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G = C3×C52⋊C6order 450 = 2·32·52

Direct product of C3 and C52⋊C6

Aliases: C3×C52⋊C6, C52⋊(C3×C6), C5⋊D5⋊C32, (C5×C15)⋊2C6, C52⋊C32C6, (C3×C5⋊D5)⋊C3, (C3×C52⋊C3)⋊4C2, SmallGroup(450,22)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C3×C52⋊C6
 Chief series C1 — C52 — C5×C15 — C3×C52⋊C3 — C3×C52⋊C6
 Lower central C52 — C3×C52⋊C6
 Upper central C1 — C3

Generators and relations for C3×C52⋊C6
G = < a,b,c,d | a3=b5=c5=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b2c3, dcd-1=b-1c-1 >

25C2
25C3
25C3
25C3
3C5
3C5
25C6
25C6
25C6
25C6
25C32
15D5
15D5
3C15
3C15
25C3×C6
15C3×D5
15C3×D5

Character table of C3×C52⋊C6

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 5A 5B 5C 5D 6A 6B 6C 6D 6E 6F 6G 6H 15A 15B 15C 15D 15E 15F 15G 15H size 1 25 1 1 25 25 25 25 25 25 6 6 6 6 25 25 25 25 25 25 25 25 6 6 6 6 6 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 ζ3 ζ32 ζ32 1 ζ3 1 ζ32 ζ3 1 1 1 1 ζ6 ζ65 ζ6 ζ65 -1 ζ65 -1 ζ6 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 linear of order 6 ρ4 1 -1 1 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 1 1 1 1 ζ6 -1 ζ65 ζ65 ζ65 ζ6 ζ6 -1 1 1 1 1 1 1 1 1 linear of order 6 ρ5 1 1 ζ3 ζ32 ζ32 1 ζ3 1 ζ32 ζ3 1 1 1 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 linear of order 3 ρ6 1 -1 ζ32 ζ3 1 ζ32 1 ζ3 ζ32 ζ3 1 1 1 1 ζ6 ζ6 -1 ζ65 ζ6 -1 ζ65 ζ65 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 linear of order 6 ρ7 1 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 1 1 1 1 ζ32 1 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 1 1 1 1 1 1 1 linear of order 3 ρ8 1 -1 ζ32 ζ3 ζ3 1 ζ32 1 ζ3 ζ32 1 1 1 1 ζ65 ζ6 ζ65 ζ6 -1 ζ6 -1 ζ65 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 linear of order 6 ρ9 1 1 ζ32 ζ3 ζ3 1 ζ32 1 ζ3 ζ32 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 linear