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G = C2×He3.S3  order 324 = 22·34

Direct product of C2 and He3.S3

direct product, metabelian, supersoluble, monomial

Aliases: C2×He3.S3, He3.3D6, C9⋊S3⋊3C6, (C3×C18)⋊2C6, (C2×He3).7S3, C32.6(S3×C6), C6.7(C32⋊C6), He3.C3⋊3C22, (C2×C9⋊S3)⋊2C3, (C3×C9)⋊3(C2×C6), (C3×C6).17(C3×S3), C3.3(C2×C32⋊C6), (C2×He3.C3)⋊2C2, SmallGroup(324,71)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C2×He3.S3
 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=bc-1, be=eb, fbf=b-1, cd=dc, ce=ec, ede-1=b-1cd, df=fd, fef=ce2 >

C1
C2
27C2
27C2
C3
3C3
9C3
27C22
C6
3C6
9S3
9C6
9S3
27C6
27C6
27S3
27S3
C32
3C9
3C32
6C9
9D6
27C2×C6
27D6
C3×C6
3C18
3C3×C6
3C3⋊S3
3C3⋊S3
6C18
9D9
9C3×S3
9D9
9C3×S3
C3×C9
He3
23- 1+2
3C2×C3⋊S3
9S3×C6
9D18
C3×C18
C9⋊S3
C9⋊S3
C2×He3
2C2×3- 1+2
3C32⋊C6
3C32⋊C6
He3.C3
C2×C9⋊S3
3C2×C32⋊C6
He3.S3
C2×He3.C3
He3.S3
C2×He3.S3

Character table of C2×He3.S3

 class 12A2B2C3A3B3C3D6A6B6C6D6E6F6G6H9A9B9C9D9E18A18B18C18D18E
 size 112727269926992727272766618186661818
ρ111111111111111111111111111    trivial
ρ211-1-111111111-1-1-1-11111111111    linear of order 2
ρ31-11-11111-1-1-1-11-11-111111-1-1-1-1-1    linear of order 2
ρ41-1-111111-1-1-1-1-11-1111111-1-1-1-1-1    linear of order 2
ρ51-1-1111ζ32ζ3-1-1ζ65ζ6ζ6ζ3ζ65ζ32111ζ3ζ32-1-1-1ζ65ζ6    linear of order 6
ρ6111111ζ32ζ311ζ3ζ32ζ32ζ3ζ3ζ32111ζ3ζ32111ζ3ζ32    linear of order 3
ρ711-1-111ζ32ζ311ζ3ζ32ζ6ζ65ζ65ζ6111ζ3ζ32111ζ3ζ32    linear of order 6
ρ81-11-111ζ32ζ3-1-1ζ65ζ6ζ32ζ65ζ3ζ6111ζ3ζ32-1-1-1ζ65ζ6    linear of order 6
ρ911-1-111ζ3ζ3211ζ32ζ3ζ65ζ6ζ6ζ65111ζ32ζ3111ζ32ζ3    linear of order 6
ρ101-11-111ζ3ζ32-1-1ζ6ζ65ζ3ζ6ζ32ζ65111ζ32ζ3-1-1-1ζ6ζ65    linear of order 6
ρ11111111ζ3ζ3211ζ32ζ3ζ3ζ32ζ32ζ3111ζ32ζ3111ζ32ζ3    linear of order 3
ρ121-1-1111ζ3ζ32-1-1ζ6ζ65ζ65ζ32ζ6ζ3111ζ32ζ3-1-1-1ζ6ζ65    linear of order 6
ρ132-2002222-2-2-2-20000-1-1-1-1-111111    orthogonal lifted from D6
ρ142200222222220000-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ15220022-1+√-3-1-√-322-1-√-3-1+√-30000-1-1-1ζ6ζ65-1-1-1ζ6ζ65    complex lifted from C3×S3
ρ162-20022-1+√-3-1-√-3-2-21+√-31-√-30000-1-1-1ζ6ζ65111ζ32ζ3    complex lifted from S3×C6
ρ17220022-1-√-3-1+√-322-1+√-3-1-√-30000-1-1-1ζ65ζ6-1-1-1ζ65ζ6    complex lifted from C3×S3
ρ182-20022-1-√-3-1+√-3-2-21-√-31+√-30000-1-1-1ζ65ζ6111ζ3ζ32    complex lifted from S3×C6
ρ196-6006-300-630000000000000000    orthogonal lifted from C2×C32⋊C6
ρ2066006-3006-30000000000000000    orthogonal lifted from C32⋊C6
ρ216-600-300030000000ζ98+ζ97-ζ94+2ζ92ζ98+ζ94-ζ92+2ζ92ζ95+ζ94+ζ92-ζ900-ζ98+2ζ97+ζ94+ζ922ζ98-ζ94+ζ92+ζ9ζ95+2ζ94-ζ92+ζ900    orthogonal faithful
ρ226600-3000-300000002ζ95+ζ94+ζ92-ζ9ζ98+ζ97-ζ94+2ζ92ζ98+ζ94-ζ92+2ζ900ζ98+ζ97-ζ94+2ζ92ζ98+ζ94-ζ92+2ζ92ζ95+ζ94+ζ92-ζ900    orthogonal lifted from He3.S3
ρ236-600-3000300000002ζ95+ζ94+ζ92-ζ9ζ98+ζ97-ζ94+2ζ92ζ98+ζ94-ζ92+2ζ900ζ95+2ζ94-ζ92+ζ9-ζ98+2ζ97+ζ94+ζ922ζ98-ζ94+ζ92+ζ900    orthogonal faithful
ρ246-600-300030000000ζ98+ζ94-ζ92+2ζ92ζ95+ζ94+ζ92-ζ9ζ98+ζ97-ζ94+2ζ92002ζ98-ζ94+ζ92+ζ9ζ95+2ζ94-ζ92+ζ9-ζ98+2ζ97+ζ94+ζ9200    orthogonal faithful
ρ256600-3000-30000000ζ98+ζ97-ζ94+2ζ92ζ98+ζ94-ζ92+2ζ92ζ95+ζ94+ζ92-ζ900ζ98+ζ94-ζ92+2ζ92ζ95+ζ94+ζ92-ζ9ζ98+ζ97-ζ94+2ζ9200    orthogonal lifted from He3.S3
ρ266600-3000-30000000ζ98+ζ94-ζ92+2ζ92ζ95+ζ94+ζ92-ζ9ζ98+ζ97-ζ94+2ζ92002ζ95+ζ94+ζ92-ζ9ζ98+ζ97-ζ94+2ζ92ζ98+ζ94-ζ92+2ζ900    orthogonal lifted from He3.S3

