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## G = C3×C27⋊C3order 243 = 35

### Direct product of C3 and C27⋊C3

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

Aliases: C3×C27⋊C3, C27⋊C32, C9.3C33, C33.3C9, (C3×C27)⋊4C3, (C3×C9).6C9, C9.4(C3×C9), C3.6(C32×C9), C32.17(C3×C9), (C32×C9).13C3, (C3×C9).27C32, SmallGroup(243,49)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C3 — C3×C27⋊C3
 Chief series C1 — C3 — C9 — C3×C9 — C32×C9 — C3×C27⋊C3
 Lower central C1 — C3 — C3×C27⋊C3
 Upper central C1 — C3×C9 — C3×C27⋊C3
 Jennings C1 — C3 — C3 — C3 — C3 — C3 — C3 — C9 — C9 — C3×C27⋊C3

Generators and relations for C3×C27⋊C3
G = < a,b,c | a3=b27=c3=1, ab=ba, ac=ca, cbc-1=b10 >

Subgroups: 72 in 60 conjugacy classes, 54 normal (8 characteristic)
C1, C3, C3, C3, C9, C9, C32, C32, C32, C27, C3×C9, C3×C9, C33, C3×C27, C27⋊C3, C32×C9, C3×C27⋊C3
Quotients: C1, C3, C9, C32, C3×C9, C33, C27⋊C3, C32×C9, C3×C27⋊C3

Smallest permutation representation of C3×C27⋊C3
On 81 points
Generators in S81
(1 54 71)(2 28 72)(3 29 73)(4 30 74)(5 31 75)(6 32 76)(7 33 77)(8 34 78)(9 35 79)(10 36 80)(11 37 81)(12 38 55)(13 39 56)(14 40 57)(15 41 58)(16 42 59)(17 43 60)(18 44 61)(19 45 62)(20 46 63)(21 47 64)(22 48 65)(23 49 66)(24 50 67)(25 51 68)(26 52 69)(27 53 70)
(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)
(2 20 11)(3 12 21)(5 23 14)(6 15 24)(8 26 17)(9 18 27)(28 46 37)(29 38 47)(31 49 40)(32 41 50)(34 52 43)(35 44 53)(55 64 73)(57 75 66)(58 67 76)(60 78 69)(61 70 79)(63 81 72)

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

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

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

C3×C27⋊C3 is a maximal subgroup of   C33.5D9

99 conjugacy classes

 class 1 3A ··· 3H 3I ··· 3N 9A ··· 9R 9S ··· 9AD 27A ··· 27BB order 1 3 ··· 3 3 ··· 3 9 ··· 9 9 ··· 9 27 ··· 27 size 1 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3

99 irreducible representations

 dim 1 1 1 1 1 1 3 type + image C1 C3 C3 C3 C9 C9 C27⋊C3 kernel C3×C27⋊C3 C3×C27 C27⋊C3 C32×C9 C3×C9 C33 C3 # reps 1 6 18 2 48 6 18

Matrix representation of C3×C27⋊C3 in GL4(𝔽109) generated by

 45 0 0 0 0 63 0 0 0 0 63 0 0 0 0 63
,
 27 0 0 0 0 45 44 0 0 63 64 45 0 42 46 0
,
 63 0 0 0 0 1 0 0 0 45 45 0 0 64 0 63
G:=sub<GL(4,GF(109))| [45,0,0,0,0,63,0,0,0,0,63,0,0,0,0,63],[27,0,0,0,0,45,63,42,0,44,64,46,0,0,45,0],[63,0,0,0,0,1,45,64,0,0,45,0,0,0,0,63] >;

C3×C27⋊C3 in GAP, Magma, Sage, TeX

C_3\times C_{27}\rtimes C_3
% in TeX

G:=Group("C3xC27:C3");
// GroupNames label

G:=SmallGroup(243,49);
// by ID

G=gap.SmallGroup(243,49);
# by ID

G:=PCGroup([5,-3,3,3,-3,-3,135,841,78]);
// Polycyclic

G:=Group<a,b,c|a^3=b^27=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations

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