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## G = He3⋊C3⋊2S3order 486 = 2·35

### 1st semidirect product of He3⋊C3 and S3 acting via S3/C3=C2

Aliases: (C32×C9)⋊10S3, He3⋊C32S3, He35S34C3, C3⋊(He3.2C6), (C3×He3).12C6, He3.10(C3×S3), C33.45(C3×S3), C3.15(He34S3), C32.13(C32⋊C6), (C3×C9)⋊3(C3⋊S3), C32.6(C3×C3⋊S3), (C3×He3⋊C3)⋊3C2, SmallGroup(486,172)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C3×He3 — He3⋊C3⋊2S3
 Chief series C1 — C3 — C32 — C33 — C3×He3 — C3×He3⋊C3 — He3⋊C3⋊2S3
 Lower central C3×He3 — He3⋊C3⋊2S3
 Upper central C1 — C3

Generators and relations for He3⋊C32S3
G = < a,b,c,d,e,f | a3=b3=c3=d3=e3=f2=1, ab=ba, cac-1=dad-1=ab-1, ae=ea, faf=a-1b, bc=cb, bd=db, be=eb, bf=fb, dcd-1=abc, ce=ec, fcf=abcd, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 704 in 96 conjugacy classes, 20 normal (12 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C32, C18, C3×S3, C3⋊S3, C3×C9, C3×C9, He3, He3, C33, C33, S3×C9, He3⋊C2, C3×C3⋊S3, He3⋊C3, He3⋊C3, C32×C9, C3×He3, C3×He3, He3.2C6, C9×C3⋊S3, He35S3, C3×He3⋊C3, He3⋊C32S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C32⋊C6, C3×C3⋊S3, He3.2C6, He34S3, He3⋊C32S3

Smallest permutation representation of He3⋊C32S3
On 54 points
Generators in S54
(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 5 4)(2 13 3)(6 15 7)(8 9 14)(10 12 11)(16 17 18)(19 20 21)(22 24 23)(25 26 27)(28 30 29)(31 32 33)(34 36 35)(37 38 39)(40 42 41)(43 44 45)(46 48 47)(49 50 51)(52 54 53)
(1 30 33)(2 41 26)(3 42 25)(4 28 32)(5 29 31)(6 53 38)(7 54 37)(8 36 51)(9 35 49)(10 47 20)(11 48 19)(12 46 21)(13 40 27)(14 34 50)(15 52 39)(16 24 43)(17 23 44)(18 22 45)
(1 41 21)(2 48 31)(3 46 33)(4 42 20)(5 40 19)(6 22 51)(7 23 50)(8 52 45)(9 54 43)(10 30 25)(11 28 27)(12 29 26)(13 47 32)(14 53 44)(15 24 49)(16 34 37)(17 36 38)(18 35 39)
(1 5 4)(2 13 3)(6 7 15)(8 14 9)(10 12 11)(16 18 17)(19 20 21)(22 23 24)(25 26 27)(28 30 29)(31 32 33)(34 35 36)(37 39 38)(40 42 41)(43 45 44)(46 48 47)(49 51 50)(52 53 54)
(1 44)(2 51)(3 50)(4 43)(5 45)(6 31)(7 33)(8 19)(9 20)(10 37)(11 39)(12 38)(13 49)(14 21)(15 32)(16 25)(17 26)(18 27)(22 48)(23 46)(24 47)(28 35)(29 36)(30 34)(40 52)(41 53)(42 54)

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 2 2 2 2 3 6 6 type + + + + + image C1 C2 C3 C6 S3 S3 C3×S3 C3×S3 He3.2C6 C32⋊C6 He3⋊C3⋊2S3 kernel He3⋊C3⋊2S3 C3×He3⋊C3 He3⋊5S3 C3×He3 He3⋊C3 C32×C9 He3 C33 C3 C32 C1 # reps 1 1 2 2 3 1 6 2 12 3 6

Matrix representation of He3⋊C32S3 in GL5(𝔽19)

 1 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 1 0 0 0 0 0 7
,
 1 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 11 0 0 0 0 0 11
,
 11 0 0 0 0 0 11 0 0 0 0 0 0 0 16 0 0 17 0 0 0 0 0 16 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 7 0 0 0 0 0 11 0
,
 0 18 0 0 0 1 18 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 18 0 0 0 18 0 0 0 0 0 0 0 8 0 0 0 12 0 0 0 0 0 0 18

G:=sub<GL(5,GF(19))| [1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11],[11,0,0,0,0,0,11,0,0,0,0,0,0,17,0,0,0,0,0,16,0,0,16,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,11,0,0,1,0,0],[0,1,0,0,0,18,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,18,0,0,0,18,0,0,0,0,0,0,0,12,0,0,0,8,0,0,0,0,0,0,18] >;

He3⋊C32S3 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes C_3\rtimes_2S_3
% in TeX

G:=Group("He3:C3:2S3");
// GroupNames label

G:=SmallGroup(486,172);
// by ID

G=gap.SmallGroup(486,172);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,218,548,867,8104,1096]);
// Polycyclic

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

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