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## G = C5×C3≀C3order 405 = 34·5

### Direct product of C5 and C3≀C3

direct product, metabelian, nilpotent (class 3), monomial, 3-elementary

Aliases: C5×C3≀C3, He31C15, C331C15, C15.2He3, 3- 1+21C15, (C5×He3)⋊1C3, C3.2(C5×He3), (C32×C15)⋊1C3, C32.1(C3×C15), (C3×C15).1C32, (C5×3- 1+2)⋊1C3, SmallGroup(405,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C5×C3≀C3
 Chief series C1 — C3 — C32 — C3×C15 — C5×He3 — C5×C3≀C3
 Lower central C1 — C3 — C32 — C5×C3≀C3
 Upper central C1 — C15 — C3×C15 — C5×C3≀C3

Generators and relations for C5×C3≀C3
G = < a,b,c,d,e | a5=b3=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, be=eb, cd=dc, ce=ec, ede-1=bc-1d >

Smallest permutation representation of C5×C3≀C3
On 45 points
Generators in S45
(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)
(6 44 15)(7 45 11)(8 41 12)(9 42 13)(10 43 14)(26 31 38)(27 32 39)(28 33 40)(29 34 36)(30 35 37)
(1 23 17)(2 24 18)(3 25 19)(4 21 20)(5 22 16)(6 15 44)(7 11 45)(8 12 41)(9 13 42)(10 14 43)(26 31 38)(27 32 39)(28 33 40)(29 34 36)(30 35 37)
(1 8 31)(2 9 32)(3 10 33)(4 6 34)(5 7 35)(11 37 22)(12 38 23)(13 39 24)(14 40 25)(15 36 21)(16 45 30)(17 41 26)(18 42 27)(19 43 28)(20 44 29)
(1 17 23)(2 18 24)(3 19 25)(4 20 21)(5 16 22)(26 38 31)(27 39 32)(28 40 33)(29 36 34)(30 37 35)

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

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

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

85 conjugacy classes

 class 1 3A 3B 3C ··· 3J 3K 3L 5A 5B 5C 5D 9A 9B 9C 9D 15A ··· 15H 15I ··· 15AN 15AO ··· 15AV 45A ··· 45P order 1 3 3 3 ··· 3 3 3 5 5 5 5 9 9 9 9 15 ··· 15 15 ··· 15 15 ··· 15 45 ··· 45 size 1 1 1 3 ··· 3 9 9 1 1 1 1 9 9 9 9 1 ··· 1 3 ··· 3 9 ··· 9 9 ··· 9

85 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + image C1 C3 C3 C3 C5 C15 C15 C15 He3 C3≀C3 C5×He3 C5×C3≀C3 kernel C5×C3≀C3 C5×He3 C5×3- 1+2 C32×C15 C3≀C3 He3 3- 1+2 C33 C15 C5 C3 C1 # reps 1 2 4 2 4 8 16 8 2 6 8 24

Matrix representation of C5×C3≀C3 in GL3(𝔽181) generated by

 125 0 0 0 125 0 0 0 125
,
 1 0 0 109 132 0 31 0 48
,
 132 0 0 0 132 0 0 0 132
,
 109 131 0 19 72 1 162 110 0
,
 48 0 0 0 48 0 150 0 1
G:=sub<GL(3,GF(181))| [125,0,0,0,125,0,0,0,125],[1,109,31,0,132,0,0,0,48],[132,0,0,0,132,0,0,0,132],[109,19,162,131,72,110,0,1,0],[48,0,150,0,48,0,0,0,1] >;

C5×C3≀C3 in GAP, Magma, Sage, TeX

C_5\times C_3\wr C_3
% in TeX

G:=Group("C5xC3wrC3");
// GroupNames label

G:=SmallGroup(405,7);
// by ID

G=gap.SmallGroup(405,7);
# by ID

G:=PCGroup([5,-3,-3,-5,-3,-3,481,3603]);
// Polycyclic

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

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