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## G = C2×He3.3S3order 324 = 22·34

### Direct product of C2 and He3.3S3

Aliases: C2×He3.3S3, He3.5D6, 3- 1+22D6, (C3×C9)⋊9D6, (C3×C18)⋊5S3, (C2×He3).9S3, He3.C32C22, C6.8(He3⋊C2), (C2×3- 1+2)⋊2S3, (C3×C6).6(C3⋊S3), C32.2(C2×C3⋊S3), (C2×He3.C3)⋊1C2, C3.3(C2×He3⋊C2), SmallGroup(324,78)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — He3.C3 — C2×He3.3S3
 Chief series C1 — C3 — C32 — C3×C9 — He3.C3 — He3.3S3 — C2×He3.3S3
 Lower central He3.C3 — C2×He3.3S3
 Upper central C1 — C2

Generators and relations for C2×He3.3S3
G = < a,b,c,d,e,f | a2=b3=c3=d3=f2=1, e3=fcf=c-1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=fbf=bc-1, be=eb, cd=dc, ce=ec, ede-1=b-1cd, fdf=d-1, fef=ce2 >

Subgroups: 436 in 70 conjugacy classes, 19 normal (13 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, 3- 1+2, D18, S3×C6, C2×C3⋊S3, C3×D9, C32⋊C6, C9⋊C6, C3×C18, C2×He3, C2×3- 1+2, He3.C3, C6×D9, C2×C32⋊C6, C2×C9⋊C6, He3.3S3, C2×He3.C3, C2×He3.3S3
Quotients: C1, C2, C22, S3, D6, C3⋊S3, C2×C3⋊S3, He3⋊C2, C2×He3⋊C2, He3.3S3, C2×He3.3S3

Character table of C2×He3.3S3

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

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

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

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

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

Matrix representation of C2×He3.3S3 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
,
 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0
,
 18 1 0 0 0 0 18 0 0 0 0 0 0 0 18 1 0 0 0 0 18 0 0 0 0 0 0 0 18 1 0 0 0 0 18 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 18 0 0 0 0 1 18 0 0 0 0 0 0 18 1 0 0 0 0 18 0
,
 1 3 1 3 15 1 16 4 16 4 18 16 15 1 1 3 1 3 18 16 16 4 16 4 1 3 15 1 1 3 16 4 18 16 16 4
,
 0 18 0 0 0 0 18 0 0 0 0 0 0 0 18 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 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],[0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[18,18,0,0,0,0,1,0,0,0,0,0,0,0,18,18,0,0,0,0,1,0,0,0,0,0,0,0,18,18,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,18,18,0,0,0,0,0,0,18,18,0,0,0,0,1,0],[1,16,15,18,1,16,3,4,1,16,3,4,1,16,1,16,15,18,3,4,3,4,1,16,15,18,1,16,1,16,1,16,3,4,3,4],[0,18,0,0,0,0,18,0,0,0,0,0,0,0,18,0,0,0,0,0,1,1,0,0,0,0,0,0,1,1,0,0,0,0,0,18] >;

C2×He3.3S3 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3._3S_3
% in TeX

G:=Group("C2xHe3.3S3");
// GroupNames label

G:=SmallGroup(324,78);
// by ID

G=gap.SmallGroup(324,78);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,146,5763,303,237,7564,1096,7781]);
// Polycyclic

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

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