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## G = C2×He3.C32order 486 = 2·35

### Direct product of C2 and He3.C32

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×He3.C32
 Chief series C1 — C3 — C32 — C33 — C3×3- 1+2 — He3.C32 — C2×He3.C32
 Lower central C1 — C3 — C32 — C2×He3.C32
 Upper central C1 — C6 — C32×C6 — C2×He3.C32

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

Subgroups: 360 in 136 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, C3×C9, He3, He3, 3- 1+2, 3- 1+2, C33, C33, C3×C18, C3×C18, C2×He3, C2×He3, C2×3- 1+2, C2×3- 1+2, C32×C6, C32×C6, He3.C3, C3×He3, C3×3- 1+2, C3×3- 1+2, C2×He3.C3, C6×He3, C6×3- 1+2, C6×3- 1+2, He3.C32, C2×He3.C32
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, C33, C2×He3, C32×C6, C3×He3, C6×He3, He3.C32, C2×He3.C32

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

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

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

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

70 conjugacy classes

 class 1 2 3A 3B 3C ··· 3J 3K ··· 3P 6A 6B 6C ··· 6J 6K ··· 6P 9A ··· 9R 18A ··· 18R 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 He3.C32 C2×He3.C32 kernel C2×He3.C32 He3.C32 C2×He3.C3 C6×He3 C6×3- 1+2 He3.C3 C3×He3 C3×3- 1+2 C3×C6 C32 C2 C1 # reps 1 1 18 2 6 18 2 6 6 6 2 2

Matrix representation of C2×He3.C32 in GL9(𝔽19)

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

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

C2×He3.C32 in GAP, Magma, Sage, TeX

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

G:=Group("C2xHe3.C3^2");
// GroupNames label

G:=SmallGroup(486,216);
// by ID

G=gap.SmallGroup(486,216);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,500,735,3250]);
// Polycyclic

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

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