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G = D5×C7⋊C3order 210 = 2·3·5·7

Direct product of D5 and C7⋊C3

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

Aliases: D5×C7⋊C3, C353C6, (C7×D5)⋊C3, C72(C3×D5), C5⋊(C2×C7⋊C3), (C5×C7⋊C3)⋊3C2, SmallGroup(210,2)

Series: Derived Chief Lower central Upper central

C1C35 — D5×C7⋊C3
C1C7C35C5×C7⋊C3 — D5×C7⋊C3
C35 — D5×C7⋊C3
C1

Generators and relations for D5×C7⋊C3
 G = < a,b,c,d | a5=b2=c7=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

5C2
7C3
35C6
5C14
7C15
7C3×D5
5C2×C7⋊C3

Character table of D5×C7⋊C3

 class 123A3B5A5B6A6B7A7B14A14B15A15B15C15D35A35B35C35D
 size 1577223535331515141414146666
ρ111111111111111111111    trivial
ρ21-11111-1-111-1-111111111    linear of order 2
ρ311ζ32ζ311ζ3ζ321111ζ3ζ3ζ32ζ321111    linear of order 3
ρ41-1ζ3ζ3211ζ6ζ6511-1-1ζ32ζ32ζ3ζ31111    linear of order 6
ρ511ζ3ζ3211ζ32ζ31111ζ32ζ32ζ3ζ31111    linear of order 3
ρ61-1ζ32ζ311ζ65ζ611-1-1ζ3ζ3ζ32ζ321111    linear of order 6
ρ72022-1+5/2-1-5/2002200-1-5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1+5/2    orthogonal lifted from D5
ρ82022-1-5/2-1+5/2002200-1+5/2-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ920-1--3-1+-3-1-5/2-1+5/2002200ζ3ζ543ζ5ζ3ζ533ζ52ζ32ζ5432ζ5ζ32ζ5332ζ52-1-5/2-1+5/2-1+5/2-1-5/2    complex lifted from C3×D5
ρ1020-1+-3-1--3-1+5/2-1-5/2002200ζ32ζ5332ζ52ζ32ζ5432ζ5ζ3ζ533ζ52ζ3ζ543ζ5-1+5/2-1-5/2-1-5/2-1+5/2    complex lifted from C3×D5
ρ1120-1+-3-1--3-1-5/2-1+5/2002200ζ32ζ5432ζ5ζ32ζ5332ζ52ζ3ζ543ζ5ζ3ζ533ζ52-1-5/2-1+5/2-1+5/2-1-5/2    complex lifted from C3×D5
ρ1220-1--3-1+-3-1+5/2-1-5/2002200ζ3ζ533ζ52ζ3ζ543ζ5ζ32ζ5332ζ52ζ32ζ5432ζ5-1+5/2-1-5/2-1-5/2-1+5/2    complex lifted from C3×D5
ρ1333003300-1--7/2-1+-7/2-1--7/2-1+-7/20000-1--7/2-1--7/2-1+-7/2-1+-7/2    complex lifted from C7⋊C3
ρ143-3003300-1--7/2-1+-7/21+-7/21--7/20000-1--7/2-1--7/2-1+-7/2-1+-7/2    complex lifted from C2×C7⋊C3
ρ1533003300-1+-7/2-1--7/2-1+-7/2-1--7/20000-1+-7/2-1+-7/2-1--7/2-1--7/2    complex lifted from C7⋊C3
ρ163-3003300-1+-7/2-1--7/21--7/21+-7/20000-1+-7/2-1+-7/2-1--7/2-1--7/2    complex lifted from C2×C7⋊C3
ρ176000-3+35/2-3-35/200-1--7-1+-7000000ζ54ζ7654ζ7554ζ735ζ765ζ755ζ73ζ53ζ7653ζ7553ζ7352ζ7652ζ7552ζ73ζ53ζ7453ζ7253ζ752ζ7452ζ7252ζ7ζ54ζ7454ζ7254ζ75ζ745ζ725ζ7    complex faithful
ρ186000-3+35/2-3-35/200-1+-7-1--7000000ζ54ζ7454ζ7254ζ75ζ745ζ725ζ7ζ53ζ7453ζ7253ζ752ζ7452ζ7252ζ7ζ53ζ7653ζ7553ζ7352ζ7652ζ7552ζ73ζ54ζ7654ζ7554ζ735ζ765ζ755ζ73    complex faithful
ρ196000-3-35/2-3+35/200-1--7-1+-7000000ζ53ζ7653ζ7553ζ7352ζ7652ζ7552ζ73ζ54ζ7654ζ7554ζ735ζ765ζ755ζ73ζ54ζ7454ζ7254ζ75ζ745ζ725ζ7ζ53ζ7453ζ7253ζ752ζ7452ζ7252ζ7    complex faithful
ρ206000-3-35/2-3+35/200-1+-7-1--7000000ζ53ζ7453ζ7253ζ752ζ7452ζ7252ζ7ζ54ζ7454ζ7254ζ75ζ745ζ725ζ7ζ54ζ7654ζ7554ζ735ζ765ζ755ζ73ζ53ζ7653ζ7553ζ7352ζ7652ζ7552ζ73    complex faithful

Smallest permutation representation of D5×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 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)
(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), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35), (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), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35), (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)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35)], [(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)]])

D5×C7⋊C3 is a maximal subgroup of   C35⋊C12

Matrix representation of D5×C7⋊C3 in GL5(𝔽211)

16000
170177000
00100
00010
00001
,
346000
124177000
00100
00010
00001
,
10000
01000
002101901
0001901
002101911
,
140000
014000
00191121
00100
001120

G:=sub<GL(5,GF(211))| [1,170,0,0,0,6,177,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[34,124,0,0,0,6,177,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,210,0,210,0,0,190,190,191,0,0,1,1,1],[14,0,0,0,0,0,14,0,0,0,0,0,191,1,1,0,0,1,0,1,0,0,21,0,20] >;

D5×C7⋊C3 in GAP, Magma, Sage, TeX

D_5\times C_7\rtimes C_3
% in TeX

G:=Group("D5xC7:C3");
// GroupNames label

G:=SmallGroup(210,2);
// by ID

G=gap.SmallGroup(210,2);
# by ID

G:=PCGroup([4,-2,-3,-5,-7,290,487]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^7=d^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

Export

Subgroup lattice of D5×C7⋊C3 in TeX
Character table of D5×C7⋊C3 in TeX

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