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## G = C32×3- 1+2order 243 = 35

### Direct product of C32 and 3- 1+2

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

Aliases: C32×3- 1+2, C9⋊C33, C34.4C3, C3.2C34, C32.11C33, C33.36C32, (C32×C9)⋊9C3, (C3×C9)⋊8C32, SmallGroup(243,63)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C3 — C32×3- 1+2
 Chief series C1 — C3 — C32 — C33 — C34 — C32×3- 1+2
 Lower central C1 — C3 — C32×3- 1+2
 Upper central C1 — C33 — C32×3- 1+2
 Jennings C1 — C3 — C3 — C32×3- 1+2

Generators and relations for C32×3- 1+2
G = < a,b,c,d | a3=b3=c9=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Subgroups: 396 in 288 conjugacy classes, 234 normal (5 characteristic)
C1, C3, C3, C3, C9, C32, C32, C3×C9, 3- 1+2, C33, C33, C33, C32×C9, C3×3- 1+2, C34, C32×3- 1+2
Quotients: C1, C3, C32, 3- 1+2, C33, C3×3- 1+2, C34, C32×3- 1+2

Smallest permutation representation of C32×3- 1+2
On 81 points
Generators in S81
(1 53 33)(2 54 34)(3 46 35)(4 47 36)(5 48 28)(6 49 29)(7 50 30)(8 51 31)(9 52 32)(10 26 40)(11 27 41)(12 19 42)(13 20 43)(14 21 44)(15 22 45)(16 23 37)(17 24 38)(18 25 39)(55 68 75)(56 69 76)(57 70 77)(58 71 78)(59 72 79)(60 64 80)(61 65 81)(62 66 73)(63 67 74)
(1 78 19)(2 79 20)(3 80 21)(4 81 22)(5 73 23)(6 74 24)(7 75 25)(8 76 26)(9 77 27)(10 31 69)(11 32 70)(12 33 71)(13 34 72)(14 35 64)(15 36 65)(16 28 66)(17 29 67)(18 30 68)(37 48 62)(38 49 63)(39 50 55)(40 51 56)(41 52 57)(42 53 58)(43 54 59)(44 46 60)(45 47 61)
(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 78 19)(2 76 23)(3 74 27)(4 81 22)(5 79 26)(6 77 21)(7 75 25)(8 73 20)(9 80 24)(10 28 72)(11 35 67)(12 33 71)(13 31 66)(14 29 70)(15 36 65)(16 34 69)(17 32 64)(18 30 68)(37 54 56)(38 52 60)(39 50 55)(40 48 59)(41 46 63)(42 53 58)(43 51 62)(44 49 57)(45 47 61)

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

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

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

C32×3- 1+2 is a maximal subgroup of   C34.11S3

99 conjugacy classes

 class 1 3A ··· 3Z 3AA ··· 3AR 9A ··· 9BB order 1 3 ··· 3 3 ··· 3 9 ··· 9 size 1 1 ··· 1 3 ··· 3 3 ··· 3

99 irreducible representations

 dim 1 1 1 1 3 type + image C1 C3 C3 C3 3- 1+2 kernel C32×3- 1+2 C32×C9 C3×3- 1+2 C34 C32 # reps 1 6 72 2 18

Matrix representation of C32×3- 1+2 in GL5(𝔽19)

 11 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7
,
 11 0 0 0 0 0 11 0 0 0 0 0 11 0 0 0 0 0 11 0 0 0 0 0 11
,
 1 0 0 0 0 0 11 0 0 0 0 0 7 9 0 0 0 1 12 1 0 0 0 18 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 7 1 0 0 0 8 0 7

G:=sub<GL(5,GF(19))| [11,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[11,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11],[1,0,0,0,0,0,11,0,0,0,0,0,7,1,0,0,0,9,12,18,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,11,7,8,0,0,0,1,0,0,0,0,0,7] >;

C32×3- 1+2 in GAP, Magma, Sage, TeX

C_3^2\times 3_-^{1+2}
% in TeX

G:=Group("C3^2xES-(3,1)");
// GroupNames label

G:=SmallGroup(243,63);
// by ID

G=gap.SmallGroup(243,63);
# by ID

G:=PCGroup([5,-3,3,3,3,-3,405,841]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^9=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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