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## G = C2×C33.S3order 324 = 22·34

### Direct product of C2 and C33.S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C2×C33.S3
 Chief series C1 — C3 — C32 — C3×C9 — C3×3- 1+2 — C33.S3 — C2×C33.S3
 Lower central C3×C9 — C2×C33.S3
 Upper central C1 — C2

Generators and relations for C2×C33.S3
G = < a,b,c,d,e,f | a2=b3=c3=d3=f2=1, e3=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=bd-1, bf=fb, cd=dc, ce=ec, fcf=c-1, de=ed, fdf=d-1, fef=d-1e2 >

Subgroups: 556 in 118 conjugacy classes, 38 normal (16 characteristic)
C1, C2, C2, C3, C3, C3, C22, S3, C6, C6, C6, C9, C9, C32, C32, D6, C2×C6, D9, C18, C18, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, D18, S3×C6, C2×C3⋊S3, C9⋊C6, C9⋊S3, C3×C18, C3×C18, C2×3- 1+2, C2×3- 1+2, C3×C3⋊S3, C32×C6, C3×3- 1+2, C2×C9⋊C6, C2×C9⋊S3, C6×C3⋊S3, C33.S3, C6×3- 1+2, C2×C33.S3
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, C3⋊S3, S3×C6, C2×C3⋊S3, C9⋊C6, C3×C3⋊S3, C2×C9⋊C6, C6×C3⋊S3, C33.S3, C2×C33.S3

Smallest permutation representation of C2×C33.S3
On 54 points
Generators in S54
(1 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 35)(11 36)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(37 52)(38 53)(39 54)(40 46)(41 47)(42 48)(43 49)(44 50)(45 51)
(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 22 25)(21 27 24)(28 34 31)(29 32 35)(37 40 43)(39 45 42)(46 49 52)(48 54 51)
(1 30 38)(2 31 39)(3 32 40)(4 33 41)(5 34 42)(6 35 43)(7 36 44)(8 28 45)(9 29 37)(10 49 19)(11 50 20)(12 51 21)(13 52 22)(14 53 23)(15 54 24)(16 46 25)(17 47 26)(18 48 27)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(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)
(1 23)(2 22)(3 21)(4 20)(5 19)(6 27)(7 26)(8 25)(9 24)(10 42)(11 41)(12 40)(13 39)(14 38)(15 37)(16 45)(17 44)(18 43)(28 46)(29 54)(30 53)(31 52)(32 51)(33 50)(34 49)(35 48)(36 47)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 9A ··· 9I 18A ··· 18I order 1 2 2 2 3 3 3 3 3 3 3 3 6 6 6 6 6 6 6 6 6 6 6 6 9 ··· 9 18 ··· 18 size 1 1 27 27 2 2 2 2 3 3 6 6 2 2 2 2 3 3 6 6 27 27 27 27 6 ··· 6 6 ··· 6

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 6 6 type + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 S3 D6 D6 C3×S3 C3×S3 S3×C6 S3×C6 C9⋊C6 C2×C9⋊C6 kernel C2×C33.S3 C33.S3 C6×3- 1+2 C2×C9⋊S3 C9⋊S3 C3×C18 C2×3- 1+2 C32×C6 3- 1+2 C33 C18 C3×C6 C9 C32 C6 C3 # reps 1 2 1 2 4 2 3 1 3 1 6 2 6 2 3 3

Matrix representation of C2×C33.S3 in GL8(𝔽19)

 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 18
,
 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 18 18 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 18 18
,
 1 3 0 0 0 0 0 0 18 17 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 18 0 0 0 0 0 0 0 18 0 0 1 0 0 0 0 0 1 18 18 0 0 0 0 18 0 0 0 0 1 0 0 0 1 0 0 18 18
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 18 0 0 0 0 0 0 0 18 0 0 1 0 0 0 0 0 1 18 18 0 0 0 0 18 0 0 0 0 1 0 0 0 1 0 0 18 18
,
 1 3 0 0 0 0 0 0 18 17 0 0 0 0 0 0 0 0 1 1 0 0 18 17 0 0 0 0 0 0 1 18 0 0 0 0 0 0 0 18 0 0 1 0 0 0 0 18 0 0 0 0 1 0 0 18 0 0 0 0 0 1 0 18
,
 18 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 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

G:=sub<GL(8,GF(19))| [18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18],[7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,1,0,1,0,0,1,0,0,0,1,1,0,0,1,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[1,18,0,0,0,0,0,0,3,17,0,0,0,0,0,0,0,0,18,18,18,0,18,0,0,0,1,0,0,1,0,1,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[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],[1,18,0,0,0,0,0,0,3,17,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,18,1,0,0,0,0,0,0,17,18,18,18,18,18],[18,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0] >;

C2×C33.S3 in GAP, Magma, Sage, TeX

C_2\times C_3^3.S_3
% in TeX

G:=Group("C2xC3^3.S3");
// GroupNames label

G:=SmallGroup(324,146);
// by ID

G=gap.SmallGroup(324,146);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,3171,735,453,2164,7781]);
// Polycyclic

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

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