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## G = C3×C3.He3order 243 = 35

### Direct product of C3 and C3.He3

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C3×C3.He3, C32.12He3, C32.4C33, C33.34C32, 3- 1+2.1C32, C3.10(C3×He3), (C32×C9).8C3, (C3×C9).23C32, (C3×3- 1+2).8C3, SmallGroup(243,54)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C32 — C3×C3.He3
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C3×C3.He3
 Lower central C1 — C3 — C32 — C3×C3.He3
 Upper central C1 — C32 — C33 — C3×C3.He3
 Jennings C1 — C3 — C32 — C3×C3.He3

Generators and relations for C3×C3.He3
G = < a,b,c,d,e | a3=b3=d3=1, c3=b-1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=bcd-1, ede-1=b-1d >

Subgroups: 126 in 66 conjugacy classes, 36 normal (7 characteristic)
C1, C3, C3, C3, C9, C32, C32, C32, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, C3.He3, C32×C9, C3×3- 1+2, C3×C3.He3
Quotients: C1, C3, C32, He3, C33, C3.He3, C3×He3, C3×C3.He3

Smallest permutation representation of C3×C3.He3
On 81 points
Generators in S81
(1 41 36)(2 42 28)(3 43 29)(4 44 30)(5 45 31)(6 37 32)(7 38 33)(8 39 34)(9 40 35)(10 81 21)(11 73 22)(12 74 23)(13 75 24)(14 76 25)(15 77 26)(16 78 27)(17 79 19)(18 80 20)(46 59 71)(47 60 72)(48 61 64)(49 62 65)(50 63 66)(51 55 67)(52 56 68)(53 57 69)(54 58 70)
(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 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 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 27 75)(11 19 76)(12 20 77)(13 21 78)(14 22 79)(15 23 80)(16 24 81)(17 25 73)(18 26 74)(46 71 59)(47 72 60)(48 64 61)(49 65 62)(50 66 63)(51 67 55)(52 68 56)(53 69 57)(54 70 58)
(1 79 54 7 76 51 4 73 48)(2 15 56 8 12 62 5 18 59)(3 24 66 9 21 72 6 27 69)(10 47 37 16 53 43 13 50 40)(11 64 36 17 70 33 14 67 30)(19 58 38 25 55 44 22 61 41)(20 46 28 26 52 34 23 49 31)(29 75 63 35 81 60 32 78 57)(39 74 65 45 80 71 42 77 68)

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

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

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

C3×C3.He3 is a maximal subgroup of   (C32×C9).S3

51 conjugacy classes

 class 1 3A ··· 3H 3I ··· 3N 9A ··· 9R 9S ··· 9AJ order 1 3 ··· 3 3 ··· 3 9 ··· 9 9 ··· 9 size 1 1 ··· 1 3 ··· 3 3 ··· 3 9 ··· 9

51 irreducible representations

 dim 1 1 1 1 3 3 type + image C1 C3 C3 C3 He3 C3.He3 kernel C3×C3.He3 C3.He3 C32×C9 C3×3- 1+2 C32 C3 # reps 1 18 2 6 6 18

Matrix representation of C3×C3.He3 in GL4(𝔽19) generated by

 11 0 0 0 0 11 0 0 0 0 11 0 0 0 0 11
,
 1 0 0 0 0 7 0 0 0 0 7 0 0 0 0 7
,
 11 0 0 0 0 17 0 0 0 0 17 0 0 0 0 5
,
 1 0 0 0 0 1 0 0 0 0 7 0 0 0 0 11
,
 7 0 0 0 0 0 1 0 0 0 0 1 0 7 0 0
G:=sub<GL(4,GF(19))| [11,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[1,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[11,0,0,0,0,17,0,0,0,0,17,0,0,0,0,5],[1,0,0,0,0,1,0,0,0,0,7,0,0,0,0,11],[7,0,0,0,0,0,0,7,0,1,0,0,0,0,1,0] >;

C3×C3.He3 in GAP, Magma, Sage, TeX

C_3\times C_3.{\rm He}_3
% in TeX

G:=Group("C3xC3.He3");
// GroupNames label

G:=SmallGroup(243,54);
// by ID

G=gap.SmallGroup(243,54);
# by ID

G:=PCGroup([5,-3,3,3,-3,-3,405,301,546,2163]);
// Polycyclic

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

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