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## G = C2×3- 1+2.S3order 324 = 22·34

### Direct product of C2 and 3- 1+2.S3

Aliases: C2×3- 1+2.S3, 3- 1+2.D6, (C3×C9).2D6, (C3×C18).10S3, C3.He3⋊C22, C6.10(He3⋊C2), (C2×3- 1+2).4S3, (C3×C6).8(C3⋊S3), (C2×C3.He3)⋊C2, C32.4(C2×C3⋊S3), C3.5(C2×He3⋊C2), SmallGroup(324,80)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3.He3 — C2×3- 1+2.S3
 Chief series C1 — C3 — C32 — C3×C9 — C3.He3 — 3- 1+2.S3 — C2×3- 1+2.S3
 Lower central C3.He3 — C2×3- 1+2.S3
 Upper central C1 — C2

Generators and relations for C2×3- 1+2.S3
G = < a,b,c,d,e | a2=b9=c3=e2=1, d3=b6, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b4, dbd-1=b7c, ebe=b5c-1, dcd-1=b3c, ce=ec, ede=b3d2 >

Subgroups: 337 in 65 conjugacy classes, 19 normal (11 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C9, C32, D6, C2×C6, D9, C18, C3×S3, C3×C6, C3×C9, 3- 1+2, D18, S3×C6, C3×D9, C9⋊C6, C3×C18, C2×3- 1+2, C3.He3, C6×D9, C2×C9⋊C6, 3- 1+2.S3, C2×C3.He3, C2×3- 1+2.S3
Quotients: C1, C2, C22, S3, D6, C3⋊S3, C2×C3⋊S3, He3⋊C2, C2×He3⋊C2, 3- 1+2.S3, C2×3- 1+2.S3

Character table of C2×3- 1+2.S3

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

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

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

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

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

Matrix representation of C2×3- 1+2.S3 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 5 12 0 0 0 0 7 17 0 0 14 14 14 14 7 16 12 12 12 12 10 7 0 7 0 7 7 5 5 0 0 7 7 5
,
 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 1 0 1 18 18 18 0 18 18
,
 18 18 18 18 18 17 0 0 0 0 1 18 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 18 18 18 18 18 18 17 0 0 1 0 0 1 0 0 0 0 0 1

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],[0,0,14,12,0,5,0,0,14,12,7,0,5,7,14,12,0,0,12,17,14,12,7,7,0,0,7,10,7,7,0,0,16,7,5,5],[1,0,0,0,0,18,0,1,0,0,0,18,0,0,0,1,0,18,0,0,18,18,1,0,0,0,0,0,0,18,0,0,0,0,1,18],[18,0,1,0,0,0,18,0,0,1,0,0,18,0,0,0,0,1,18,0,0,0,0,0,18,1,0,0,0,0,17,18,0,0,1,1],[0,1,0,18,0,0,1,0,0,18,0,0,0,0,0,18,1,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,18,17,1,1] >;

C2×3- 1+2.S3 in GAP, Magma, Sage, TeX

C_2\times 3_-^{1+2}.S_3
% in TeX

G:=Group("C2xES-(3,1).S3");
// GroupNames label

G:=SmallGroup(324,80);
// by ID

G=gap.SmallGroup(324,80);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,2090,986,3171,303,453,7564,1096,7781]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^9=c^3=e^2=1,d^3=b^6,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^4,d*b*d^-1=b^7*c,e*b*e=b^5*c^-1,d*c*d^-1=b^3*c,c*e=e*c,e*d*e=b^3*d^2>;
// generators/relations

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