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## G = C2×S3×3- 1+2order 324 = 22·34

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×S3×3- 1+2
 Chief series C1 — C3 — C32 — C3×C9 — C3×3- 1+2 — S3×3- 1+2 — C2×S3×3- 1+2
 Lower central C3 — C32 — C2×S3×3- 1+2
 Upper central C1 — C6 — C2×3- 1+2

Generators and relations for C2×S3×3- 1+2
G = < a,b,c,d,e | a2=b3=c2=d9=e3=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >

Subgroups: 220 in 102 conjugacy classes, 49 normal (20 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C9, C9, C32, C32, D6, C2×C6, C18, C18, C3×S3, C3×S3, C3×C6, C3×C6, C3×C9, 3- 1+2, 3- 1+2, C33, C2×C18, S3×C6, S3×C6, C62, S3×C9, C3×C18, C2×3- 1+2, C2×3- 1+2, S3×C32, C32×C6, C3×3- 1+2, S3×C18, C22×3- 1+2, S3×C3×C6, S3×3- 1+2, C6×3- 1+2, C2×S3×3- 1+2
Quotients: C1, C2, C3, C22, S3, C6, C32, D6, C2×C6, C3×S3, C3×C6, 3- 1+2, S3×C6, C62, C2×3- 1+2, S3×C32, C22×3- 1+2, S3×C3×C6, S3×3- 1+2, C2×S3×3- 1+2

Smallest permutation representation of C2×S3×3- 1+2
On 36 points
Generators in S36
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 10)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 19)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)
(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)
(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 22 25)(21 27 24)(28 31 34)(30 36 33)

G:=sub<Sym(36)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,10)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,19)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (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), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,22,25)(21,27,24)(28,31,34)(30,36,33)>;

G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,10)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,19)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (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), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,22,25)(21,27,24)(28,31,34)(30,36,33) );

G=PermutationGroup([[(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,10),(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,19),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36)], [(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)], [(2,8,5),(3,6,9),(10,13,16),(12,18,15),(19,22,25),(21,27,24),(28,31,34),(30,36,33)]])

66 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 3H 3I 6A 6B 6C 6D 6E 6F ··· 6K 6L 6M 6N 6O 6P 6Q 9A ··· 9F 9G ··· 9L 18A ··· 18F 18G ··· 18L 18M ··· 18X order 1 2 2 2 3 3 3 3 3 3 3 3 3 6 6 6 6 6 6 ··· 6 6 6 6 6 6 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 18 ··· 18 size 1 1 3 3 1 1 2 2 2 3 3 6 6 1 1 2 2 2 3 ··· 3 6 6 9 9 9 9 3 ··· 3 6 ··· 6 3 ··· 3 6 ··· 6 9 ··· 9

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 6 6 type + + + + + image C1 C2 C2 C3 C3 C6 C6 C6 C6 S3 D6 C3×S3 C3×S3 S3×C6 S3×C6 3- 1+2 C2×3- 1+2 C2×3- 1+2 S3×3- 1+2 C2×S3×3- 1+2 kernel C2×S3×3- 1+2 S3×3- 1+2 C6×3- 1+2 S3×C18 S3×C3×C6 S3×C9 C3×C18 S3×C32 C32×C6 C2×3- 1+2 3- 1+2 C18 C3×C6 C9 C32 D6 S3 C6 C2 C1 # reps 1 2 1 6 2 12 6 4 2 1 1 6 2 6 2 2 4 2 2 2

Matrix representation of C2×S3×3- 1+2 in GL5(𝔽19)

 18 0 0 0 0 0 18 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 18 1 0 0 0 18 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 7 0 0 0 0 0 7 0 0 0 0 0 0 0 11 0 0 11 0 0 0 0 0 1 0
,
 7 0 0 0 0 0 7 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 7

G:=sub<GL(5,GF(19))| [18,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[18,18,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[7,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,1,0,0,11,0,0],[7,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,7] >;

C2×S3×3- 1+2 in GAP, Magma, Sage, TeX

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

G:=Group("C2xS3xES-(3,1)");
// GroupNames label

G:=SmallGroup(324,141);
// by ID

G=gap.SmallGroup(324,141);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,303,93,7781]);
// Polycyclic

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

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