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## G = C9×He3order 243 = 35

### Direct product of C9 and He3

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C9×He3, C33.25C32, C32.18C33, C32⋊C99C3, C321(C3×C9), (C32×C9)⋊3C3, C3.2(C3×He3), (C3×He3).9C3, C3.4(C32×C9), C3.2(C9○He3), (C3×C9).21C32, (C3×C9)(C3×He3), (C3×C9)(C32⋊C9), SmallGroup(243,35)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C9×He3
G = < a,b,c,d | a9=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >

Subgroups: 189 in 101 conjugacy classes, 57 normal (7 characteristic)
C1, C3, C3, C3, C9, C9, C32, C32, C32, C3×C9, C3×C9, C3×C9, He3, C33, C32⋊C9, C32×C9, C3×He3, C9×He3
Quotients: C1, C3, C9, C32, C3×C9, He3, C33, C32×C9, C3×He3, C9○He3, C9×He3

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

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

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

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

C9×He3 is a maximal subgroup of   He33D9  He34D9

99 conjugacy classes

 class 1 3A ··· 3H 3I ··· 3AF 9A ··· 9R 9S ··· 9BN order 1 3 ··· 3 3 ··· 3 9 ··· 9 9 ··· 9 size 1 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3

99 irreducible representations

 dim 1 1 1 1 1 3 3 type + image C1 C3 C3 C3 C9 He3 C9○He3 kernel C9×He3 C32⋊C9 C32×C9 C3×He3 He3 C9 C3 # reps 1 16 8 2 54 6 12

Matrix representation of C9×He3 in GL4(𝔽19) generated by

 5 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 7 13 0 0 0 12 1 0 0 8 0
,
 1 0 0 0 0 11 0 0 0 0 11 0 0 0 0 11
,
 11 0 0 0 0 11 15 0 0 18 8 11 0 1 18 0
G:=sub<GL(4,GF(19))| [5,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,7,0,0,0,13,12,8,0,0,1,0],[1,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[11,0,0,0,0,11,18,1,0,15,8,18,0,0,11,0] >;

C9×He3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(243,35);
// by ID

G=gap.SmallGroup(243,35);
# by ID

G:=PCGroup([5,-3,3,3,-3,3,405,301,147]);
// Polycyclic

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

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