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G = C9⋊3- 1+2  order 243 = 35

The semidirect product of C9 and 3- 1+2 acting via 3- 1+2/C32=C3

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

Aliases: C9⋊33- 1+2, C33.30C32, C32.24C33, C32.33- 1+2, C9⋊C9⋊3C3, C32⋊C9.8C3, (C3×C9).7C32, C3.6(C9○He3), (C32×C9).12C3, C3.6(C3×3- 1+2), (C3×3- 1+2).4C3, SmallGroup(243,41)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C32 — C9⋊3- 1+2
Chief series
C1 — C3 — C32 — C33 — C32×C9 — C9⋊3- 1+2
Lower central
C1 — C32 — C9⋊3- 1+2
Upper central
C1 — C32 — 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, bab-1=a7, ac=ca, cbc-1=b4 >

Subgroups: 108 in 62 conjugacy classes, 39 normal (9 characteristic)
C1, C3, C3, C3, C9, C9, C32, C32, C32, C3×C9, C3×C9, C3×C9, 3- 1+2, C33, C32⋊C9, C9⋊C9, C32×C9, C3×3- 1+2, C9⋊3- 1+2
Quotients: C1, C3, C32, 3- 1+2, C33, 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 71 41 27 58 28 79 49)(2 18 69 42 22 56 29 74 47)(3 13 67 43 26 63 30 78 54)(4 17 65 44 21 61 31 73 52)(5 12 72 45 25 59 32 77 50)(6 16 70 37 20 57 33 81 48)(7 11 68 38 24 55 34 76 46)(8 15 66 39 19 62 35 80 53)(9 10 64 40 23 60 36 75 51)

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

C9⋊3- 1+2 is a maximal subgroup of   D9⋊3- 1+2

51 conjugacy classes

class 1 3A···3H3I···3N9A···9R9S···9AJ
order13···33···39···99···9
size11···13···33···39···9

51 irreducible representations

dim11111333
type+
imageC1C3C3C3C33- 1+23- 1+2C9○He3
kernelC9⋊3- 1+2C32⋊C9C9⋊C9C32×C9C3×3- 1+2C9C32C3
# reps1418226612

Matrix representation of C9⋊3- 1+2 ►in GL6(𝔽19)

1700000
0160000
005000
0001100
0000110
0000011
,
070000
001000
700000
000010
000007
000100
,
700000
010000
0011000
000100
000070
0000011

G:=sub<GL(6,GF(19))| [17,0,0,0,0,0,0,16,0,0,0,0,0,0,5,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[0,0,7,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,7,0],[7,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,11] >;
 

C9⋊3- 1+2 in GAP, Magma, Sage, TeX 

C_9\rtimes 3_-^{1+2}
 
% in TeX
 
G:=Group("C9:ES-(3,1)");
 
// GroupNames label
 
G:=SmallGroup(243,41);
 
// by ID
 
G=gap.SmallGroup(243,41);
 
# by ID
 
G:=PCGroup([5,-3,3,3,-3,3,405,301,96,1352]);
 
// Polycyclic
 
G:=Group<a,b,c|a^9=b^9=c^3=1,b*a*b^-1=a^7,a*c=c*a,c*b*c^-1=b^4>;
 
// generators/relations
 

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