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G = C9⋊C9order 81 = 34

The semidirect product of C9 and C9 acting via C9/C3=C3

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

Aliases: C9⋊C9, C32.7C32, C3.23- 1+2, C3.2(C3×C9), (C3×C9).1C3, SmallGroup(81,4)

Series: Derived Chief Lower central Upper central Jennings

C1C3 — C9⋊C9
C1C3C32C3×C9 — C9⋊C9
C1C3 — C9⋊C9
C1C32 — C9⋊C9
C1C32C32 — C9⋊C9

Generators and relations for C9⋊C9
 G = < a,b | a9=b9=1, bab-1=a7 >

3C9
3C9
3C9

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

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

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

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

C9⋊C9 is a maximal subgroup of
C9⋊C18  C27⋊C9  C32.He3  C32.5He3  C32.6He3  C923C3  C9×3- 1+2  C9⋊3- 1+2  C33.31C32  C927C3  C924C3  C925C3  C928C3  C929C3  C62.11C32  C62.12C32
C9⋊C9 is a maximal quotient of
C3.C92  C9⋊C27  C27⋊C9  C62.11C32  C62.12C32

33 conjugacy classes

class 1 3A···3H9A···9X
order13···39···9
size11···13···3

33 irreducible representations

dim1113
type+
imageC1C3C93- 1+2
kernelC9⋊C9C3×C9C9C3
# reps18186

Matrix representation of C9⋊C9 in GL4(𝔽19) generated by

7000
01190
0081
07120
,
16000
013124
011314
013412
G:=sub<GL(4,GF(19))| [7,0,0,0,0,11,0,7,0,9,8,12,0,0,1,0],[16,0,0,0,0,13,1,13,0,12,13,4,0,4,14,12] >;

C9⋊C9 in GAP, Magma, Sage, TeX

C_9\rtimes C_9
% in TeX

G:=Group("C9:C9");
// GroupNames label

G:=SmallGroup(81,4);
// by ID

G=gap.SmallGroup(81,4);
# by ID

G:=PCGroup([4,-3,3,-3,3,108,97,29]);
// Polycyclic

G:=Group<a,b|a^9=b^9=1,b*a*b^-1=a^7>;
// generators/relations

Export

Subgroup lattice of C9⋊C9 in TeX

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