of order 3 ρ10 1 -1 ζ3 ζ32 1 ζ3 1 ζ32 ζ3 ζ32 1 1 1 1 ζ65 ζ65 -1 ζ6 ζ65 -1 ζ6 ζ6 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 linear of order 6 ρ11 1 -1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 -1 ζ6 ζ6 -1 ζ65 ζ65 ζ6 ζ65 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 linear of order 6 ρ12 1 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 1 1 1 1 ζ3 1 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 1 1 1 1 1 1 1 linear of order 3 ρ13 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 1 ζ32 ζ32 1 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 linear of order 3 ρ14 1 -1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 -1 ζ65 ζ65 -1 ζ6 ζ6 ζ65 ζ6 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 linear of order 6 ρ15 1 -1 1 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 1 1 1 1 ζ65 -1 ζ6 ζ6 ζ6 ζ65 ζ65 -1 1 1 1 1 1 1 1 1 linear of order 6 ρ16 1 1 ζ32 ζ3 1 ζ32 1 ζ3 ζ32 ζ3 1 1 1 1 ζ32 ζ32 1 ζ3 ζ32 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 linear of order 3 ρ17 1 1 ζ3 ζ32 1 ζ3 1 ζ32 ζ3 ζ32 1 1 1 1 ζ3 ζ3 1 ζ32 ζ3 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 linear of order 3 ρ18 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 1 ζ3 ζ3 1 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 linear of order 3 ρ19 6 0 6 6 0 0 0 0 0 0 -3-√5/2 1+√5 1-√5 -3+√5/2 0 0 0 0 0 0 0 0 1-√5 -3+√5/2 -3-√5/2 1+√5 -3+√5/2 -3-√5/2 1+√5 1-√5 orthogonal lifted from C52⋊C6 ρ20 6 0 6 6 0 0 0 0 0 0 -3+√5/2 1-√5 1+√5 -3-√5/2 0 0 0 0 0 0 0 0 1+√5 -3-√5/2 -3+√5/2 1-√5 -3-√5/2 -3+√5/2 1-√5 1+√5 orthogonal lifted from C52⋊C6 ρ21 6 0 6 6 0 0 0 0 0 0 1+√5 -3+√5/2 -3-√5/2 1-√5 0 0 0 0 0 0 0 0 -3-√5/2 1-√5 1+√5 -3+√5/2 1-√5 1+√5 -3+√5/2 -3-√5/2 orthogonal lifted from C52⋊C6 ρ22 6 0 6 6 0 0 0 0 0 0 1-√5 -3-√5/2 -3+√5/2 1+√5 0 0 0 0 0 0 0 0 -3+√5/2 1+√5 1-√5 -3-√5/2 1+√5 1-√5 -3-√5/2 -3+√5/2 orthogonal lifted from C52⋊C6 ρ23 6 0 -3+3√-3 -3-3√-3 0 0 0 0 0 0 1+√5 -3+√5/2 -3-√5/2 1-√5 0 0 0 0 0 0 0 0 ζ3ζ53+ζ3ζ52-ζ3 -2ζ3ζ54-2ζ3ζ5 -2ζ3ζ53-2ζ3ζ52 ζ3ζ54+ζ3ζ5-ζ3 -2ζ32ζ54-2ζ32ζ5 -2ζ32ζ53-2ζ32ζ52 ζ32ζ54+ζ32ζ5-ζ32 ζ32ζ53+ζ32ζ52-ζ32 complex faithful ρ24 6 0 -3+3√-3 -3-3√-3 0 0 0 0 0 0 1-√5 -3-√5/2 -3+√5/2 1+√5 0 0 0 0 0 0 0 0 ζ3ζ54+ζ3ζ5-ζ3 -2ζ3ζ53-2ζ3ζ52 -2ζ3ζ54-2ζ3ζ5 ζ3ζ53+ζ3ζ52-ζ3 -2ζ32ζ53-2ζ32ζ52 -2ζ32ζ54-2ζ32ζ5 ζ32ζ53+ζ32ζ52-ζ32 ζ32ζ54+ζ32ζ5-ζ32 complex faithful ρ25 6 0 -3+3√-3 -3-3√-3 0 0 0 0 0 0 -3+√5/2 1-√5 1+√5 -3-√5/2 0 0 0 0 0 0 0 0 -2ζ3ζ53-2ζ3ζ52 ζ3ζ53+ζ3ζ52-ζ3 ζ3ζ54+ζ3ζ5-ζ3 -2ζ3ζ54-2ζ3ζ5 ζ32ζ53+ζ32ζ52-ζ32 ζ32ζ54+ζ32ζ5-ζ32 -2ζ32ζ54-2ζ32ζ5 -2ζ32ζ53-2ζ32ζ52 complex faithful ρ26 6 0 -3-3√-3 -3+3√-3 0 0 0 0 0 0 -3+√5/2 1-√5 1+√5 -3-√5/2 0 0 0 0 0 0 0 0 -2ζ32ζ53-2ζ32ζ52 ζ32ζ53+ζ32ζ52-ζ32 ζ32ζ54+ζ32ζ5-ζ32 -2ζ32ζ54-2ζ32ζ5 ζ3ζ53+ζ3ζ52-ζ3 ζ3ζ54+ζ3ζ5-ζ3 -2ζ3ζ54-2ζ3ζ5 -2ζ3ζ53-2ζ3ζ52 complex faithful ρ27 6 0 -3-3√-3 -3+3√-3 0 0 0 0 0 0 1-√5 -3-√5/2 -3+√5/2 1+√5 0 0 0 0 0 0 0 0 ζ32ζ54+ζ32ζ5-ζ32 -2ζ32ζ53-2ζ32ζ52 -2ζ32ζ54-2ζ32ζ5 ζ32ζ53+ζ32ζ52-ζ32 -2ζ3ζ53-2ζ3ζ52 -2ζ3ζ54-2ζ3ζ5 ζ3ζ53+ζ3ζ52-ζ3 ζ3ζ54+ζ3ζ5-ζ3 complex faithful ρ28 6 0 -3+3√-3 -3-3√-3 0 0 0 0 0 0 -3-√5/2 1+√5 1-√5 -3+√5/2 0 0 0 0 0 0 0 0 -2ζ3ζ54-2ζ3ζ5 ζ3ζ54+ζ3ζ5-ζ3 ζ3ζ53+ζ3ζ52-ζ3 -2ζ3ζ53-2ζ3ζ52 ζ32ζ54+ζ32ζ5-ζ32 ζ32ζ53+ζ32ζ52-ζ32 -2ζ32ζ53-2ζ32ζ52 -2ζ32ζ54-2ζ32ζ5 complex faithful ρ29 6 0 -3-3√-3 -3+3√-3 0 0 0 0 0 0 -3-√5/2 1+√5 1-√5 -3+√5/2 0 0 0 0 0 0 0 0 -2ζ32ζ54-2ζ32ζ5 ζ32ζ54+ζ32ζ5-ζ32 ζ32ζ53+ζ32ζ52-ζ32 -2ζ32ζ53-2ζ32ζ52 ζ3ζ54+ζ3ζ5-ζ3 ζ3ζ53+ζ3ζ52-ζ3 -2ζ3ζ53-2ζ3ζ52 -2ζ3ζ54-2ζ3ζ5 complex faithful ρ30 6 0 -3-3√-3 -3+3√-3 0 0 0 0 0 0 1+√5 -3+√5/2 -3-√5/2 1-√5 0 0 0 0 0 0 0 0 ζ32ζ53+ζ32ζ52-ζ32 -2ζ32ζ54-2ζ32ζ5 -2ζ32ζ53-2ζ32ζ52 ζ32ζ54+ζ32ζ5-ζ32 -2ζ3ζ54-2ζ3ζ5 -2ζ3ζ53-2ζ3ζ52 ζ3ζ54+ζ3ζ5-ζ3 ζ3ζ53+ζ3ζ52-ζ3 complex faithful

