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## G = C9○He3⋊3S3order 486 = 2·35

### 1st semidirect product of C9○He3 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C9 — C9○He3⋊3S3
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C3×C9○He3 — C9○He3⋊3S3
 Lower central C32×C9 — C9○He3⋊3S3
 Upper central C1

Generators and relations for C9○He33S3
G = < a,b,c,d,e,f | a9=b3=d3=e3=f2=1, c1=a6, ab=ba, ac=ca, ad=da, ae=ea, faf=a-1, bc=cb, dbd-1=a3b, be=eb, fbf=b-1, cd=dc, ce=ec, fcf=a3, de=ed, fdf=bd, fef=e-1 >

Subgroups: 1304 in 225 conjugacy classes, 62 normal (13 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, C3×C9, He3, He3, 3- 1+2, 3- 1+2, C33, C33, C3×D9, C32⋊C6, C9⋊C6, C9⋊S3, C3×C3⋊S3, C33⋊C2, C32×C9, C32×C9, C3×He3, C3×3- 1+2, C3×3- 1+2, C9○He3, C9○He3, C3×C9⋊S3, He34S3, C33.S3, He3.4S3, C324D9, C3×C9○He3, C9○He33S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C3×C3⋊S3, C33⋊C2, He3.4S3, C3×C33⋊C2, C9○He33S3

Smallest permutation representation of C9○He33S3
On 81 points
Generators in S81
(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)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81)
(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 79 76)(74 80 77)(75 81 78)
(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)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)(73 79 76)(74 80 77)(75 81 78)
(1 64 24)(2 65 25)(3 66 26)(4 67 27)(5 68 19)(6 69 20)(7 70 21)(8 71 22)(9 72 23)(10 32 59)(11 33 60)(12 34 61)(13 35 62)(14 36 63)(15 28 55)(16 29 56)(17 30 57)(18 31 58)(37 51 81)(38 52 73)(39 53 74)(40 54 75)(41 46 76)(42 47 77)(43 48 78)(44 49 79)(45 50 80)
(1 39 30)(2 40 31)(3 41 32)(4 42 33)(5 43 34)(6 44 35)(7 45 36)(8 37 28)(9 38 29)(10 26 76)(11 27 77)(12 19 78)(13 20 79)(14 21 80)(15 22 81)(16 23 73)(17 24 74)(18 25 75)(46 59 66)(47 60 67)(48 61 68)(49 62 69)(50 63 70)(51 55 71)(52 56 72)(53 57 64)(54 58 65)
(2 9)(3 8)(4 7)(5 6)(10 75)(11 74)(12 73)(13 81)(14 80)(15 79)(16 78)(17 77)(18 76)(19 23)(20 22)(24 27)(25 26)(28 41)(29 40)(30 39)(31 38)(32 37)(33 45)(34 44)(35 43)(36 42)(46 55)(47 63)(48 62)(49 61)(50 60)(51 59)(52 58)(53 57)(54 56)(65 72)(66 71)(67 70)(68 69)

G:=sub<Sym(81)| (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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81), (10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,79,76)(74,80,77)(75,81,78), (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)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69)(73,79,76)(74,80,77)(75,81,78), (1,64,24)(2,65,25)(3,66,26)(4,67,27)(5,68,19)(6,69,20)(7,70,21)(8,71,22)(9,72,23)(10,32,59)(11,33,60)(12,34,61)(13,35,62)(14,36,63)(15,28,55)(16,29,56)(17,30,57)(18,31,58)(37,51,81)(38,52,73)(39,53,74)(40,54,75)(41,46,76)(42,47,77)(43,48,78)(44,49,79)(45,50,80), (1,39,30)(2,40,31)(3,41,32)(4,42,33)(5,43,34)(6,44,35)(7,45,36)(8,37,28)(9,38,29)(10,26,76)(11,27,77)(12,19,78)(13,20,79)(14,21,80)(15,22,81)(16,23,73)(17,24,74)(18,25,75)(46,59,66)(47,60,67)(48,61,68)(49,62,69)(50,63,70)(51,55,71)(52,56,72)(53,57,64)(54,58,65), (2,9)(3,8)(4,7)(5,6)(10,75)(11,74)(12,73)(13,81)(14,80)(15,79)(16,78)(17,77)(18,76)(19,23)(20,22)(24,27)(25,26)(28,41)(29,40)(30,39)(31,38)(32,37)(33,45)(34,44)(35,43)(36,42)(46,55)(47,63)(48,62)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56)(65,72)(66,71)(67,70)(68,69)>;

G:=Group( (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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81), (10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,79,76)(74,80,77)(75,81,78), (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)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69)(73,79,76)(74,80,77)(75,81,78), (1,64,24)(2,65,25)(3,66,26)(4,67,27)(5,68,19)(6,69,20)(7,70,21)(8,71,22)(9,72,23)(10,32,59)(11,33,60)(12,34,61)(13,35,62)(14,36,63)(15,28,55)(16,29,56)(17,30,57)(18,31,58)(37,51,81)(38,52,73)(39,53,74)(40,54,75)(41,46,76)(42,47,77)(43,48,78)(44,49,79)(45,50,80), (1,39,30)(2,40,31)(3,41,32)(4,42,33)(5,43,34)(6,44,35)(7,45,36)(8,37,28)(9,38,29)(10,26,76)(11,27,77)(12,19,78)(13,20,79)(14,21,80)(15,22,81)(16,23,73)(17,24,74)(18,25,75)(46,59,66)(47,60,67)(48,61,68)(49,62,69)(50,63,70)(51,55,71)(52,56,72)(53,57,64)(54,58,65), (2,9)(3,8)(4,7)(5,6)(10,75)(11,74)(12,73)(13,81)(14,80)(15,79)(16,78)(17,77)(18,76)(19,23)(20,22)(24,27)(25,26)(28,41)(29,40)(30,39)(31,38)(32,37)(33,45)(34,44)(35,43)(36,42)(46,55)(47,63)(48,62)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56)(65,72)(66,71)(67,70)(68,69) );

