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## G = C2×C34.C3order 486 = 2·35

### Direct product of C2 and C34.C3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C34.C3
 Chief series C1 — C3 — C32 — C33 — C34 — C34.C3 — C2×C34.C3
 Lower central C1 — C32 — C2×C34.C3
 Upper central C1 — C3×C6 — C2×C34.C3

Generators and relations for C2×C34.C3
G = < a,b,c,d,e,f | a2=b3=c3=d3=e3=1, f3=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bd-1, cd=dc, ce=ec, fcf-1=ce-1, de=ed, df=fd, ef=fe >

Subgroups: 576 in 252 conjugacy classes, 90 normal (12 characteristic)
C1, C2, C3, C3, C3, C6, C6, C6, C9, C32, C32, C32, C18, C3×C6, C3×C6, C3×C6, C3×C9, 3- 1+2, C33, C33, C33, C3×C18, C2×3- 1+2, C32×C6, C32×C6, C32×C6, C32⋊C9, C3×3- 1+2, C34, C2×C32⋊C9, C6×3- 1+2, C33×C6, C34.C3, C2×C34.C3
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, 3- 1+2, C33, C2×He3, C2×3- 1+2, C32×C6, C3×He3, C3×3- 1+2, C6×He3, C6×3- 1+2, C34.C3, C2×C34.C3

Smallest permutation representation of C2×C34.C3
On 54 points
Generators in S54
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 28)(18 29)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(26 37)(27 38)
(1 38 31)(2 45 35)(3 43 30)(4 41 34)(5 39 29)(6 37 33)(7 44 28)(8 42 32)(9 40 36)(10 48 23)(11 46 27)(12 53 22)(13 51 26)(14 49 21)(15 47 25)(16 54 20)(17 52 24)(18 50 19)
(1 4 7)(2 29 42)(3 40 33)(5 32 45)(6 43 36)(8 35 39)(9 37 30)(10 54 26)(11 14 17)(12 25 50)(13 48 20)(15 19 53)(16 51 23)(18 22 47)(21 24 27)(28 31 34)(38 41 44)(46 49 52)
(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 44 34)(2 45 35)(3 37 36)(4 38 28)(5 39 29)(6 40 30)(7 41 31)(8 42 32)(9 43 33)(10 51 20)(11 52 21)(12 53 22)(13 54 23)(14 46 24)(15 47 25)(16 48 26)(17 49 27)(18 50 19)
(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)| (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,28)(18,29)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,37)(27,38), (1,38,31)(2,45,35)(3,43,30)(4,41,34)(5,39,29)(6,37,33)(7,44,28)(8,42,32)(9,40,36)(10,48,23)(11,46,27)(12,53,22)(13,51,26)(14,49,21)(15,47,25)(16,54,20)(17,52,24)(18,50,19), (1,4,7)(2,29,42)(3,40,33)(5,32,45)(6,43,36)(8,35,39)(9,37,30)(10,54,26)(11,14,17)(12,25,50)(13,48,20)(15,19,53)(16,51,23)(18,22,47)(21,24,27)(28,31,34)(38,41,44)(46,49,52), (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), (1,44,34)(2,45,35)(3,37,36)(4,38,28)(5,39,29)(6,40,30)(7,41,31)(8,42,32)(9,43,33)(10,51,20)(11,52,21)(12,53,22)(13,54,23)(14,46,24)(15,47,25)(16,48,26)(17,49,27)(18,50,19), (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( (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,28)(18,29)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,37)(27,38), (1,38,31)(2,45,35)(3,43,30)(4,41,34)(5,39,29)(6,37,33)(7,44,28)(8,42,32)(9,40,36)(10,48,23)(11,46,27)(12,53,22)(13,51,26)(14,49,21)(15,47,25)(16,54,20)(17,52,24)(18,50,19), (1,4,7)(2,29,42)(3,40,33)(5,32,45)(6,43,36)(8,35,39)(9,37,30)(10,54,26)(11,14,17)(12,25,50)(13,48,20)(15,19,53)(16,51,23)(18,22,47)(21,24,27)(28,31,34)(38,41,44)(46,49,52), (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), (1,44,34)(2,45,35)(3,37,36)(4,38,28)(5,39,29)(6,40,30)(7,41,31)(8,42,32)(9,43,33)(10,51,20)(11,52,21)(12,53,22)(13,54,23)(14,46,24)(15,47,25)(16,48,26)(17,49,27)(18,50,19), (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([[(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,28),(18,29),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44),(25,45),(26,37),(27,38)], [(1,38,31),(2,45,35),(3,43,30),(4,41,34),(5,39,29),(6,37,33),(7,44,28),(8,42,32),(9,40,36),(10,48,23),(11,46,27),(12,53,22),(13,51,26),(14,49,21),(15,47,25),(16,54,20),(17,52,24),(18,50,19)], [(1,4,7),(2,29,42),(3,40,33),(5,32,45),(6,43,36),(8,35,39),(9,37,30),(10,54,26),(11,14,17),(12,25,50),(13,48,20),(15,19,53),(16,51,23),(18,22,47),(21,24,27),(28,31,34),(38,41,44),(46,49,52)], [(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)], [(1,44,34),(2,45,35),(3,37,36),(4,38,28),(5,39,29),(6,40,30),(7,41,31),(8,42,32),(9,43,33),(10,51,20),(11,52,21),(12,53,22),(13,54,23),(14,46,24),(15,47,25),(16,48,26),(17,49,27),(18,50,19)], [(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)]])

102 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3AF 6A ··· 6H 6I ··· 6AF 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 9 ··· 9 9 ··· 9

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C3 C3 C6 C6 C6 He3 3- 1+2 C2×He3 C2×3- 1+2 kernel C2×C34.C3 C34.C3 C2×C32⋊C9 C6×3- 1+2 C33×C6 C32⋊C9 C3×3- 1+2 C34 C3×C6 C3×C6 C32 C32 # reps 1 1 18 6 2 18 6 2 6 18 6 18

Matrix representation of C2×C34.C3 in GL6(𝔽19)

 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18
,
 1 0 0 0 0 0 0 7 0 0 0 0 8 1 11 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 7
,
 7 0 0 0 0 0 0 1 0 0 0 0 7 12 11 0 0 0 0 0 0 7 0 0 0 0 0 0 11 0 0 0 0 0 0 1
,
 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7
,
 0 1 0 0 0 0 8 12 10 0 0 0 12 0 7 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 7 0 0

G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[1,0,8,0,0,0,0,7,1,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,7],[7,0,7,0,0,0,0,1,12,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[0,8,12,0,0,0,1,12,0,0,0,0,0,10,7,0,0,0,0,0,0,0,0,7,0,0,0,1,0,0,0,0,0,0,1,0] >;

C2×C34.C3 in GAP, Magma, Sage, TeX

C_2\times C_3^4.C_3
% in TeX

G:=Group("C2xC3^4.C3");
// GroupNames label

G:=SmallGroup(486,197);
// by ID

G=gap.SmallGroup(486,197);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,548,2169]);
// Polycyclic

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

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