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## G = C9×3- 1+2order 243 = 35

### Direct product of C9 and 3- 1+2

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

Aliases: C9×3- 1+2, C926C3, C32.19C33, C33.26C32, C91(C3×C9), C9⋊C911C3, C3.5(C32×C9), C32.5(C3×C9), (C32×C9).6C3, C32⋊C9.15C3, C3.3(C9○He3), (C3×C9).22C32, C3.3(C3×3- 1+2), (C3×3- 1+2).10C3, (C3×C9)(C9⋊C9), (C3×C9)(C3×3- 1+2), SmallGroup(243,36)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C3 — C9×3- 1+2
 Chief series C1 — C3 — C32 — C3×C9 — C92 — C9×3- 1+2
 Lower central C1 — C3 — C9×3- 1+2
 Upper central C1 — C3×C9 — C9×3- 1+2
 Jennings C1 — C32 — C32 — C9×3- 1+2

Generators and relations for C9×3- 1+2
G = < a,b,c | a9=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Subgroups: 108 in 74 conjugacy classes, 57 normal (9 characteristic)
C1, C3, C3, C3, C9, C9, C32, C32, C32, C3×C9, C3×C9, C3×C9, 3- 1+2, C33, C92, C32⋊C9, C9⋊C9, C32×C9, C3×3- 1+2, C9×3- 1+2
Quotients: C1, C3, C9, C32, C3×C9, 3- 1+2, C33, C32×C9, C3×3- 1+2, C9○He3, C9×3- 1+2

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

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

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

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

C9×3- 1+2 is a maximal subgroup of   C929C6

99 conjugacy classes

 class 1 3A ··· 3H 3I ··· 3N 9A ··· 9R 9S ··· 9CF 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 1 1 3 3 type + image C1 C3 C3 C3 C3 C3 C9 3- 1+2 C9○He3 kernel C9×3- 1+2 C92 C32⋊C9 C9⋊C9 C32×C9 C3×3- 1+2 3- 1+2 C9 C3 # reps 1 6 4 12 2 2 54 6 12

Matrix representation of C9×3- 1+2 in GL4(𝔽19) generated by

 17 0 0 0 0 7 0 0 0 0 7 0 0 0 0 7
,
 7 0 0 0 0 5 4 0 0 5 14 1 0 12 12 0
,
 11 0 0 0 0 1 0 0 0 17 7 0 0 8 0 11
G:=sub<GL(4,GF(19))| [17,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[7,0,0,0,0,5,5,12,0,4,14,12,0,0,1,0],[11,0,0,0,0,1,17,8,0,0,7,0,0,0,0,11] >;

C9×3- 1+2 in GAP, Magma, Sage, TeX

C_9\times 3_-^{1+2}
% in TeX

G:=Group("C9xES-(3,1)");
// GroupNames label

G:=SmallGroup(243,36);
// by ID

G=gap.SmallGroup(243,36);
# by ID

G:=PCGroup([5,-3,3,3,-3,3,405,301,96,147]);
// Polycyclic

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

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