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G = S3×C11⋊C5order 330 = 2·3·5·11

Direct product of S3 and C11⋊C5

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

Aliases: S3×C11⋊C5, C333C10, (S3×C11)⋊C5, C112(C5×S3), C3⋊(C2×C11⋊C5), (C3×C11⋊C5)⋊3C2, SmallGroup(330,2)

Series: Derived Chief Lower central Upper central

C1C33 — S3×C11⋊C5
C1C11C33C3×C11⋊C5 — S3×C11⋊C5
C33 — S3×C11⋊C5
C1

Generators and relations for S3×C11⋊C5
 G = < a,b,c,d | a3=b2=c11=d5=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

3C2
11C5
33C10
11C15
3C22
11C5×S3
3C2×C11⋊C5

Character table of S3×C11⋊C5

 class 1235A5B5C5D10A10B10C10D11A11B15A15B15C15D22A22B33A33B
 size 1321111111133333333552222222215151010
ρ1111111111111111111111    trivial
ρ21-111111-1-1-1-1111111-1-111    linear of order 2
ρ31-11ζ5ζ52ζ53ζ54535254511ζ53ζ52ζ54ζ5-1-111    linear of order 10
ρ4111ζ53ζ5ζ54ζ52ζ54ζ5ζ52ζ5311ζ54ζ5ζ52ζ531111    linear of order 5
ρ5111ζ5ζ52ζ53ζ54ζ53ζ52ζ54ζ511ζ53ζ52ζ54ζ51111    linear of order 5
ρ61-11ζ53ζ5ζ54ζ52545525311ζ54ζ5ζ52ζ53-1-111    linear of order 10
ρ7111ζ54ζ53ζ52ζ5ζ52ζ53ζ5ζ5411ζ52ζ53ζ5ζ541111    linear of order 5
ρ8111ζ52ζ54ζ5ζ53ζ5ζ54ζ53ζ5211ζ5ζ54ζ53ζ521111    linear of order 5
ρ91-11ζ52ζ54ζ5ζ53554535211ζ5ζ54ζ53ζ52-1-111    linear of order 10
ρ101-11ζ54ζ53ζ52ζ5525355411ζ52ζ53ζ5ζ54-1-111    linear of order 10
ρ1120-12222000022-1-1-1-100-1-1    orthogonal lifted from S3
ρ1220-15453525000022525355400-1-1    complex lifted from C5×S3
ρ1320-15355452000022545525300-1-1    complex lifted from C5×S3
ρ1420-15525354000022535254500-1-1    complex lifted from C5×S3
ρ1520-15254553000022554535200-1-1    complex lifted from C5×S3
ρ165-5500000000-1--11/2-1+-11/200001--11/21+-11/2-1+-11/2-1--11/2    complex lifted from C2×C11⋊C5
ρ175-5500000000-1+-11/2-1--11/200001+-11/21--11/2-1--11/2-1+-11/2    complex lifted from C2×C11⋊C5
ρ1855500000000-1+-11/2-1--11/20000-1--11/2-1+-11/2-1--11/2-1+-11/2    complex lifted from C11⋊C5
ρ1955500000000-1--11/2-1+-11/20000-1+-11/2-1--11/2-1+-11/2-1--11/2    complex lifted from C11⋊C5
ρ20100-500000000-1+-11-1--110000001+-11/21--11/2    complex faithful
ρ21100-500000000-1--11-1+-110000001--11/21+-11/2    complex faithful

Smallest permutation representation of S3×C11⋊C5
On 33 points
Generators in S33
(1 12 23)(2 13 24)(3 14 25)(4 15 26)(5 16 27)(6 17 28)(7 18 29)(8 19 30)(9 20 31)(10 21 32)(11 22 33)
(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)
(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)
(2 5 6 10 4)(3 9 11 8 7)(13 16 17 21 15)(14 20 22 19 18)(24 27 28 32 26)(25 31 33 30 29)

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

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

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

Matrix representation of S3×C11⋊C5 in GL7(𝔽331)

033000000
133000000
0010000
0001000
0000100
0000010
0000001
,
0100000
1000000
0010000
0001000
0000100
0000010
0000001
,
1000000
0100000
00330330330330104
001000228
000100329
000010106
000001227
,
150000000
015000000
00033010106
00010500227
00022700103
001330002
000101226

G:=sub<GL(7,GF(331))| [0,1,0,0,0,0,0,330,330,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,330,1,0,0,0,0,0,330,0,1,0,0,0,0,330,0,0,1,0,0,0,330,0,0,0,1,0,0,104,228,329,106,227],[150,0,0,0,0,0,0,0,150,0,0,0,0,0,0,0,0,0,0,1,0,0,0,330,105,227,330,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,106,227,103,2,226] >;

S3×C11⋊C5 in GAP, Magma, Sage, TeX

S_3\times C_{11}\rtimes C_5
% in TeX

G:=Group("S3xC11:C5");
// GroupNames label

G:=SmallGroup(330,2);
// by ID

G=gap.SmallGroup(330,2);
# by ID

G:=PCGroup([4,-2,-5,-3,-11,242,967]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^11=d^5=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^3>;
// generators/relations

Export

Subgroup lattice of S3×C11⋊C5 in TeX
Character table of S3×C11⋊C5 in TeX

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