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G = C3×D15order 90 = 2·32·5

Direct product of C3 and D15

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

Aliases: C3×D15, C151C6, C152S3, C321D5, C5⋊(C3×S3), C3⋊(C3×D5), (C3×C15)⋊2C2, SmallGroup(90,7)

Series: Derived Chief Lower central Upper central

C1C15 — C3×D15
C1C5C15C3×C15 — C3×D15
C15 — C3×D15
C1C3

Generators and relations for C3×D15
 G = < a,b,c | a3=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >

15C2
2C3
5S3
15C6
3D5
2C15
5C3×S3
3C3×D5

Character table of C3×D15

 class 123A3B3C3D3E5A5B6A6B15A15B15C15D15E15F15G15H15I15J15K15L15M15N15O15P
 size 115112222215152222222222222222
ρ1111111111111111111111111111    trivial
ρ21-11111111-1-11111111111111111    linear of order 2
ρ311ζ32ζ3ζ3ζ32111ζ3ζ321ζ32ζ32ζ32ζ321ζ3ζ31ζ3ζ3ζ3ζ31ζ32ζ32    linear of order 3
ρ41-1ζ32ζ3ζ3ζ32111ζ65ζ61ζ32ζ32ζ32ζ321ζ3ζ31ζ3ζ3ζ3ζ31ζ32ζ32    linear of order 6
ρ511ζ3ζ32ζ32ζ3111ζ32ζ31ζ3ζ3ζ3ζ31ζ32ζ321ζ32ζ32ζ32ζ321ζ3ζ3    linear of order 3
ρ61-1ζ3ζ32ζ32ζ3111ζ6ζ651ζ3ζ3ζ3ζ31ζ32ζ321ζ32ζ32ζ32ζ321ζ3ζ3    linear of order 6
ρ72022-1-1-12200-1-1-1-1-1-122-1-1-1-1-1-122    orthogonal lifted from S3
ρ82022-1-1-1-1+5/2-1-5/20032ζ5432ζ5543ζ533ζ525332ζ5432ζ5543ζ543ζ554ζ3ζ533ζ52523ζ543ζ554-1-5/2-1+5/2ζ3ζ533ζ52523ζ533ζ525332ζ5432ζ5543ζ543ζ554ζ3ζ533ζ52523ζ533ζ5253-1-5/2-1+5/2    orthogonal lifted from D15
ρ92022222-1+5/2-1-5/200-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1-5/2-1+5/2    orthogonal lifted from D5
ρ102022222-1-5/2-1+5/200-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+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
ρ112022-1-1-1-1-5/2-1+5/2003ζ533ζ52533ζ543ζ5543ζ533ζ5253ζ3ζ533ζ525232ζ5432ζ554ζ3ζ533ζ5252-1+5/2-1-5/232ζ5432ζ5543ζ543ζ5543ζ533ζ5253ζ3ζ533ζ525232ζ5432ζ5543ζ543ζ554-1+5/2-1-5/2    orthogonal lifted from D15
ρ122022-1-1-1-1-5/2-1+5/200ζ3ζ533ζ525232ζ5432ζ554ζ3ζ533ζ52523ζ533ζ52533ζ543ζ5543ζ533ζ5253-1+5/2-1-5/23ζ543ζ55432ζ5432ζ554ζ3ζ533ζ52523ζ533ζ52533ζ543ζ55432ζ5432ζ554-1+5/2-1-5/2    orthogonal lifted from D15
ρ132022-1-1-1-1+5/2-1-5/2003ζ543ζ554ζ3ζ533ζ52523ζ543ζ55432ζ5432ζ5543ζ533ζ525332ζ5432ζ554-1-5/2-1+5/23ζ533ζ5253ζ3ζ533ζ52523ζ543ζ55432ζ5432ζ5543ζ533ζ5253ζ3ζ533ζ5252-1-5/2-1+5/2    orthogonal lifted from D15
ρ1420-1+-3-1--3ζ6ζ65-12200-1ζ65ζ65ζ65ζ65-1-1--3-1--3-1ζ6ζ6ζ6ζ6-1-1+-3-1+-3    complex lifted from C3×S3
ρ1520-1--3-1+-3ζ65ζ6-12200-1ζ6ζ6ζ6ζ6-1-1+-3-1+-3-1ζ65ζ65ζ65ζ65-1-1--3-1--3    complex lifted from C3×S3
