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## G = C2×C32.C33order 486 = 2·35

### Direct product of C2 and C32.C33

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C32.C33
 Chief series C1 — C3 — C32 — C33 — C3×3- 1+2 — C32.C33 — C2×C32.C33
 Lower central C1 — C3 — C32 — C2×C32.C33
 Upper central C1 — C6 — C32×C6 — C2×C32.C33

Generators and relations for C2×C32.C33
G = < a,b,c,d,e,f | a2=b3=c3=f3=1, d3=c, e3=c-1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=bc-1, be=eb, bf=fb, cd=dc, fef-1=ce=ec, cf=fc, ede-1=bc-1d, df=fd >

Subgroups: 252 in 124 conjugacy classes, 66 normal (12 characteristic)
C1, C2, C3, C3, C6, C6, C9, C32, C32, C32, C18, C3×C6, C3×C6, C3×C6, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, C3×C18, C3×C18, C2×3- 1+2, C2×3- 1+2, C32×C6, C3.He3, C3×3- 1+2, C3×3- 1+2, C2×C3.He3, C6×3- 1+2, C6×3- 1+2, C32.C33, C2×C32.C33
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, C33, C2×He3, C32×C6, C3×He3, C6×He3, C32.C33, C2×C32.C33

Smallest permutation representation of C2×C32.C33
On 54 points
Generators in S54
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)(19 43)(20 44)(21 45)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 47)(29 48)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 46)
(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 22 25)(21 27 24)(28 34 31)(29 32 35)(37 40 43)(39 45 42)(47 53 50)(48 51 54)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)
(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)(46 47 48 49 50 51 52 53 54)
(1 41 36 7 38 33 4 44 30)(2 45 34 8 42 31 5 39 28)(3 43 29 9 40 35 6 37 32)(10 22 51 16 19 48 13 25 54)(11 23 52 17 20 49 14 26 46)(12 27 50 18 24 47 15 21 53)
(19 25 22)(20 26 23)(21 27 24)(28 31 34)(29 32 35)(30 33 36)(37 43 40)(38 44 41)(39 45 42)(46 49 52)(47 50 53)(48 51 54)

G:=sub<Sym(54)| (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13)(19,43)(20,44)(21,45)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,47)(29,48)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,46), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,22,25)(21,27,24)(28,34,31)(29,32,35)(37,40,43)(39,45,42)(47,53,50)(48,51,54), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (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)(46,47,48,49,50,51,52,53,54), (1,41,36,7,38,33,4,44,30)(2,45,34,8,42,31,5,39,28)(3,43,29,9,40,35,6,37,32)(10,22,51,16,19,48,13,25,54)(11,23,52,17,20,49,14,26,46)(12,27,50,18,24,47,15,21,53), (19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,43,40)(38,44,41)(39,45,42)(46,49,52)(47,50,53)(48,51,54)>;

G:=Group( (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13)(19,43)(20,44)(21,45)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,47)(29,48)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,46), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,22,25)(21,27,24)(28,34,31)(29,32,35)(37,40,43)(39,45,42)(47,53,50)(48,51,54), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (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)(46,47,48,49,50,51,52,53,54), (1,41,36,7,38,33,4,44,30)(2,45,34,8,42,31,5,39,28)(3,43,29,9,40,35,6,37,32)(10,22,51,16,19,48,13,25,54)(11,23,52,17,20,49,14,26,46)(12,27,50,18,24,47,15,21,53), (19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,43,40)(38,44,41)(39,45,42)(46,49,52)(47,50,53)(48,51,54) );

G=PermutationGroup([[(1,14),(2,15),(3,16),(4,17),(5,18),(6,10),(7,11),(8,12),(9,13),(19,43),(20,44),(21,45),(22,37),(23,38),(24,39),(25,40),(26,41),(27,42),(28,47),(29,48),(30,49),(31,50),(32,51),(33,52),(34,53),(35,54),(36,46)], [(2,8,5),(3,6,9),(10,13,16),(12,18,15),(19,22,25),(21,27,24),(28,34,31),(29,32,35),(37,40,43),(39,45,42),(47,53,50),(48,51,54)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54)], [(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),(46,47,48,49,50,51,52,53,54)], [(1,41,36,7,38,33,4,44,30),(2,45,34,8,42,31,5,39,28),(3,43,29,9,40,35,6,37,32),(10,22,51,16,19,48,13,25,54),(11,23,52,17,20,49,14,26,46),(12,27,50,18,24,47,15,21,53)], [(19,25,22),(20,26,23),(21,27,24),(28,31,34),(29,32,35),(30,33,36),(37,43,40),(38,44,41),(39,45,42),(46,49,52),(47,50,53),(48,51,54)]])

70 conjugacy classes

 class 1 2 3A 3B 3C ··· 3J 6A 6B 6C ··· 6J 9A ··· 9X 18A ··· 18X order 1 2 3 3 3 ··· 3 6 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 1 3 ··· 3 1 1 3 ··· 3 9 ··· 9 9 ··· 9

70 irreducible representations

 dim 1 1 1 1 1 1 3 3 9 9 type + + image C1 C2 C3 C3 C6 C6 He3 C2×He3 C32.C33 C2×C32.C33 kernel C2×C32.C33 C32.C33 C2×C3.He3 C6×3- 1+2 C3.He3 C3×3- 1+2 C3×C6 C32 C2 C1 # reps 1 1 18 8 18 8 6 6 2 2

Matrix representation of C2×C32.C33 in GL9(𝔽19)

 18 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 18
,
 1 0 0 18 18 18 7 7 7 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 11
,
 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7
,
 11 0 0 7 7 7 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 13 12 12 18 18 18 8 8 8 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0
,
 11 18 18 0 8 8 12 1 1 0 0 1 0 0 0 0 0 0 15 8 8 12 12 12 18 18 18 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 1 0 0
,
 1 12 1 0 18 11 0 8 7 0 7 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 11

G:=sub<GL(9,GF(19))| [18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,7,0,0,0,0,0,18,0,0,0,7,0,0,0,0,18,0,0,0,0,7,0,0,0,7,0,0,0,0,0,11,0,0,7,0,0,0,0,0,0,11,0,7,0,0,0,0,0,0,0,11],[7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7],[11,0,0,0,0,0,13,0,0,0,0,0,0,0,0,12,7,0,0,0,0,0,0,0,12,0,7,7,0,0,0,0,0,18,0,0,7,1,0,0,0,0,18,0,0,7,0,1,0,0,0,18,0,0,1,0,0,1,0,0,8,0,0,1,0,0,0,1,0,8,0,0,1,0,0,0,0,1,8,0,0],[11,0,15,0,0,0,0,0,0,18,0,8,0,0,0,0,0,0,18,1,8,0,0,0,0,0,0,0,0,12,0,0,11,0,0,0,8,0,12,1,0,0,0,0,0,8,0,12,0,1,0,0,0,0,12,0,18,0,0,0,0,0,1,1,0,18,0,0,0,7,0,0,1,0,18,0,0,0,0,7,0],[1,0,0,0,0,0,0,0,0,12,7,0,0,0,0,0,0,0,1,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,7,0,0,0,0,11,0,0,0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,0,7,0,7,0,0,0,0,0,0,0,11] >;

C2×C32.C33 in GAP, Magma, Sage, TeX

C_2\times C_3^2.C_3^3
% in TeX

G:=Group("C2xC3^2.C3^3");
// GroupNames label

G:=SmallGroup(486,218);
// by ID

G=gap.SmallGroup(486,218);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,548,986,735,3250]);
// Polycyclic

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

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