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## G = C9×He3⋊C2order 486 = 2·35

### Direct product of C9 and He3⋊C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for C9×He3⋊C2
G = < a,b,c,d,e | a9=b3=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe=b-1, cd=dc, ce=ec, ede=d-1 >

Subgroups: 402 in 130 conjugacy classes, 28 normal (11 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, C18, C3×S3, C3×C6, C3×C9, C3×C9, He3, He3, C33, S3×C9, He3⋊C2, C3×C18, S3×C32, C32⋊C9, C32×C9, C3×He3, S3×C3×C9, C3×He3⋊C2, C9×He3, C9×He3⋊C2
Quotients: C1, C2, C3, S3, C6, C9, C18, C3×S3, C3⋊S3, S3×C9, He3⋊C2, C3×C3⋊S3, C9×C3⋊S3, C3×He3⋊C2, He3.4C6, C9×He3⋊C2

Smallest permutation representation of C9×He3⋊C2
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)
(1 33 27)(2 34 19)(3 35 20)(4 36 21)(5 28 22)(6 29 23)(7 30 24)(8 31 25)(9 32 26)(10 64 76)(11 65 77)(12 66 78)(13 67 79)(14 68 80)(15 69 81)(16 70 73)(17 71 74)(18 72 75)(37 55 46)(38 56 47)(39 57 48)(40 58 49)(41 59 50)(42 60 51)(43 61 52)(44 62 53)(45 63 54)
(1 69 51)(2 70 52)(3 71 53)(4 72 54)(5 64 46)(6 65 47)(7 66 48)(8 67 49)(9 68 50)(10 55 22)(11 56 23)(12 57 24)(13 58 25)(14 59 26)(15 60 27)(16 61 19)(17 62 20)(18 63 21)(28 76 37)(29 77 38)(30 78 39)(31 79 40)(32 80 41)(33 81 42)(34 73 43)(35 74 44)(36 75 45)
(10 22 55)(11 23 56)(12 24 57)(13 25 58)(14 26 59)(15 27 60)(16 19 61)(17 20 62)(18 21 63)(28 76 37)(29 77 38)(30 78 39)(31 79 40)(32 80 41)(33 81 42)(34 73 43)(35 74 44)(36 75 45)
(10 76)(11 77)(12 78)(13 79)(14 80)(15 81)(16 73)(17 74)(18 75)(19 34)(20 35)(21 36)(22 28)(23 29)(24 30)(25 31)(26 32)(27 33)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)

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

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

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

90 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3T 6A ··· 6H 9A ··· 9R 9S ··· 9AP 18A ··· 18R order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 9 1 ··· 1 6 ··· 6 9 ··· 9 1 ··· 1 6 ··· 6 9 ··· 9

90 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 3 type + + + image C1 C2 C3 C6 C9 C18 S3 C3×S3 S3×C9 He3⋊C2 He3.4C6 kernel C9×He3⋊C2 C9×He3 C3×He3⋊C2 C3×He3 He3⋊C2 He3 C32×C9 C33 C32 C9 C3 # reps 1 1 2 2 6 6 4 8 24 12 24

Matrix representation of C9×He3⋊C2 in GL5(𝔽19)

 9 0 0 0 0 0 9 0 0 0 0 0 6 0 0 0 0 0 6 0 0 0 0 0 6
,
 18 18 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 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
,
 18 18 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 7 0 0 0 0 0 11
,
 1 0 0 0 0 18 18 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(19))| [9,0,0,0,0,0,9,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,6],[18,1,0,0,0,18,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[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],[18,1,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,11],[1,18,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C9×He3⋊C2 in GAP, Magma, Sage, TeX

C_9\times {\rm He}_3\rtimes C_2
% in TeX

G:=Group("C9xHe3:C2");
// GroupNames label

G:=SmallGroup(486,143);
// by ID

G=gap.SmallGroup(486,143);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,867,3244,382]);
// Polycyclic

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

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