Smallest permutation representation of C3×C52⋊C6
On 45 points
Generators in S45
(1 6 7)(2 4 8)(3 5 9)(10 24 38)(11 25 39)(12 26 34)(13 27 35)(14 22 36)(15 23 37)(16 32 40)(17 33 41)(18 28 42)(19 29 43)(20 30 44)(21 31 45)
(1 18 25 22 21)(2 26 16 19 23)(3 20 27 24 17)(4 34 32 29 37)(5 30 35 38 33)(6 28 39 36 31)(7 42 11 14 45)(8 12 40 43 15)(9 44 13 10 41)
(2 16 23 26 19)(3 17 24 27 20)(4 32 37 34 29)(5 33 38 35 30)(8 40 15 12 43)(9 41 10 13 44)
(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 36 37 38 39)(40 41 42 43 44 45)

G:=sub<Sym(45)| (1,6,7)(2,4,8)(3,5,9)(10,24,38)(11,25,39)(12,26,34)(13,27,35)(14,22,36)(15,23,37)(16,32,40)(17,33,41)(18,28,42)(19,29,43)(20,30,44)(21,31,45), (1,18,25,22,21)(2,26,16,19,23)(3,20,27,24,17)(4,34,32,29,37)(5,30,35,38,33)(6,28,39,36,31)(7,42,11,14,45)(8,12,40,43,15)(9,44,13,10,41), (2,16,23,26,19)(3,17,24,27,20)(4,32,37,34,29)(5,33,38,35,30)(8,40,15,12,43)(9,41,10,13,44), (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)(34,35,36,37,38,39)(40,41,42,43,44,45)>;

G:=Group( (1,6,7)(2,4,8)(3,5,9)(10,24,38)(11,25,39)(12,26,34)(13,27,35)(14,22,36)(15,23,37)(16,32,40)(17,33,41)(18,28,42)(19,29,43)(20,30,44)(21,31,45), (1,18,25,22,21)(2,26,16,19,23)(3,20,27,24,17)(4,34,32,29,37)(5,30,35,38,33)(6,28,39,36,31)(7,42,11,14,45)(8,12,40,43,15)(9,44,13,10,41), (2,16,23,26,19)(3,17,24,27,20)(4,32,37,34,29)(5,33,38,35,30)(8,40,15,12,43)(9,41,10,13,44), (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)(34,35,36,37,38,39)(40,41,42,43,44,45) );

G=PermutationGroup([[(1,6,7),(2,4,8),(3,5,9),(10,24,38),(11,25,39),(12,26,34),(13,27,35),(14,22,36),(15,23,37),(16,32,40),(17,33,41),(18,28,42),(19,29,43),(20,30,44),(21,31,45)], [(1,18,25,22,21),(2,26,16,19,23),(3,20,27,24,17),(4,34,32,29,37),(5,30,35,38,33),(6,28,39,36,31),(7,42,11,14,45),(8,12,40,43,15),(9,44,13,10,41)], [(2,16,23,26,19),(3,17,24,27,20),(4,32,37,34,29),(5,33,38,35,30),(8,40,15,12,43),(9,41,10,13,44)], [(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),(34,35,36,37,38,39),(40,41,42,43,44,45)]])

Matrix representation of C3×C52⋊C6 in GL6(𝔽31)

 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5
,
 13 30 0 0 0 0 14 30 0 0 0 0 25 0 30 12 0 0 21 21 19 19 0 0 5 0 0 0 12 30 3 26 0 0 1 0
,
 30 1 0 0 0 0 17 13 0 0 0 0 16 25 12 30 0 0 6 0 1 0 0 0 26 0 0 0 1 0 26 0 0 0 0 1
,
 19 1 24 26 0 0 30 13 28 2 0 0 0 0 30 0 5 0 10 25 30 0 29 26 0 0 6 0 0 0 28 5 6 0 0 0

G:=sub<GL(6,GF(31))| [5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[13,14,25,21,5,3,30,30,0,21,0,26,0,0,30,19,0,0,0,0,12,19,0,0,0,0,0,0,12,1,0,0,0,0,30,0],[30,17,16,6,26,26,1,13,25,0,0,0,0,0,12,1,0,0,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[19,30,0,10,0,28,1,13,0,25,0,5,24,28,30,30,6,6,26,2,0,0,0,0,0,0,5,29,0,0,0,0,0,26,0,0] >;

C3×C52⋊C6 in GAP, Magma, Sage, TeX

C_3\times C_5^2\rtimes C_6
% in TeX

G:=Group("C3xC5^2:C6");
// GroupNames label

G:=SmallGroup(450,22);
// by ID

G=gap.SmallGroup(450,22);
# by ID

G:=PCGroup([5,-2,-3,-3,-5,5,1443,2348,9004,1359]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^5=c^5=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^2*c^3,d*c*d^-1=b^-1*c^-1>;
// generators/relations

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