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## G = C2×C33⋊C32order 486 = 2·35

### Direct product of C2 and C33⋊C32

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C33⋊C32
 Chief series C1 — C3 — C32 — C33 — C3×He3 — C33⋊C32 — C2×C33⋊C32
 Lower central C1 — C3 — C32 — C2×C33⋊C32
 Upper central C1 — C6 — C32×C6 — C2×C33⋊C32

Generators and relations for C2×C33⋊C32
G = < a,b,c,d,e,f | a2=b3=c3=d3=e3=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, ebe-1=bc-1d, cd=dc, ece-1=cd-1, cf=fc, de=ed, df=fd, ef=fe >

Subgroups: 468 in 148 conjugacy classes, 66 normal (16 characteristic)
C1, C2, C3, C3, C6, C6, C9, C32, C32, C32, C18, C3×C6, C3×C6, C3×C6, C3×C9, He3, He3, 3- 1+2, 3- 1+2, C33, C33, C33, C3×C18, C2×He3, C2×He3, C2×3- 1+2, C2×3- 1+2, C32×C6, C32×C6, C32×C6, C3≀C3, C3×He3, C3×3- 1+2, C2×C3≀C3, C6×He3, C6×3- 1+2, C33⋊C32, C2×C33⋊C32
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, C33, C2×He3, C32×C6, C3×He3, C6×He3, C33⋊C32, C2×C33⋊C32

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

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

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

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

70 conjugacy classes

 class 1 2 3A 3B 3C ··· 3J 3K ··· 3V 6A 6B 6C ··· 6J 6K ··· 6V 9A ··· 9L 18A ··· 18L order 1 2 3 3 3 ··· 3 3 ··· 3 6 6 6 ··· 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 1 3 ··· 3 9 ··· 9 1 1 3 ··· 3 9 ··· 9 9 ··· 9 9 ··· 9

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 9 9 type + + image C1 C2 C3 C3 C3 C6 C6 C6 He3 C2×He3 C33⋊C32 C2×C33⋊C32 kernel C2×C33⋊C32 C33⋊C32 C2×C3≀C3 C6×He3 C6×3- 1+2 C3≀C3 C3×He3 C3×3- 1+2 C3×C6 C32 C2 C1 # reps 1 1 18 4 4 18 4 4 6 6 2 2

Matrix representation of C2×C33⋊C32 in GL12(𝔽19)

 18 0 0 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1
,
 0 7 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 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 8 1 0 11 6 0 0 0 0 0 0 0 0 0 0 0 7 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 18 0 0 1 0 0 0 0 0 0 0 0 7 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 1 0 7 0 0 0 0
,
 7 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 8 12 6 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 8 1 0 11 6 0 0 0 0 0 0 0 0 0 18 0 8 1 0 0 0 0 0 0 0 0 7 0 12 0 0 0 0 0 0 0 8 1 0 0 0 0 11 6 0 0 0 0 0 0 18 0 0 0 0 8 1 0 0 0 0 0 7 0 0 0 0 12 0
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 7
,
 11 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 8 11 7 0 0 0 0 0 0 0 0 0 7 8 0 11 0 4 0 0 0 0 0 0 8 12 0 18 0 18 0 0 0 0 0 0 0 18 0 1 11 8 0 0 0 0 0 0 12 11 0 0 0 0 7 9 0 0 0 0 0 11 0 0 0 0 8 12 7 0 0 0 8 0 0 0 0 0 11 18 0
,
 7 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 8 1 0 0 7 0 0 0 0 0 0 0 1 12 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 7 8 0 0 0 0 0 11 0 0 0 0 8 1 0 0 0 0 0 0 11

G:=sub<GL(12,GF(19))| [18,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,11,0,0,0,0,0,0,0,0,0,0,0,6,7,0,18,7,0,18,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0],[7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,8,0,0,8,0,0,0,0,0,1,12,0,1,0,0,1,0,0,0,0,0,0,6,7,0,18,7,0,18,7,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,6,8,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,6,8,12,0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7],[11,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,8,7,8,0,12,0,8,0,0,0,0,11,11,8,12,18,11,11,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,11,18,1,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,4,18,8,0,0,0,0,0,0,0,0,0,0,0,0,7,8,11,0,0,0,0,0,0,0,0,0,9,12,18,0,0,0,0,0,0,0,0,0,0,7,0],[7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,8,1,0,7,8,0,0,0,0,1,0,0,1,12,0,8,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11] >;

C2×C33⋊C32 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(486,215);
// by ID

G=gap.SmallGroup(486,215);
# by ID

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

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

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