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## G = C2×C32.6He3order 486 = 2·35

### Direct product of C2 and C32.6He3

direct product, metabelian, nilpotent (class 4), monomial, 3-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C2×C32.6He3
 Chief series C1 — C3 — C32 — C3×C9 — C9⋊C9 — C32.6He3 — C2×C32.6He3
 Lower central C1 — C3 — C32 — C3×C9 — C2×C32.6He3
 Upper central C1 — C6 — C3×C6 — C3×C18 — C2×C32.6He3

Generators and relations for C2×C32.6He3
G = < a,b,c,d,e,f | a2=b3=c3=1, d3=b-1, e3=f3=c-1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bc-1, ede-1=cd=dc, ce=ec, cf=fc, fdf-1=bde2, fef-1=b-1ce >

Smallest permutation representation of C2×C32.6He3
On 54 points
Generators in S54
(1 5)(2 4)(3 6)(7 14)(8 15)(9 13)(10 17)(11 18)(12 16)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 38)(29 39)(30 40)(31 41)(32 42)(33 43)(34 44)(35 45)(36 37)
(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)
(1 3 2)(4 5 6)(7 9 8)(10 11 12)(13 15 14)(16 17 18)(19 22 25)(20 23 26)(21 24 27)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 49 52)(47 50 53)(48 51 54)
(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 11 9 2 10 7 3 12 8)(4 17 14 6 16 15 5 18 13)(19 21 26 25 27 23 22 24 20)(28 29 33 31 32 36 34 35 30)(37 44 45 40 38 39 43 41 42)(46 48 53 52 54 50 49 51 47)
(1 19 34 2 25 28 3 22 31)(4 52 38 6 49 41 5 46 44)(7 26 33 8 23 36 9 20 30)(10 24 29 12 21 32 11 27 35)(13 47 40 14 53 43 15 50 37)(16 48 42 18 54 45 17 51 39)

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

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

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

38 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 6A 6B 6C 6D 6E 6F 9A ··· 9H 9I 9J 9K 9L 18A ··· 18H 18I 18J 18K 18L order 1 2 3 3 3 3 3 3 6 6 6 6 6 6 9 ··· 9 9 9 9 9 18 ··· 18 18 18 18 18 size 1 1 1 1 3 3 27 27 1 1 3 3 27 27 9 ··· 9 27 27 27 27 9 ··· 9 27 27 27 27

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 9 9 type + + image C1 C2 C3 C3 C3 C6 C6 C6 He3 C2×He3 He3⋊C3 C2×He3⋊C3 C32.6He3 C2×C32.6He3 kernel C2×C32.6He3 C32.6He3 C2×C9⋊C9 C2×He3⋊C3 C2×C3.He3 C9⋊C9 He3⋊C3 C3.He3 C3×C6 C32 C6 C3 C2 C1 # reps 1 1 2 2 4 2 2 4 2 2 6 6 2 2

Matrix representation of C2×C32.6He3 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
,
 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 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 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
,
 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 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 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 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
,
 5 0 0 0 0 0 0 0 0 0 0 0 15 16 7 0 0 0 0 0 0 0 0 0 0 0 5 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 11 0 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 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 11 0
,
 7 0 0 0 0 0 0 0 0 0 0 0 2 11 15 0 0 0 0 0 0 0 0 0 0 0 1 0 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 7 0 0 0 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 7 0 0 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 0 11 0 0 0 0 0 0 0 0 0 1 0 0
,
 16 13 10 0 0 0 0 0 0 0 0 0 10 3 9 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 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 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

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],[7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11],[5,15,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,7,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0],[7,2,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0],[16,10,7,0,0,0,0,0,0,0,0,0,13,3,0,0,0,0,0,0,0,0,0,0,10,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0] >;

C2×C32.6He3 in GAP, Magma, Sage, TeX

C_2\times C_3^2._6{\rm He}_3
% in TeX

G:=Group("C2xC3^2.6He3");
// GroupNames label

G:=SmallGroup(486,90);
// by ID

G=gap.SmallGroup(486,90);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1951,224,338,873,735,453,3250]);
// Polycyclic

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

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