Smallest permutation representation of C2×He3.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 30)(20 31)(21 32)(22 33)(23 34)(24 35)(25 36)(26 28)(27 29)(37 53)(38 54)(39 46)(40 47)(41 48)(42 49)(43 50)(44 51)(45 52)

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

(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 42 33)(3 31 40)(5 45 36)(6 34 43)(8 39 30)(9 28 37)(10 23 50)(12 46 19)(13 26 53)(15 49 22)(16 20 47)(18 52 25)(21 27 24)(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)

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

Matrix representation of C2×He3.S3 ►in GL8(𝔽19)

180000000
018000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
000000181
0011001718
000000180
001000180
000010180
000001180
,
10000000
01000000
001810000
001800000
001800100
0001181800
001800001
0001001818
,
70000000
07000000
000000181
0011001718
001000180
000100180
00111818180
000010180
,
1016000000
188000000
00572141217
00175721412
0000721217
0057001412
001707200
0007001217
,
93000000
510000000
00721751412
00214571217
0000571412
0072001217
0070001412
00221751217

G:=sub<GL(8,GF(19))| [18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,18,17,18,18,18,18,0,0,1,18,0,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,18,18,0,18,0,0,0,1,0,0,1,0,1,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,1,0,1,0,0,0,0,1,0,1,1,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0,18,17,18,18,18,18,0,0,1,18,0,0,0,0],[10,18,0,0,0,0,0,0,16,8,0,0,0,0,0,0,0,0,5,17,0,5,17,0,0,0,7,5,0,7,0,7,0,0,2,7,7,0,7,0,0,0,14,2,2,0,2,0,0,0,12,14,12,14,0,12,0,0,17,12,17,12,0,17],[9,5,0,0,0,0,0,0,3,10,0,0,0,0,0,0,0,0,7,2,0,7,7,2,0,0,2,14,0,2,0,2,0,0,17,5,5,0,0,17,0,0,5,7,7,0,0,5,0,0,14,12,14,12,14,12,0,0,12,17,12,17,12,17] >;
 

C2×He3.S3 in GAP, Magma, Sage, TeX 

C_2\times {\rm He}_3.S_3
 
% in TeX
 
G:=Group("C2xHe3.S3");
 
// GroupNames label
 
G:=SmallGroup(324,71);
 
// by ID
 
G=gap.SmallGroup(324,71);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,5763,303,237,2164,1096,7781]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=f^2=1,e^3=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=b*c^-1,b*e=e*b,f*b*f=b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=b^-1*c*d,d*f=f*d,f*e*f=c*e^2>;
 
// generators/relations
 

Export

Subgroup lattice of C2×He3.S3 in TeX
Character table of C2×He3.S3 in TeX

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