ρ1620-1+-3-1--3ζ6ζ65-1-1+5/2-1-5/20032ζ5432ζ554ζ32ζ5253ζ32ζ545ζ32ζ554ζ32ζ53523ζ543ζ554ζ32ζ5332ζ52ζ32ζ5432ζ5ζ3ζ533ζ5252ζ3ζ5352ζ3ζ554ζ3ζ545ζ3ζ52533ζ533ζ5253ζ3ζ533ζ52ζ3ζ543ζ5    complex faithful
ρ1720-1+-3-1--3-1--3-1+-32-1-5/2-1+5/200-1-5/2ζ3ζ543ζ5ζ3ζ533ζ52ζ3ζ533ζ52ζ3ζ543ζ5-1-5/2ζ32ζ5432ζ5ζ32ζ5332ζ52-1+5/2ζ32ζ5432ζ5ζ32ζ5332ζ52ζ32ζ5332ζ52ζ32ζ5432ζ5-1+5/2ζ3ζ543ζ5ζ3ζ533ζ52    complex lifted from C3×D5
ρ1820-1--3-1+-3-1+-3-1--32-1+5/2-1-5/200-1+5/2ζ32ζ5332ζ52ζ32ζ5432ζ5ζ32ζ5432ζ5ζ32ζ5332ζ52-1+5/2ζ3ζ533ζ52ζ3ζ543ζ5-1-5/2ζ3ζ533ζ52ζ3ζ543ζ5ζ3ζ543ζ5ζ3ζ533ζ52-1-5/2ζ32ζ5332ζ52ζ32ζ5432ζ5    complex lifted from C3×D5
ρ1920-1--3-1+-3ζ65ζ6-1-1+5/2-1-5/20032ζ5432ζ554ζ3ζ5352ζ3ζ554ζ3ζ545ζ3ζ52533ζ543ζ554ζ3ζ533ζ52ζ3ζ543ζ5ζ3ζ533ζ5252ζ32ζ5253ζ32ζ545ζ32ζ554ζ32ζ53523ζ533ζ5253ζ32ζ5332ζ52ζ32ζ5432ζ5    complex faithful
ρ2020-1--3-1+-3ζ65ζ6-1-1-5/2-1+5/2003ζ533ζ5253ζ3ζ545ζ3ζ5352ζ3ζ5253ζ3ζ554ζ3ζ533ζ5252ζ3ζ543ζ5ζ3ζ533ζ5232ζ5432ζ554ζ32ζ554ζ32ζ5253ζ32ζ5352ζ32ζ5453ζ543ζ554ζ32ζ5432ζ5ζ32ζ5332ζ52    complex faithful
ρ2120-1--3-1+-3ζ65ζ6-1-1-5/2-1+5/200ζ3ζ533ζ5252ζ3ζ554ζ3ζ5253ζ3ζ5352ζ3ζ5453ζ533ζ5253ζ3ζ543ζ5ζ3ζ533ζ523ζ543ζ554ζ32ζ545ζ32ζ5352ζ32ζ5253ζ32ζ55432ζ5432ζ554ζ32ζ5432ζ5ζ32ζ5332ζ52    complex faithful
ρ2220-1+-3-1--3ζ6ζ65-1-1-5/2-1+5/2003ζ533ζ5253ζ32ζ554ζ32ζ5253ζ32ζ5352ζ32ζ545ζ3ζ533ζ5252ζ32ζ5432ζ5ζ32ζ5332ζ5232ζ5432ζ554ζ3ζ545ζ3ζ5352ζ3ζ5253ζ3ζ5543ζ543ζ554ζ3ζ543ζ5ζ3ζ533ζ52    complex faithful
ρ2320-1+-3-1--3ζ6ζ65-1-1-5/2-1+5/200ζ3ζ533ζ5252ζ32ζ545ζ32ζ5352ζ32ζ5253ζ32ζ5543ζ533ζ5253ζ32ζ5432ζ5ζ32ζ5332ζ523ζ543ζ554ζ3ζ554ζ3ζ5253ζ3ζ5352ζ3ζ54532ζ5432ζ554ζ3ζ543ζ5ζ3ζ533ζ52    complex faithful
ρ2420-1--3-1+-3ζ65ζ6-1-1+5/2-1-5/2003ζ543ζ554ζ3ζ5253ζ3ζ545ζ3ζ554ζ3ζ535232ζ5432ζ554ζ3ζ533ζ52ζ3ζ543ζ53ζ533ζ5253ζ32ζ5352ζ32ζ554ζ32ζ545ζ32ζ5253ζ3ζ533ζ5252ζ32ζ5332ζ52ζ32ζ5432ζ5    complex faithful
ρ2520-1+-3-1--3ζ6ζ65-1-1+5/2-1-5/2003ζ543ζ554ζ32ζ5352ζ32ζ554ζ32ζ545ζ32ζ525332ζ5432ζ554ζ32ζ5332ζ52ζ32ζ5432ζ53ζ533ζ5253ζ3ζ5253ζ3ζ545ζ3ζ554ζ3ζ5352ζ3ζ533ζ5252ζ3ζ533ζ52ζ3ζ543ζ5    complex faithful
ρ2620-1--3-1+-3-1+-3-1--32-1-5/2-1+5/200-1-5/2ζ32ζ5432ζ5ζ32ζ5332ζ52ζ32ζ5332ζ52ζ32ζ5432ζ5-1-5/2ζ3ζ543ζ5ζ3ζ533ζ52-1+5/2ζ3ζ543ζ5ζ3ζ533ζ52ζ3ζ533ζ52ζ3ζ543ζ5-1+5/2ζ32ζ5432ζ5ζ32ζ5332ζ52    complex lifted from C3×D5
ρ2720-1+-3-1--3-1--3-1+-32-1+5/2-1-5/200-1+5/2ζ3ζ533ζ52ζ3ζ543ζ5ζ3ζ543ζ5ζ3ζ533ζ52-1+5/2ζ32ζ5332ζ52ζ32ζ5432ζ5-1-5/2ζ32ζ5332ζ52ζ32ζ5432ζ5ζ32ζ5432ζ5ζ32ζ5332ζ52-1-5/2ζ3ζ533ζ52ζ3ζ543ζ5    complex lifted from C3×D5

