,6,21],[42,24,1,0,24,1,42,25,1],[7,15,33,28,8,16,8,30,28],[42,0,0,11,25,5,16,30,18] >;

C72⋊S3 in GAP, Magma, Sage, TeX

C_7^2\rtimes S_3
% in TeX

G:=Group("C7^2:S3");
// GroupNames label

G:=SmallGroup(294,7);
// by ID

G=gap.SmallGroup(294,7);
# by ID

G:=PCGroup([4,-2,-3,-7,7,33,506,78,99,1351]);
// Polycyclic

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

Export

Subgroup lattice of C72⋊S3 in TeX
Character table of C72⋊S3 in TeX

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