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## G = C3×He3.C3order 243 = 35

### Direct product of C3 and He3.C3

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C3×He3.C3, He3.1C32, C32.10He3, C32.2C33, C33.32C32, 3- 1+22C32, (C32×C9)⋊5C3, (C3×C9)⋊6C32, C3.8(C3×He3), (C3×He3).6C3, (C3×3- 1+2)⋊6C3, SmallGroup(243,52)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C32 — C3×He3.C3
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C3×He3.C3
 Lower central C1 — C3 — C32 — C3×He3.C3
 Upper central C1 — C32 — C33 — C3×He3.C3
 Jennings C1 — C3 — C32 — C3×He3.C3

Generators and relations for C3×He3.C3
G = < a,b,c,d,e | a3=b3=c3=d3=1, e3=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, be=eb, cd=dc, ce=ec, ede-1=bc-1d >

Subgroups: 180 in 72 conjugacy classes, 36 normal (8 characteristic)
C1, C3, C3, C3, C9, C32, C32, C32, C3×C9, C3×C9, He3, He3, 3- 1+2, 3- 1+2, C33, C33, He3.C3, C32×C9, C3×He3, C3×3- 1+2, C3×He3.C3
Quotients: C1, C3, C32, He3, C33, He3.C3, C3×He3, C3×He3.C3

Smallest permutation representation of C3×He3.C3
On 81 points
Generators in S81
(1 14 41)(2 15 42)(3 16 43)(4 17 44)(5 18 45)(6 10 37)(7 11 38)(8 12 39)(9 13 40)(19 36 61)(20 28 62)(21 29 63)(22 30 55)(23 31 56)(24 32 57)(25 33 58)(26 34 59)(27 35 60)(46 76 67)(47 77 68)(48 78 69)(49 79 70)(50 80 71)(51 81 72)(52 73 64)(53 74 65)(54 75 66)
(1 47 31)(2 48 32)(3 49 33)(4 50 34)(5 51 35)(6 52 36)(7 53 28)(8 54 29)(9 46 30)(10 73 61)(11 74 62)(12 75 63)(13 76 55)(14 77 56)(15 78 57)(16 79 58)(17 80 59)(18 81 60)(19 37 64)(20 38 65)(21 39 66)(22 40 67)(23 41 68)(24 42 69)(25 43 70)(26 44 71)(27 45 72)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)
(1 38 17)(2 66 63)(3 22 79)(4 41 11)(5 69 57)(6 25 73)(7 44 14)(8 72 60)(9 19 76)(10 33 64)(12 51 27)(13 36 67)(15 54 21)(16 30 70)(18 48 24)(20 56 34)(23 59 28)(26 62 31)(29 42 75)(32 45 78)(35 39 81)(37 58 52)(40 61 46)(43 55 49)(47 68 77)(50 71 80)(53 65 74)
(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)

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

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

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

C3×He3.C3 is a maximal subgroup of   He3.C3⋊S3  C324D9⋊C3  He3.(C3⋊S3)

51 conjugacy classes

 class 1 3A ··· 3H 3I ··· 3N 3O ··· 3T 9A ··· 9R 9S ··· 9AD order 1 3 ··· 3 3 ··· 3 3 ··· 3 9 ··· 9 9 ··· 9 size 1 1 ··· 1 3 ··· 3 9 ··· 9 3 ··· 3 9 ··· 9

51 irreducible representations

 dim 1 1 1 1 1 3 3 type + image C1 C3 C3 C3 C3 He3 He3.C3 kernel C3×He3.C3 He3.C3 C32×C9 C3×He3 C3×3- 1+2 C32 C3 # reps 1 18 2 2 4 6 18

Matrix representation of C3×He3.C3 in GL6(𝔽19)

 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7
,
 7 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 11 0 0 0 0 0 0 7
,
 0 7 0 0 0 0 0 0 7 0 0 0 7 0 0 0 0 0 0 0 0 8 12 8 0 0 0 8 8 12 0 0 0 12 8 8

G:=sub<GL(6,GF(19))| [11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7],[0,0,7,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,0,8,8,12,0,0,0,12,8,8,0,0,0,8,12,8] >;

C3×He3.C3 in GAP, Magma, Sage, TeX

C_3\times {\rm He}_3.C_3
% in TeX

G:=Group("C3xHe3.C3");
// GroupNames label

G:=SmallGroup(243,52);
// by ID

G=gap.SmallGroup(243,52);
# by ID

G:=PCGroup([5,-3,3,3,-3,-3,301,276,2163]);
// Polycyclic

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

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