Permutation representations of C3×D15
On 30 points - transitive group 30T16
Generators in S30
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)
(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)
(1 17)(2 16)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 24)(10 23)(11 22)(12 21)(13 20)(14 19)(15 18)

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

G:=Group( (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (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), (1,17)(2,16)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18) );

G=PermutationGroup([(1,6,11),(2,7,12),(3,8,13),(4,9,14),(5,10,15),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25)], [(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)], [(1,17),(2,16),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,24),(10,23),(11,22),(12,21),(13,20),(14,19),(15,18)])

G:=TransitiveGroup(30,16);

C3×D15 is a maximal subgroup of   C3×S3×D5  D15⋊S3  He3⋊D5  D45⋊C3  C32⋊D15
C3×D15 is a maximal quotient of   He3⋊D5  D45⋊C3

Matrix representation of C3×D15 in GL2(𝔽31) generated by

50
05
,
190
918
,
132
918
G:=sub<GL(2,GF(31))| [5,0,0,5],[19,9,0,18],[13,9,2,18] >;

C3×D15 in GAP, Magma, Sage, TeX

C_3\times D_{15}
% in TeX

G:=Group("C3xD15");
// GroupNames label

G:=SmallGroup(90,7);
// by ID

G=gap.SmallGroup(90,7);
# by ID

G:=PCGroup([4,-2,-3,-3,-5,146,1155]);
// Polycyclic

G:=Group<a,b,c|a^3=b^15=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D15 in TeX
Character table of C3×D15 in TeX

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