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## G = C2×He3.2S3order 324 = 22·34

### Direct product of C2 and He3.2S3

Aliases: C2×He3.2S3, He3.4D6, C9⋊S34C6, (C3×C18)⋊3C6, (C2×He3).8S3, C32.7(S3×C6), C6.8(C32⋊C6), He3⋊C33C22, (C2×C9⋊S3)⋊3C3, (C3×C9)⋊4(C2×C6), (C3×C6).18(C3×S3), C3.4(C2×C32⋊C6), (C2×He3⋊C3)⋊2C2, SmallGroup(324,73)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C2×He3.2S3
 Chief series C1 — C3 — C32 — C3×C9 — He3⋊C3 — He3.2S3 — C2×He3.2S3
 Lower central C3×C9 — C2×He3.2S3
 Upper central C1 — C2

Generators and relations for C2×He3.2S3
G = < a,b,c,d,e,f | a2=b3=c3=d3=f2=1, e3=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=bc-1, be=eb, fbf=b-1, cd=dc, ce=ec, fcf=c-1, ede-1=b-1cd, df=fd, fef=c-1e2 >

Subgroups: 448 in 56 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C9, C32, C32, D6, C2×C6, D9, C18, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C9, He3, He3, D18, S3×C6, C2×C3⋊S3, C32⋊C6, C9⋊S3, C3×C18, C2×He3, C2×He3, He3⋊C3, C2×C32⋊C6, C2×C9⋊S3, He3.2S3, C2×He3⋊C3, C2×He3.2S3
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, S3×C6, C32⋊C6, C2×C32⋊C6, He3.2S3, C2×He3.2S3

Character table of C2×He3.2S3

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 9A 9B 9C 18A 18B 18C size 1 1 27 27 2 6 9 9 18 18 2 6 9 9 18 18 27 27 27 27 6 6 6 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -1 1 -1 1 1 1 -1 -1 -1 linear of order 2 ρ3 1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 -1 1 1 1 1 -1 -1 -1 linear of order 2 ρ4 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 linear of order 3 ρ6 1 -1 1 -1 1 1 ζ3 ζ32 ζ3 ζ32 -1 -1 ζ65 ζ6 ζ65 ζ6 ζ32 ζ65 ζ3 ζ6 1 1 1 -1 -1 -1 linear of order 6 ρ7 1 1 -1 -1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 ζ3 ζ32 ζ3 ζ32 ζ6 ζ65 ζ65 ζ6 1 1 1 1 1 1 linear of order 6 ρ8 1 -1 -1 1 1 1 ζ3 ζ32 ζ3 ζ32 -1 -1 ζ65 ζ6 ζ65 ζ6 ζ6 ζ3 ζ65 ζ32 1 1 1 -1 -1 -1 linear of order 6 ρ9 1 -1 -1 1 1 1 ζ32 ζ3 ζ32 ζ3 -1 -1 ζ6 ζ65 ζ6 ζ65 ζ65 ζ32 ζ6 ζ3 1 1 1 -1 -1 -1 linear of order 6 ρ10 1 -1 1 -1 1 1 ζ32 ζ3 ζ32 ζ3 -1 -1 ζ6 ζ65 ζ6 ζ65 ζ3 ζ6 ζ32 ζ65 1 1 1 -1 -1 -1 linear of order 6 ρ11 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 linear of order 3 ρ12 1 1 -1 -1 1 1 ζ32 ζ3 ζ32 ζ3 1 1 ζ32 ζ3 ζ32 ζ3 ζ65 ζ6 ζ6 ζ65 1 1 1 1 1 1 linear of order 6 ρ13 2 -2 0 0 2 2 2 2 -1 -1 -2 -2 -2 -2 1 1 0 0 0 0 -1 -1 -1 1 1 1 orthogonal lifted from D6 ρ14 2 2 0 0 2 2 2 2 -1 -1 2 2 2 2 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ15 2 2 0 0 2 2 -1-√-3 -1+√-3 ζ6 ζ65 2 2 -1-√-3 -1+√-3 ζ6 ζ65 0 0 0 0 -1 -1 -1 -1 -1 -1 complex lifted from C3×S3 ρ16 2 -2 0 0 2 2 -1-√-3 -1+√-3 ζ6 ζ65 -2 -2 1+√-3 1-√-3 ζ32 ζ3 0 0 0 0 -1 -1 -1 1 1 1 complex lifted from S3×C6 ρ17 2 2 0 0 2 2 -1+√-3 -1-√-3 ζ65 ζ6 2 2 -1+√-3 -1-√-3 ζ65 ζ6 0 0 0 0 -1 -1 -1 -1 -1 -1 complex lifted from C3×S3 ρ18 2 -2 0 0 2 2 -1+√-3 -1-√-3 ζ65 ζ6 -2 -2 1-√-3 1+√-3 ζ3 ζ32 0 0 0 0 -1 -1 -1 1 1 1 complex lifted from S3×C6 ρ19 6 6 0 0 6 -3 0 0 0 0 6 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ20 6 -6 0 0 6 -3 0 0 0 0 -6 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×C32⋊C6 ρ21 6 -6 0 0 -3 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 orthogonal faithful ρ22 6 6 0 0 -3 0 0 0 0 0 -3 0 0 0 0 0 0 0 0 0 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 orthogonal lifted from He3.2S3 ρ23 6 -6 0 0 -3 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 ζ98+ζ97-ζ94+2ζ92 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 orthogonal faithful ρ24 6 6 0 0 -3 0 0 0 0 0 -3 0 0 0 0 0 0 0 0 0 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 orthogonal lifted from He3.2S3 ρ25 6 6 0 0 -3 0 0 0 0 0 -3 0 0 0 0 0 0 0 0 0 -ζ98+2ζ97+ζ94+ζ92 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 orthogonal lifted from He3.2S3 ρ26 6 -6 0 0 -3 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 ζ95+2ζ94-ζ92+ζ9 2ζ98-ζ94+ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 2ζ95+ζ94+ζ92-ζ9 ζ98+ζ94-ζ92+2ζ9 ζ98+ζ97-ζ94+2ζ92 orthogonal faithful

