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G = C5×C7⋊C3  order 105 = 3·5·7

Direct product of C5 and C7⋊C3

direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C5×C7⋊C3, C7⋊C15, C35⋊C3, SmallGroup(105,1)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C7 — C5×C7⋊C3
Chief series
C1 — C7 — C35 — C5×C7⋊C3
Lower central
C7 — C5×C7⋊C3
Upper central
C1 — C5

Generators and relations for C5×C7⋊C3
 G = < a,b,c | a5=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

C1
7C3
C5
C7
7C15
C7⋊C3
C35
C5×C7⋊C3

Character table of C5×C7⋊C3

 class 13A3B5A5B5C5D7A7B15A15B15C15D15E15F15G15H35A35B35C35D35E35F35G35H
 size 1771111337777777733333333
ρ11111111111111111111111111    trivial
ρ21ζ32ζ3111111ζ32ζ3ζ3ζ3ζ3ζ32ζ32ζ3211111111    linear of order 3
ρ31ζ3ζ32111111ζ3ζ32ζ32ζ32ζ32ζ3ζ3ζ311111111    linear of order 3
ρ4111ζ54ζ5ζ52ζ5311ζ54ζ52ζ53ζ54ζ5ζ5ζ52ζ53ζ53ζ53ζ5ζ54ζ54ζ5ζ52ζ52    linear of order 5
ρ5111ζ5ζ54ζ53ζ5211ζ5ζ53ζ52ζ5ζ54ζ54ζ53ζ52ζ52ζ52ζ54ζ5ζ5ζ54ζ53ζ53    linear of order 5
ρ6111ζ52ζ53ζ5ζ5411ζ52ζ5ζ54ζ52ζ53ζ53ζ5ζ54ζ54ζ54ζ53ζ52ζ52ζ53ζ5ζ5    linear of order 5
ρ7111ζ53ζ52ζ54ζ511ζ53ζ54ζ5ζ53ζ52ζ52ζ54ζ5ζ5ζ5ζ52ζ53ζ53ζ52ζ54ζ54    linear of order 5
ρ81ζ32ζ3ζ54ζ5ζ52ζ5311ζ32ζ54ζ3ζ52ζ3ζ53ζ3ζ54ζ3ζ5ζ32ζ5ζ32ζ52ζ32ζ53ζ53ζ53ζ5ζ54ζ54ζ5ζ52ζ52    linear of order 15
ρ91ζ3ζ32ζ5ζ54ζ53ζ5211ζ3ζ5ζ32ζ53ζ32ζ52ζ32ζ5ζ32ζ54ζ3ζ54ζ3ζ53ζ3ζ52ζ52ζ52ζ54ζ5ζ5ζ54ζ53ζ53    linear of order 15
ρ101ζ32ζ3ζ52ζ53ζ5ζ5411ζ32ζ52ζ3ζ5ζ3ζ54ζ3ζ52ζ3ζ53ζ32ζ53ζ32ζ5ζ32ζ54ζ54ζ54ζ53ζ52ζ52ζ53ζ5ζ5    linear of order 15
ρ111ζ32ζ3ζ53ζ52ζ54ζ511ζ32ζ53ζ3ζ54ζ3ζ5ζ3ζ53ζ3ζ52ζ32ζ52ζ32ζ54ζ32ζ5ζ5ζ5ζ52ζ53ζ53ζ52ζ54ζ54    linear of order 15
ρ121ζ3ζ32ζ53ζ52ζ54ζ511ζ3ζ53ζ32ζ54ζ32ζ5ζ32ζ53ζ32ζ52ζ3ζ52ζ3ζ54ζ3ζ5ζ5ζ5ζ52ζ53ζ53ζ52ζ54ζ54    linear of order 15
ρ131ζ32ζ3ζ5ζ54ζ53ζ5211ζ32ζ5ζ3ζ53ζ3ζ52ζ3ζ5ζ3ζ54ζ32ζ54ζ32ζ53ζ32ζ52ζ52ζ52ζ54ζ5ζ5ζ54ζ53ζ53    linear of order 15
ρ141ζ3ζ32ζ52ζ53ζ5ζ5411ζ3ζ52ζ32ζ5ζ32ζ54ζ32ζ52ζ32ζ53ζ3ζ53ζ3ζ5ζ3ζ54ζ54ζ54ζ53ζ52ζ52ζ53ζ5ζ5    linear of order 15
