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G = C2×C9⋊S3order 108 = 22·33

Direct product of C2 and C9⋊S3

Aliases: C2×C9⋊S3, C6⋊D9, C18⋊S3, C92D6, C32D18, C32.4D6, (C3×C18)⋊3C2, (C3×C6).8S3, (C3×C9)⋊4C22, C6.2(C3⋊S3), C3.(C2×C3⋊S3), SmallGroup(108,27)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C9⋊S3
G = < a,b,c,d | a2=b9=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Character table of C2×C9⋊S3

 class 1 2A 2B 2C 3A 3B 3C 3D 6A 6B 6C 6D 9A 9B 9C 9D 9E 9F 9G 9H 9I 18A 18B 18C 18D 18E 18F 18G 18H 18I size 1 1 27 27 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 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 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 -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 -1 -1 -1 -1 linear of order 2 ρ5 2 2 0 0 -1 2 -1 -1 -1 2 -1 -1 -1 2 -1 -1 -1 2 2 -1 -1 -1 -1 -1 -1 2 2 2 -1 -1 orthogonal lifted from S3 ρ6 2 -2 0 0 -1 2 -1 -1 1 -2 1 1 -1 -1 2 -1 -1 -1 -1 2 2 -2 1 1 1 1 1 1 -2 -2 orthogonal lifted from D6 ρ7 2 2 0 0 2 2 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 -2 0 0 -1 2 -1 -1 1 -2 1 1 -1 2 -1 -1 -1 2 2 -1 -1 1 1 1 1 -2 -2 -2 1 1 orthogonal lifted from D6 ρ9 2 2 0 0 -1 2 -1 -1 -1 2 -1 -1 -1 -1 2 -1 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 -1 2 2 orthogonal lifted from S3 ρ10 2 -2 0 0 2 2 2 2 -2 -2 -2 -2 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 orthogonal lifted from D6 ρ11 2 2 0 0 -1 2 -1 -1 -1 2 -1 -1 2 -1 -1 2 2 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 -2 0 0 -1 2 -1 -1 1 -2 1 1 2 -1 -1 2 2 -1 -1 -1 -1 1 -2 -2 -2 1 1 1 1 1 orthogonal lifted from D6 ρ13 2 -2 0 0 2 -1 -1 -1 -2 1 1 1 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -ζ97-ζ92 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ95-ζ94 -ζ98-ζ9 orthogonal lifted from D18 ρ14 2 2 0 0 -1 -1 2 -1 -1 -1 2 -1 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ15 2 -2 0 0 -1 -1 2 -1 1 1 -2 1 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 orthogonal lifted from D18 ρ16 2 2 0 0 -1 -1 -1 2 -1 -1 -1 2 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ17 2 2 0 0 2 -1 -1 -1 2 -1 -1 -1 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ18 2 -2 0 0 2 -1 -1 -1 -2 1 1 1 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -ζ95-ζ94 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ98-ζ9 -ζ97-ζ92 orthogonal lifted from D18 ρ19 2 2 0 0 2 -1 -1 -1 2 -1 -1 -1 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ20 2 -2 0 0 -1 -1 -1 2 1 1 1 -2 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 -ζ98-ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ97-ζ92 -ζ95-ζ94 orthogonal lifted from D18 ρ21 2 -2 0 0 2 -1 -1 -1 -2 1 1 1 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -ζ98-ζ9 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ97-ζ92 -ζ95-ζ94 orthogonal lifted from D18 ρ22 2 2 0 0 -1 -1 2 -1 -1 -1 2 -1 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ23 2 2 0 0 2 -1 -1 -1 2 -1 -1 -1 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ24 2 -2 0 0 -1 -1 2 -1 1 1 -2 1 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 orthogonal lifted from D18 ρ25 2 -2 0 0 -1 -1 -1 2 1 1 1 -2 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 -ζ97-ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ95-ζ94 -ζ98-ζ9 orthogonal lifted from D18 ρ26 2 -2 0 0 -1 -1 -1 2 1 1 1 -2 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 -ζ95-ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ98-ζ9 -ζ97-ζ92 orthogonal lifted from D18 ρ27 2 2 0 0 -1 -1 -1 2 -1 -1 -1 2 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ28 2 2 0 0 -1 -1 2 -1 -1 -1 2 -1 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ29 2 2 0 0 -1 -1 -1 2 -1 -1 -1 2 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ30 2 -2 0 0 -1 -1 2 -1 1 1 -2 1 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 orthogonal lifted from D18

Smallest permutation representation of C2×C9⋊S3
On 54 points
Generators in S54
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 10)(9 11)(19 44)(20 45)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 54)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 53)
(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 35 37)(2 36 38)(3 28 39)(4 29 40)(5 30 41)(6 31 42)(7 32 43)(8 33 44)(9 34 45)(10 50 19)(11 51 20)(12 52 21)(13 53 22)(14 54 23)(15 46 24)(16 47 25)(17 48 26)(18 49 27)
(2 9)(3 8)(4 7)(5 6)(10 14)(11 13)(15 18)(16 17)(19 54)(20 53)(21 52)(22 51)(23 50)(24 49)(25 48)(26 47)(27 46)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(36 45)

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

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

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

C2×C9⋊S3 is a maximal subgroup of
C18.D6  C3⋊D36  C9⋊D12  C36⋊S3  C6.D18  C2×S3×D9  C18.6S4  C32.3GL2(𝔽3)
C2×C9⋊S3 is a maximal quotient of
C12.D9  C36⋊S3  C6.D18

Matrix representation of C2×C9⋊S3 in GL4(𝔽19) generated by

 1 0 0 0 0 1 0 0 0 0 18 0 0 0 0 18
,
 5 7 0 0 12 17 0 0 0 0 12 17 0 0 2 14
,
 0 1 0 0 18 18 0 0 0 0 18 18 0 0 1 0
,
 1 0 0 0 18 18 0 0 0 0 18 0 0 0 1 1
G:=sub<GL(4,GF(19))| [1,0,0,0,0,1,0,0,0,0,18,0,0,0,0,18],[5,12,0,0,7,17,0,0,0,0,12,2,0,0,17,14],[0,18,0,0,1,18,0,0,0,0,18,1,0,0,18,0],[1,18,0,0,0,18,0,0,0,0,18,1,0,0,0,1] >;

C2×C9⋊S3 in GAP, Magma, Sage, TeX

C_2\times C_9\rtimes S_3
% in TeX

G:=Group("C2xC9:S3");
// GroupNames label

G:=SmallGroup(108,27);
// by ID

G=gap.SmallGroup(108,27);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-3,662,282,483,1804]);
// Polycyclic

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

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