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G = C2×C9⋊S3order 108 = 22·33

Direct product of C2 and C9⋊S3

direct product, metabelian, supersoluble, monomial, A-group

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

C1C3×C9 — C2×C9⋊S3
C1C3C32C3×C9C9⋊S3 — C2×C9⋊S3
C3×C9 — C2×C9⋊S3
C1C2

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 >

27C2
27C2
27C22
9S3
9S3
9S3
9S3
9S3
9S3
9S3
9S3
9D6
9D6
9D6
9D6
3C3⋊S3
3D9
3D9
3D9
3D9
3C3⋊S3
3D9
3D9
3C2×C3⋊S3
3D18
3D18
3D18

Character table of C2×C9⋊S3

 class 12A2B2C3A3B3C3D6A6B6C6D9A9B9C9D9E9F9G9H9I18A18B18C18D18E18F18G18H18I
 size 11272722222222222222222222222222
ρ1111111111111111111111111111111    trivial
ρ211-1-111111111111111111111111111    linear of order 2
ρ31-1-111111-1-1-1-1111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ41-11-11111-1-1-1-1111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ52200-12-1-1-12-1-1-12-1-1-122-1-1-1-1-1-1222-1-1    orthogonal lifted from S3
ρ62-200-12-1-11-211-1-12-1-1-1-122-2111111-2-2    orthogonal lifted from D6
ρ7220022222222-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ82-200-12-1-11-211-12-1-1-122-1-11111-2-2-211    orthogonal lifted from D6
ρ92200-12-1-1-12-1-1-1-12-1-1-1-1222-1-1-1-1-1-122    orthogonal lifted from S3
ρ102-2002222-2-2-2-2-1-1-1-1-1-1-1-1-1111111111    orthogonal lifted from D6
ρ112200-12-1-1-12-1-12-1-122-1-1-1-1-1222-1-1-1-1-1    orthogonal lifted from S3
ρ122-200-12-1-11-2112-1-122-1-1-1-11-2-2-211111    orthogonal lifted from D6
ρ132-2002-1-1-1-2111ζ989ζ989ζ9594ζ9792ζ9594ζ9792ζ9594ζ989ζ9792979298997929594979295949899594989    orthogonal lifted from D18
ρ142200-1-12-1-1-12-1ζ989ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9792ζ9594ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ152-200-1-12-111-21ζ989ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9792ζ9594959498997929594989979295949899792    orthogonal lifted from D18
ρ162200-1-1-12-1-1-12ζ9594ζ989ζ989ζ989ζ9792ζ9792ζ9594ζ9792ζ9594ζ9594ζ9594ζ989ζ9792ζ9792ζ9594ζ989ζ989ζ9792    orthogonal lifted from D9
ρ1722002-1-1-12-1-1-1ζ9594ζ9594ζ9792ζ989ζ9792ζ989ζ9792ζ9594ζ989ζ989ζ9594ζ989ζ9792ζ989ζ9792ζ9594ζ9792ζ9594    orthogonal lifted from D9
ρ182-2002-1-1-1-2111ζ9792ζ9792ζ989ζ9594ζ989ζ9594ζ989ζ9792ζ9594959497929594989959498997929899792    orthogonal lifted from D18
ρ1922002-1-1-12-1-1-1ζ989ζ989ζ9594ζ9792ζ9594ζ9792ζ9594ζ989ζ9792ζ9792ζ989ζ9792ζ9594ζ9792ζ9594ζ989ζ9594ζ989    orthogonal lifted from D9
ρ202-200-1-1-12111-2ζ989ζ9792ζ9792ζ9792ζ9594ζ9594ζ989ζ9594ζ989989989979295949594989979297929594    orthogonal lifted from D18
ρ212-2002-1-1-1-2111ζ9594ζ9594ζ9792ζ989ζ9792ζ989ζ9792ζ9594ζ989989959498997929899792959497929594    orthogonal lifted from D18
ρ222200-1-12-1-1-12-1ζ9594ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ989ζ9792ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ2322002-1-1-12-1-1-1ζ9792ζ9792ζ989ζ9594ζ989ζ9594ζ989ζ9792ζ9594ζ9594ζ9792ζ9594ζ989ζ9594ζ989ζ9792ζ989ζ9792    orthogonal lifted from D9
ρ242-200-1-12-111-21ζ9792ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ9594ζ989989979295949899792959498997929594    orthogonal lifted from D18
ρ252-200-1-1-12111-2ζ9792ζ9594ζ9594ζ9594ζ989ζ989ζ9792ζ989ζ9792979297929594989989979295949594989    orthogonal lifted from D18
ρ262-200-1-1-12111-2ζ9594ζ989ζ989ζ989ζ9792ζ9792ζ9594ζ9792ζ9594959495949899792979295949899899792    orthogonal lifted from D18
ρ272200-1-1-12-1-1-12ζ9792ζ9594ζ9594ζ9594ζ989ζ989ζ9792ζ989ζ9792ζ9792ζ9792ζ9594ζ989ζ989ζ9792ζ9594ζ9594ζ989    orthogonal lifted from D9
ρ282200-1-12-1-1-12-1ζ9792ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ9594ζ989ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ292200-1-1-12-1-1-12ζ989ζ9792ζ9792ζ9792ζ9594ζ9594ζ989ζ9594ζ989ζ989ζ989ζ9792ζ9594ζ9594ζ989ζ9792ζ9792ζ9594    orthogonal lifted from D9
ρ302-200-1-12-111-21ζ9594ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ989ζ9792979295949899792959498997929594989    orthogonal lifted from D18

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

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

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

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

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

1000
0100
00180
00018
,
5700
121700
001217
00214
,
0100
181800
001818
0010
,
1000
181800
00180
0011
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

Export

Subgroup lattice of C2×C9⋊S3 in TeX
Character table of C2×C9⋊S3 in TeX

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