G=PermutationGroup([[(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),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81)], [(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72),(73,79,76),(74,80,77),(75,81,78)], [(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),(55,61,58),(56,62,59),(57,63,60),(64,70,67),(65,71,68),(66,72,69),(73,79,76),(74,80,77),(75,81,78)], [(1,64,24),(2,65,25),(3,66,26),(4,67,27),(5,68,19),(6,69,20),(7,70,21),(8,71,22),(9,72,23),(10,32,59),(11,33,60),(12,34,61),(13,35,62),(14,36,63),(15,28,55),(16,29,56),(17,30,57),(18,31,58),(37,51,81),(38,52,73),(39,53,74),(40,54,75),(41,46,76),(42,47,77),(43,48,78),(44,49,79),(45,50,80)], [(1,39,30),(2,40,31),(3,41,32),(4,42,33),(5,43,34),(6,44,35),(7,45,36),(8,37,28),(9,38,29),(10,26,76),(11,27,77),(12,19,78),(13,20,79),(14,21,80),(15,22,81),(16,23,73),(17,24,74),(18,25,75),(46,59,66),(47,60,67),(48,61,68),(49,62,69),(50,63,70),(51,55,71),(52,56,72),(53,57,64),(54,58,65)], [(2,9),(3,8),(4,7),(5,6),(10,75),(11,74),(12,73),(13,81),(14,80),(15,79),(16,78),(17,77),(18,76),(19,23),(20,22),(24,27),(25,26),(28,41),(29,40),(30,39),(31,38),(32,37),(33,45),(34,44),(35,43),(36,42),(46,55),(47,63),(48,62),(49,61),(50,60),(51,59),(52,58),(53,57),(54,56),(65,72),(66,71),(67,70),(68,69)]])

54 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G ··· 3Q 6A 6B 9A ··· 9I 9J ··· 9AG order 1 2 3 3 3 3 3 3 3 ··· 3 6 6 9 ··· 9 9 ··· 9 size 1 81 2 2 2 2 3 3 6 ··· 6 81 81 2 ··· 2 6 ··· 6

54 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 6 type + + + + + + + image C1 C2 C3 C6 S3 S3 S3 S3 C3×S3 C3×S3 He3.4S3 kernel C9○He3⋊3S3 C3×C9○He3 C32⋊4D9 C32×C9 C32×C9 C3×He3 C3×3- 1+2 C9○He3 C3×C9 C33 C3 # reps 1 1 2 2 1 1 2 9 24 2 9

Matrix representation of C9○He33S3 in GL8(𝔽19)

 0 1 0 0 0 0 0 0 18 18 0 0 0 0 0 0 0 0 12 2 0 0 0 0 0 0 17 14 0 0 0 0 0 0 0 0 12 2 0 0 0 0 0 0 17 14 0 0 0 0 0 13 2 0 14 2 0 0 13 6 0 17 17 12
,
 18 18 0 0 0 0 0 0 1 0 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 18 1 0 0 0 0 0 0 18 0 0 0 0 0 16 6 0 18 18 18 0 0 0 0 1 0 1 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 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 0 0 0 0 0 3 18 0 18 18 0 0 3 16 0 1 1 0
,
 18 18 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 3 13 1 1 2 1 0 0 3 13 1 1 1 2 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 8 7 0 3 0 0 9 0 7 8 0 3
,
 0 1 0 0 0 0 0 0 18 18 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 18 0 0 0 0 0 0 16 1 0 0 1 0 0 16 3 0 18 18 18
,
 1 0 0 0 0 0 0 0 18 18 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 3 16 0 0 1 0 0 0 0 3 18 18 18 18

G:=sub<GL(8,GF(19))| [0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,12,17,0,0,0,13,0,0,2,14,0,0,13,6,0,0,0,0,12,17,2,0,0,0,0,0,2,14,0,17,0,0,0,0,0,0,14,17,0,0,0,0,0,0,2,12],[18,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,1,0,0,6,0,0,0,0,0,18,18,0,1,0,0,0,0,1,0,18,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,3,0,0,18,18,0,0,3,16,0,0,0,0,0,1,18,0,0,0,0,0,18,18,0,1,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0],[18,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,3,3,1,0,9,9,0,0,13,13,0,1,0,0,0,0,1,1,0,0,8,7,0,0,1,1,0,0,7,8,0,0,2,1,0,0,0,0,0,0,1,2,0,0,3,3],[0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,18,18,0,0,0,16,0,0,1,0,0,0,16,3,0,0,0,0,18,18,1,0,0,0,0,0,1,0,0,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[1,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,1,0,0,0,16,3,0,0,0,0,0,1,0,18,0,0,0,0,1,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,18] >;

C9○He33S3 in GAP, Magma, Sage, TeX

C_9\circ {\rm He}_3\rtimes_3S_3
% in TeX

G:=Group("C9oHe3:3S3");
// GroupNames label

G:=SmallGroup(486,245);
// by ID

G=gap.SmallGroup(486,245);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,3134,986,867,2169,3244,11669]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^9=b^3=d^3=e^3=f^2=1,c^1=a^6,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f=a^-1,b*c=c*b,d*b*d^-1=a^3*b,b*e=e*b,f*b*f=b^-1,c*d=d*c,c*e=e*c,f*c*f=a^3,d*e=e*d,f*d*f=b*d,f*e*f=e^-1>;
// generators/relations

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