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## G = C2×C32⋊C9order 162 = 2·34

### Direct product of C2 and C32⋊C9

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Aliases: C2×C32⋊C9, C6.1He3, C322C18, C33.3C6, C6.13- 1+2, (C3×C6)⋊C9, (C3×C9)⋊8C6, (C3×C18)⋊1C3, C6.1(C3×C9), C3.1(C3×C18), C3.1(C2×He3), C32.9(C3×C6), (C3×C6).6C32, (C32×C6).1C3, C3.1(C2×3- 1+2), SmallGroup(162,24)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×C32⋊C9
 Chief series C1 — C3 — C32 — C33 — C32⋊C9 — C2×C32⋊C9
 Lower central C1 — C3 — C2×C32⋊C9
 Upper central C1 — C3×C6 — C2×C32⋊C9

Generators and relations for C2×C32⋊C9
G = < a,b,c,d | a2=b3=c3=d9=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >

Smallest permutation representation of C2×C32⋊C9
On 54 points
Generators in S54
(1 51)(2 52)(3 53)(4 54)(5 46)(6 47)(7 48)(8 49)(9 50)(10 34)(11 35)(12 36)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 43)(20 44)(21 45)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)
(1 4 7)(2 31 42)(3 40 35)(5 34 45)(6 43 29)(8 28 39)(9 37 32)(10 21 46)(11 53 25)(12 15 18)(13 24 49)(14 47 19)(16 27 52)(17 50 22)(20 23 26)(30 33 36)(38 41 44)(48 51 54)
(1 44 36)(2 45 28)(3 37 29)(4 38 30)(5 39 31)(6 40 32)(7 41 33)(8 42 34)(9 43 35)(10 49 27)(11 50 19)(12 51 20)(13 52 21)(14 53 22)(15 54 23)(16 46 24)(17 47 25)(18 48 26)
(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)

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

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

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

C2×C32⋊C9 is a maximal subgroup of   C32⋊C36  C32⋊Dic9  C322Dic9  C18×He3  C18×3- 1+2

66 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3N 6A ··· 6H 6I ··· 6N 9A ··· 9R 18A ··· 18R 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 3 ··· 3 3 ··· 3

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C3 C6 C6 C9 C18 He3 3- 1+2 C2×He3 C2×3- 1+2 kernel C2×C32⋊C9 C32⋊C9 C3×C18 C32×C6 C3×C9 C33 C3×C6 C32 C6 C6 C3 C3 # reps 1 1 6 2 6 2 18 18 2 4 2 4

Matrix representation of C2×C32⋊C9 in GL4(𝔽19) generated by

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

C2×C32⋊C9 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes C_9
% in TeX

G:=Group("C2xC3^2:C9");
// GroupNames label

G:=SmallGroup(162,24);
// by ID

G=gap.SmallGroup(162,24);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,276,187]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^3=c^3=d^9=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
// generators/relations

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