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G = C5×C7⋊C3order 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

C1C7 — C5×C7⋊C3
C1C7C35 — C5×C7⋊C3
C7 — C5×C7⋊C3
C1C5

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

7C3
7C15

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
ρ183005455253-1+-7/2-1--7/200000000ζ53ζ7653ζ7553ζ73ζ53ζ7453ζ7253ζ7ζ5ζ765ζ755ζ73ζ54ζ7654ζ7554ζ73ζ54ζ7454ζ7254ζ7ζ5ζ745ζ725ζ7ζ52ζ7452ζ7252ζ7ζ52ζ7652ζ7552ζ73    complex faithful
ρ193005455253-1--7/2-1+-7/200000000ζ53ζ7453ζ7253ζ7ζ53ζ7653ζ7553ζ73ζ5ζ745ζ725ζ7ζ54ζ7454ζ7254ζ7ζ54ζ7654ζ7554ζ73ζ5ζ765ζ755ζ73ζ52ζ7652ζ7552ζ73ζ52ζ7452ζ7252ζ7    complex faithful
ρ203005352545-1+-7/2-1--7/200000000ζ5ζ765ζ755ζ73ζ5ζ745ζ725ζ7ζ52ζ7652ζ7552ζ73ζ53ζ7653ζ7553ζ73ζ53ζ7453ζ7253ζ7ζ52ζ7452ζ7252ζ7ζ54ζ7454ζ7254ζ7ζ54ζ7654ζ7554ζ73    complex faithful
ρ213005545352-1--7/2-1+-7/200000000ζ52ζ7452ζ7252ζ7ζ52ζ7652ζ7552ζ73ζ54ζ7454ζ7254ζ7ζ5ζ745ζ725ζ7ζ5ζ765ζ755ζ73ζ54ζ7654ζ7554ζ73ζ53ζ7653ζ7553ζ73ζ53ζ7453ζ7253ζ7    complex faithful
ρ223005253554-1+-7/2-1--7/200000000ζ54ζ7654ζ7554ζ73ζ54ζ7454ζ7254ζ7ζ53ζ7653ζ7553ζ73ζ52ζ7652ζ7552ζ73ζ52ζ7452ζ7252ζ7ζ53ζ7453ζ7253ζ7ζ5ζ745ζ725ζ7ζ5ζ765ζ755ζ73    complex faithful
ρ233005253554-1--7/2-1+-7/200000000ζ54ζ7454ζ7254ζ7ζ54ζ7654ζ7554ζ73ζ53ζ7453ζ7253ζ7ζ52ζ7452ζ7252ζ7ζ52ζ7652ζ7552ζ73ζ53ζ7653ζ7553ζ73ζ5ζ765ζ755ζ73ζ5ζ745ζ725ζ7    complex faithful
ρ243005352545-1--7/2-1+-7/200000000ζ5ζ745ζ725ζ7ζ5ζ765ζ755ζ73ζ52ζ7452ζ7252ζ7ζ53ζ7453ζ7253ζ7ζ53ζ7653ζ7553ζ73ζ52ζ7652ζ7552ζ73ζ54ζ7654ζ7554ζ73ζ54ζ7454ζ7254ζ7    complex faithful
ρ253005545352-1+-7/2-1--7/200000000ζ52ζ7652ζ7552ζ73ζ52ζ7452ζ7252ζ7ζ54ζ7654ζ7554ζ73ζ5ζ765ζ755ζ73ζ5ζ745ζ725ζ7ζ54ζ7454ζ7254ζ7ζ53ζ7453ζ7253ζ7ζ53ζ7653ζ7553ζ73    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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