ρ151ζ3ζ32ζ54ζ5ζ52ζ5311ζ3ζ54ζ32ζ52ζ32ζ53ζ32ζ54ζ32ζ5ζ3ζ5ζ3ζ52ζ3ζ53ζ53ζ53ζ5ζ54ζ54ζ5ζ52ζ52    linear of order 15
ρ163003333-1-√-7/2-1+√-7/200000000-1+√-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
ρ173003333-1+√-7/2-1-√-7/200000000-1-√-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
ρ183003ζ543ζ53ζ523ζ53-1+√-7/2-1-√-7/200000000ζ53ζ76+ζ53ζ75+ζ53ζ73ζ53ζ74+ζ53ζ72+ζ53ζ7ζ5ζ76+ζ5ζ75+ζ5ζ73ζ54ζ76+ζ54ζ75+ζ54ζ73ζ54ζ74+ζ54ζ72+ζ54ζ7ζ5ζ74+ζ5ζ72+ζ5ζ7ζ52ζ74+ζ52ζ72+ζ52ζ7ζ52ζ76+ζ52ζ75+ζ52ζ73    complex faithful
ρ193003ζ543ζ53ζ523ζ53-1-√-7/2-1+√-7/200000000ζ53ζ74+ζ53ζ72+ζ53ζ7ζ53ζ76+ζ53ζ75+ζ53ζ73ζ5ζ74+ζ5ζ72+ζ5ζ7ζ54ζ74+ζ54ζ72+ζ54ζ7ζ54ζ76+ζ54ζ75+ζ54ζ73ζ5ζ76+ζ5ζ75+ζ5ζ73ζ52ζ76+ζ52ζ75+ζ52ζ73ζ52ζ74+ζ52ζ72+ζ52ζ7    complex faithful
ρ203003ζ533ζ523ζ543ζ5-1+√-7/2-1-√-7/200000000ζ5ζ76+ζ5ζ75+ζ5ζ73ζ5ζ74+ζ5ζ72+ζ5ζ7ζ52ζ76+ζ52ζ75+ζ52ζ73ζ53ζ76+ζ53ζ75+ζ53ζ73ζ53ζ74+ζ53ζ72+ζ53ζ7ζ52ζ74+ζ52ζ72+ζ52ζ7ζ54ζ74+ζ54ζ72+ζ54ζ7ζ54ζ76+ζ54ζ75+ζ54ζ73    complex faithful
ρ213003ζ53ζ543ζ533ζ52-1-√-7/2-1+√-7/200000000ζ52ζ74+ζ52ζ72+ζ52ζ7ζ52ζ76+ζ52ζ75+ζ52ζ73ζ54ζ74+ζ54ζ72+ζ54ζ7ζ5ζ74+ζ5ζ72+ζ5ζ7ζ5ζ76+ζ5ζ75+ζ5ζ73ζ54ζ76+ζ54ζ75+ζ54ζ73ζ53ζ76+ζ53ζ75+ζ53ζ73ζ53ζ74+ζ53ζ72+ζ53ζ7    complex faithful
ρ223003ζ523ζ533ζ53ζ54-1+√-7/2-1-√-7/200000000ζ54ζ76+ζ54ζ75+ζ54ζ73ζ54ζ74+ζ54ζ72+ζ54ζ7ζ53ζ76+ζ53ζ75+ζ53ζ73ζ52ζ76+ζ52ζ75+ζ52ζ73ζ52ζ74+ζ52ζ72+ζ52ζ7ζ53ζ74+ζ53ζ72+ζ53ζ7ζ5ζ74+ζ5ζ72+ζ5ζ7ζ5ζ76+ζ5ζ75+ζ5ζ73    complex faithful
ρ233003ζ523ζ533ζ53ζ54-1-√-7/2-1+√-7/200000000ζ54ζ74+ζ54ζ72+ζ54ζ7ζ54ζ76+ζ54ζ75+ζ54ζ73ζ53ζ74+ζ53ζ72+ζ53ζ7ζ52ζ74+ζ52ζ72+ζ52ζ7ζ52ζ76+ζ52ζ75+ζ52ζ73ζ53ζ76+ζ53ζ75+ζ53ζ73ζ5ζ76+ζ5ζ75+ζ5ζ73ζ5ζ74+ζ5ζ72+ζ5ζ7    complex faithful
ρ243003ζ533ζ523ζ543ζ5-1-√-7/2-1+√-7/200000000ζ5ζ74+ζ5ζ72+ζ5ζ7ζ5ζ76+ζ5ζ75+ζ5ζ73ζ52ζ74+ζ52ζ72+ζ52ζ7ζ53ζ74+ζ53ζ72+ζ53ζ7ζ53ζ76+ζ53ζ75+ζ53ζ73ζ52ζ76+ζ52ζ75+ζ52ζ73ζ54ζ76+ζ54ζ75+ζ54ζ73ζ54ζ74+ζ54ζ72+ζ54ζ7    complex faithful
ρ253003ζ53ζ543ζ533ζ52-1+√-7/2-1-√-7/200000000ζ52ζ76+ζ52ζ75+ζ52ζ73ζ52ζ74+ζ52ζ72+ζ52ζ7ζ54ζ76+ζ54ζ75+ζ54ζ73ζ5ζ76+ζ5ζ75+ζ5ζ73ζ5ζ74+ζ5ζ72+ζ5ζ7ζ54ζ74+ζ54ζ72+ζ54ζ7ζ53ζ74+ζ53ζ72+ζ53ζ7ζ53ζ76+ζ53ζ75+ζ53ζ73    complex faithful

