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G = C2×C27⋊C9  order 486 = 2·35

Direct product of C2 and C27⋊C9

direct product, metacyclic, nilpotent (class 3), monomial, 3-elementary

Aliases: C2×C27⋊C9, C54⋊C9, C27⋊2C18, C18.53- 1+2, C9⋊C9.1C6, C27⋊C3.2C6, C6.3(C9⋊C9), C18.2(C3×C9), C9.2(C3×C18), (C3×C18).1C32, C9.5(C2×3- 1+2), (C3×C6).73- 1+2, C32.7(C2×3- 1+2), (C2×C27⋊C3).C3, C3.3(C2×C9⋊C9), (C2×C9⋊C9).1C3, (C3×C9).1(C3×C6), SmallGroup(486,82)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C9 — C2×C27⋊C9
Chief series
C1 — C3 — C32 — C3×C9 — C27⋊C3 — C27⋊C9 — C2×C27⋊C9
Lower central
C1 — C3 — C9 — C2×C27⋊C9
Upper central
C1 — C6 — C3×C18 — C2×C27⋊C9

Generators and relations for C2×C27⋊C9
 G = < a,b,c | a2=b27=c9=1, ab=ba, ac=ca, cbc-1=b7 >

C1
C2
C3
3C3
C6
3C6
C9
C9
C32
C9
9C9
C18
C3×C6
C18
C18
9C18
C27
C3×C9
C27
C27
3C27
3C3×C9
3C27
C54
C54
C3×C18
C54
3C54
3C54
3C3×C18
C27⋊C3
C27⋊C3
C9⋊C9
C27⋊C3
C2×C27⋊C3
C2×C27⋊C3
C2×C9⋊C9
C2×C27⋊C3
C27⋊C9
C2×C27⋊C9

Smallest permutation representation of C2×C27⋊C9
►On 54 points
Generators in S54
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 28)(26 29)(27 30)

(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)

(2 5 17 11 14 26 20 23 8)(3 9 6 21 27 24 12 18 15)(4 13 22)(7 25 16)(28 46 37)(29 50 53 38 32 35 47 41 44)(30 54 42 48 45 33 39 36 51)(34 43 52)
 
G:=sub<Sym(54)| (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,28)(26,29)(27,30), (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), (2,5,17,11,14,26,20,23,8)(3,9,6,21,27,24,12,18,15)(4,13,22)(7,25,16)(28,46,37)(29,50,53,38,32,35,47,41,44)(30,54,42,48,45,33,39,36,51)(34,43,52)>;
 
G:=Group( (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,28)(26,29)(27,30), (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), (2,5,17,11,14,26,20,23,8)(3,9,6,21,27,24,12,18,15)(4,13,22)(7,25,16)(28,46,37)(29,50,53,38,32,35,47,41,44)(30,54,42,48,45,33,39,36,51)(34,43,52) );
 
G=PermutationGroup([[(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,28),(26,29),(27,30)], [(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)], [(2,5,17,11,14,26,20,23,8),(3,9,6,21,27,24,12,18,15),(4,13,22),(7,25,16),(28,46,37),(29,50,53,38,32,35,47,41,44),(30,54,42,48,45,33,39,36,51),(34,43,52)]])
 

70 conjugacy classes

class 1  2 3A3B3C3D6A6B6C6D9A···9F9G···9L18A···18F18G···18L27A···27R54A···54R
order12333366669···99···918···1818···1827···2754···54
size11113311333···39···93···39···99···99···9

70 irreducible representations

dim11111111333399
type++
imageC1C2C3C3C6C6C9C183- 1+23- 1+2C2×3- 1+2C2×3- 1+2C27⋊C9C2×C27⋊C9
kernelC2×C27⋊C9C27⋊C9C2×C9⋊C9C2×C27⋊C3C9⋊C9C27⋊C3C54C27C18C3×C6C9C32C2C1
# reps1126261818424222

Matrix representation of C2×C27⋊C9 ►in GL9(𝔽109)

10800000000
01080000000
00108000000
00010800000
00001080000
00000108000
00000010800
00000001080
00000000108
,
000100000
4645045620000
0000631000
0000006300
10863006400630
636400100063
0450000000
6364450460000
0100450000
,
100000000
0450000000
644563000000
45108063062000
06300064000
110464546000
464500001910
4500000461081
6310000108640

G:=sub<GL(9,GF(109))| [108,0,0,0,0,0,0,0,0,0,108,0,0,0,0,0,0,0,0,0,108,0,0,0,0,0,0,0,0,0,108,0,0,0,0,0,0,0,0,0,108,0,0,0,0,0,0,0,0,0,108,0,0,0,0,0,0,0,0,0,108,0,0,0,0,0,0,0,0,0,108,0,0,0,0,0,0,0,0,0,108],[0,46,0,0,108,63,0,63,0,0,45,0,0,63,64,45,64,1,0,0,0,0,0,0,0,45,0,1,45,0,0,0,0,0,0,0,0,62,63,0,64,1,0,46,45,0,0,1,0,0,0,0,0,0,0,0,0,63,0,0,0,0,0,0,0,0,0,63,0,0,0,0,0,0,0,0,0,63,0,0,0],[1,0,64,45,0,1,46,45,63,0,45,45,108,63,1,45,0,1,0,0,63,0,0,0,0,0,0,0,0,0,63,0,46,0,0,0,0,0,0,0,0,45,0,0,0,0,0,0,62,64,46,0,0,0,0,0,0,0,0,0,1,46,108,0,0,0,0,0,0,91,108,64,0,0,0,0,0,0,0,1,0] >;
 

C2×C27⋊C9 in GAP, Magma, Sage, TeX 

C_2\times C_{27}\rtimes C_9
 
% in TeX
 
G:=Group("C2xC27:C9");
 
// GroupNames label
 
G:=SmallGroup(486,82);
 
// by ID
 
G=gap.SmallGroup(486,82);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,224,68,2169,237,1906]);
 
// Polycyclic
 
G:=Group<a,b,c|a^2=b^27=c^9=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^7>;
 
// generators/relations
 

Export

Subgroup lattice of C2×C27⋊C9 in TeX

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