Smallest permutation representation of C2×He3.2S3
On 54 points
Generators in S54
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 28)(8 29)(9 30)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)
(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)(37 43 40)(38 44 41)(39 45 42)(46 49 52)(47 50 53)(48 51 54)
(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 16 22)(2 11 23)(3 15 24)(4 10 25)(5 14 26)(6 18 27)(7 13 19)(8 17 20)(9 12 21)(28 40 46)(29 44 47)(30 39 48)(31 43 49)(32 38 50)(33 42 51)(34 37 52)(35 41 53)(36 45 54)
(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 9)(3 8)(4 7)(5 6)(10 13)(11 12)(14 18)(15 17)(19 25)(20 24)(21 23)(26 27)(28 34)(29 33)(30 32)(35 36)(37 40)(38 39)(41 45)(42 44)(46 52)(47 51)(48 50)(53 54)

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

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

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

Matrix representation of C2×He3.2S3 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 1 0 0 0 0 11 11 18 18 0 0 0 0 1 0 0 0 0 0 0 0 0 1 7 7 0 0 18 18
,
 18 1 0 0 0 0 18 0 0 0 0 0 8 0 0 1 0 0 0 11 18 18 0 0 12 0 0 0 0 1 0 7 0 0 18 18
,
 0 0 0 0 18 1 7 7 0 0 17 18 0 0 0 0 8 0 1 0 0 0 8 0 0 0 1 0 12 0 0 0 0 1 12 0
,
 12 2 0 0 0 0 17 14 0 0 0 0 18 2 7 5 0 0 16 0 14 2 0 0 5 0 0 0 14 2 0 14 0 0 17 12
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 11 11 18 18 0 0 0 0 0 0 1 0 7 7 0 0 18 18

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],[1,0,11,0,0,7,0,1,11,0,0,7,0,0,18,1,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,1,18],[18,18,8,0,12,0,1,0,0,11,0,7,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,0,18,0,0,0,0,1,18],[0,7,0,1,0,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,18,17,8,8,12,12,1,18,0,0,0,0],[12,17,18,16,5,0,2,14,2,0,0,14,0,0,7,14,0,0,0,0,5,2,0,0,0,0,0,0,14,17,0,0,0,0,2,12],[0,1,0,11,0,7,1,0,0,11,0,7,0,0,1,18,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,18] >;

C2×He3.2S3 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3._2S_3
% in TeX

G:=Group("C2xHe3.2S3");
// GroupNames label

G:=SmallGroup(324,73);
// by ID

G=gap.SmallGroup(324,73);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,3171,303,453,2164,1096,7781]);
// Polycyclic

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

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