Smallest permutation representation of C5×C7⋊C3
►On 35 points
Generators in S35
(1 29 22 15 8)(2 30 23 16 9)(3 31 24 17 10)(4 32 25 18 11)(5 33 26 19 12)(6 34 27 20 13)(7 35 28 21 14)

(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)(29 30 31 32 33 34 35)

(2 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)(23 24 26)(25 28 27)(30 31 33)(32 35 34)
 
G:=sub<Sym(35)| (1,29,22,15,8)(2,30,23,16,9)(3,31,24,17,10)(4,32,25,18,11)(5,33,26,19,12)(6,34,27,20,13)(7,35,28,21,14), (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)(29,30,31,32,33,34,35), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27)(30,31,33)(32,35,34)>;
 
G:=Group( (1,29,22,15,8)(2,30,23,16,9)(3,31,24,17,10)(4,32,25,18,11)(5,33,26,19,12)(6,34,27,20,13)(7,35,28,21,14), (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)(29,30,31,32,33,34,35), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27)(30,31,33)(32,35,34) );
 
G=PermutationGroup([[(1,29,22,15,8),(2,30,23,16,9),(3,31,24,17,10),(4,32,25,18,11),(5,33,26,19,12),(6,34,27,20,13),(7,35,28,21,14)], [(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),(29,30,31,32,33,34,35)], [(2,3,5),(4,7,6),(9,10,12),(11,14,13),(16,17,19),(18,21,20),(23,24,26),(25,28,27),(30,31,33),(32,35,34)]])
 

C5×C7⋊C3 is a maximal subgroup of   C5⋊F7

Matrix representation of C5×C7⋊C3 ►in GL3(𝔽11) generated by

300
030
003
,
850
479
1070
,
1510
035
047
G:=sub<GL(3,GF(11))| [3,0,0,0,3,0,0,0,3],[8,4,10,5,7,7,0,9,0],[1,0,0,5,3,4,10,5,7] >;
 

C5×C7⋊C3 in GAP, Magma, Sage, TeX 

C_5\times C_7\rtimes C_3
 
% in TeX
 
G:=Group("C5xC7:C3");
 
// GroupNames label
 
G:=SmallGroup(105,1);
 
// by ID
 
G=gap.SmallGroup(105,1);
 
# by ID
 
G:=PCGroup([3,-3,-5,-7,272]);
 
// Polycyclic
 
G:=Group<a,b,c|a^5=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
 
// generators/relations
 

Export

Subgroup lattice of C5×C7⋊C3 in TeX
Character table of C5×C7⋊C3 in TeX

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