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## G = C2×C32.He3order 486 = 2·35

### Direct product of C2 and C32.He3

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

Aliases: C2×C32.He3, C9⋊C94C6, (C3×C6).4He3, He3⋊C36C6, C3.He33C6, (C3×C18).2C32, C32.4(C2×He3), C6.9(He3⋊C3), (C2×C9⋊C9)⋊1C3, (C3×C9).2(C3×C6), (C2×He3⋊C3)⋊2C3, C3.9(C2×He3⋊C3), (C2×C3.He3)⋊2C3, SmallGroup(486,88)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C32.He3
G = < a,b,c,d,e,f | a2=b3=c3=f3=1, d3=b-1, e3=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.He3
On 54 points
Generators in S54
(1 4)(2 5)(3 6)(7 15)(8 13)(9 14)(10 16)(11 17)(12 18)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 46)(26 47)(27 48)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)
(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 6 5)(2 4 3)(7 8 9)(10 12 11)(13 14 15)(16 18 17)(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 15 12 5 14 10 6 13 11)(2 9 16 3 8 17 4 7 18)(19 21 26 25 27 23 22 24 20)(28 35 36 31 29 30 34 32 33)(37 44 45 40 38 39 43 41 42)(46 48 53 52 54 50 49 51 47)
(1 49 41)(2 25 35)(3 22 29)(4 19 32)(5 46 44)(6 52 38)(7 27 33)(8 21 30)(9 24 36)(10 47 40)(11 53 43)(12 50 37)(13 51 39)(14 54 45)(15 48 42)(16 26 31)(17 23 34)(18 20 28)

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

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

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

38 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 6C 6D 6E 6F 6G 6H 9A ··· 9H 9I 9J 18A ··· 18H 18I 18J order 1 2 3 3 3 3 3 3 3 3 6 6 6 6 6 6 6 6 9 ··· 9 9 9 18 ··· 18 18 18 size 1 1 1 1 3 3 27 27 27 27 1 1 3 3 27 27 27 27 9 ··· 9 27 27 9 ··· 9 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.He3 C2×C32.He3 kernel C2×C32.He3 C32.He3 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 4 2 2 4 2 2 2 6 6 2 2

Matrix representation of C2×C32.He3 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
,
 17 0 0 0 0 0 0 0 0 0 0 0 0 17 0 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 12 7 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 18 12 15 0 0 0 0 0 0 0 0 0 11 1 7 0 0 0 0 0 0 0 0 0 0 0 0 18 12 15 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 7 0 1
,
 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 11 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 8 10 0 0 0 0 0 0 0 0 0 12 0 11 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 8 10 0 0 0 0 0 0 0 0 0 12 0 11 0 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 18 12 15 0 0 0 0 0 0 0 0 0 18 0 7
,
 0 0 11 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 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 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

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],[17,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,0,0,0,0,7,7,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,18,11,0,0,0,0,0,0,0,0,0,7,12,1,0,0,0,0,0,0,0,0,0,0,15,7,0,0,0,0,0,0,0,0,0,0,0,0,18,7,7,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,15,0,1],[7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,0,0,0,0,10,11,0,0,0,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,0,0,0,0,10,11,0,0,0,0,0,0,0,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,0,11,12,0,0,0,0,0,0,0,0,0,0,0,15,7],[0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0] >;

C2×C32.He3 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC3^2.He3");
// GroupNames label

G:=SmallGroup(486,88);
// by ID

G=gap.SmallGroup(486,88);
# by ID

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

G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=f^3=1,d^3=b^-1,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,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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