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G = C75F7order 294 = 2·3·72

The semidirect product of C7 and F7 acting via F7/C7⋊C3=C2

metabelian, supersoluble, monomial, A-group

Aliases: C75F7, C723C6, C7⋊C3⋊D7, C7⋊(C3×D7), C7⋊D71C3, (C7×C7⋊C3)⋊3C2, SmallGroup(294,10)

Series: Derived Chief Lower central Upper central

C1C72 — C75F7
C1C7C72C7×C7⋊C3 — C75F7
C72 — C75F7
C1

Generators and relations for C75F7
 G = < a,b,c | a7=b7=c6=1, ab=ba, cac-1=a-1, cbc-1=b5 >

49C2
7C3
3C7
3C7
49C6
7D7
7D7
21D7
21D7
7C21
7F7
7C3×D7

Character table of C75F7

 class 123A3B6A6B7A7B7C7D7E7F7G7H7I7J21A21B21C21D21E21F
 size 1497749492226666666141414141414
ρ11111111111111111111111    trivial
ρ21-111-1-11111111111111111    linear of order 2
ρ311ζ32ζ3ζ3ζ321111111111ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ411ζ3ζ32ζ32ζ31111111111ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ51-1ζ3ζ32ζ6ζ651111111111ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 6
ρ61-1ζ32ζ3ζ65ζ61111111111ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 6
ρ7202200ζ7572ζ7473ζ767ζ7473ζ75722ζ767ζ7473ζ767ζ7572ζ7473ζ767ζ7572ζ767ζ7572ζ7473    orthogonal lifted from D7
ρ8202200ζ767ζ7572ζ7473ζ7572ζ7672ζ7473ζ7572ζ7473ζ767ζ7572ζ7473ζ767ζ7473ζ767ζ7572    orthogonal lifted from D7
ρ9202200ζ7473ζ767ζ7572ζ767ζ74732ζ7572ζ767ζ7572ζ7473ζ767ζ7572ζ7473ζ7572ζ7473ζ767    orthogonal lifted from D7
ρ1020-1--3-1+-300ζ7572ζ7473ζ767ζ7473ζ75722ζ767ζ7473ζ767ζ7572ζ3ζ743ζ73ζ3ζ763ζ7ζ3ζ753ζ72ζ32ζ7632ζ7ζ32ζ7532ζ72ζ32ζ7432ζ73    complex lifted from C3×D7
ρ1120-1+-3-1--300ζ7572ζ7473ζ767ζ7473ζ75722ζ767ζ7473ζ767ζ7572ζ32ζ7432ζ73ζ32ζ7632ζ7ζ32ζ7532ζ72ζ3ζ763ζ7ζ3ζ753ζ72ζ3ζ743ζ73    complex lifted from C3×D7
ρ1220-1+-3-1--300ζ7473ζ767ζ7572ζ767ζ74732ζ7572ζ767ζ7572ζ7473ζ32ζ7632ζ7ζ32ζ7532ζ72ζ32ζ7432ζ73ζ3ζ753ζ72ζ3ζ743ζ73ζ3ζ763ζ7    complex lifted from C3×D7
ρ1320-1--3-1+-300ζ767ζ7572ζ7473ζ7572ζ7672ζ7473ζ7572ζ7473ζ767ζ3ζ753ζ72ζ3ζ743ζ73ζ3ζ763ζ7ζ32ζ7432ζ73ζ32ζ7632ζ7ζ32ζ7532ζ72    complex lifted from C3×D7
ρ1420-1--3-1+-300ζ7473ζ767ζ7572ζ767ζ74732ζ7572ζ767ζ7572ζ7473ζ3ζ763ζ7ζ3ζ753ζ72ζ3ζ743ζ73ζ32ζ7532ζ72ζ32ζ7432ζ73ζ32ζ7632ζ7    complex lifted from C3×D7
ρ1520-1+-3-1--300ζ767ζ7572ζ7473ζ7572ζ7672ζ7473ζ7572ζ7473ζ767ζ32ζ7532ζ72ζ32ζ7432ζ73ζ32ζ7632ζ7ζ3ζ743ζ73ζ3ζ763ζ7ζ3ζ753ζ72    complex lifted from C3×D7
ρ16600000666-1-1-1-1-1-1-1000000    orthogonal lifted from F7
ρ1760000075+3ζ7274+3ζ7376+3ζ7767+1ζ76+2ζ74+2ζ737-17572+1767572+2ζ7757473+2ζ727473+1000000    orthogonal faithful
ρ1860000075+3ζ7274+3ζ7376+3ζ7767572+2ζ77473+1-1757473+2ζ72767+17572+1ζ76+2ζ74+2ζ737000000    orthogonal faithful
ρ1960000074+3ζ7376+3ζ775+3ζ72757473+2ζ72767+1-1ζ76+2ζ74+2ζ7377572+17473+1767572+2ζ7000000    orthogonal faithful
ρ2060000074+3ζ7376+3ζ775+3ζ727572+1767572+2ζ7-17473+1757473+2ζ72ζ76+2ζ74+2ζ737767+1000000    orthogonal faithful
ρ2160000076+3ζ775+3ζ7274+3ζ737473+1757473+2ζ72-1767+1ζ76+2ζ74+2ζ737767572+2ζ77572+1000000    orthogonal faithful
ρ2260000076+3ζ775+3ζ7274+3ζ73ζ76+2ζ74+2ζ7377572+1-1767572+2ζ77473+1767+1757473+2ζ72000000    orthogonal faithful

Permutation representations of C75F7
On 21 points - transitive group 21T16
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 8 15)(2 14 16 7 9 21)(3 13 17 6 10 20)(4 12 18 5 11 19)

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

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

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

G:=TransitiveGroup(21,16);

Matrix representation of C75F7 in GL6(𝔽43)

23230000
20350000
00232300
00203500
00002323
00002035
,
8420000
100000
00203500
0084200
00003520
00002323
,
000010
0000842
100000
8420000
001000
0084200

G:=sub<GL(6,GF(43))| [23,20,0,0,0,0,23,35,0,0,0,0,0,0,23,20,0,0,0,0,23,35,0,0,0,0,0,0,23,20,0,0,0,0,23,35],[8,1,0,0,0,0,42,0,0,0,0,0,0,0,20,8,0,0,0,0,35,42,0,0,0,0,0,0,35,23,0,0,0,0,20,23],[0,0,1,8,0,0,0,0,0,42,0,0,0,0,0,0,1,8,0,0,0,0,0,42,1,8,0,0,0,0,0,42,0,0,0,0] >;

C75F7 in GAP, Magma, Sage, TeX

C_7\rtimes_5F_7
% in TeX

G:=Group("C7:5F7");
// GroupNames label

G:=SmallGroup(294,10);
// by ID

G=gap.SmallGroup(294,10);
# by ID

G:=PCGroup([4,-2,-3,-7,-7,434,4035,679]);
// Polycyclic

G:=Group<a,b,c|a^7=b^7=c^6=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations

Export

Subgroup lattice of C75F7 in TeX
Character table of C75F7 in TeX

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