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## G = C3×C9○He3order 243 = 35

### Direct product of C3 and C9○He3

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

Aliases: C3×C9○He3, C3.3C34, C9.1C33, He3.6C32, C32.12C33, C33.37C32, 3- 1+25C32, (C3×C9)He3, (C3×C9)⋊9C32, (C32×C9)⋊10C3, (C3×He3).10C3, (C3×C9)3- 1+2, (C3×3- 1+2)⋊12C3, SmallGroup(243,64)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C3×C9○He3
G = < a,b,c,d,e | a3=b9=c3=e3=1, d1=b6, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=b3c, de=ed >

Subgroups: 288 in 240 conjugacy classes, 216 normal (6 characteristic)
C1, C3, C3, C3, C9, C32, C32, C32, C3×C9, C3×C9, He3, 3- 1+2, C33, C32×C9, C3×He3, C3×3- 1+2, C9○He3, C3×C9○He3
Quotients: C1, C3, C32, C33, C9○He3, C34, C3×C9○He3

Smallest permutation representation of C3×C9○He3
On 81 points
Generators in S81
(1 31 41)(2 32 42)(3 33 43)(4 34 44)(5 35 45)(6 36 37)(7 28 38)(8 29 39)(9 30 40)(10 79 21)(11 80 22)(12 81 23)(13 73 24)(14 74 25)(15 75 26)(16 76 27)(17 77 19)(18 78 20)(46 66 55)(47 67 56)(48 68 57)(49 69 58)(50 70 59)(51 71 60)(52 72 61)(53 64 62)(54 65 63)
(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)
(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(46 52 49)(47 53 50)(48 54 51)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)(73 76 79)(74 77 80)(75 78 81)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)(73 79 76)(74 80 77)(75 81 78)
(1 23 54)(2 24 46)(3 25 47)(4 26 48)(5 27 49)(6 19 50)(7 20 51)(8 21 52)(9 22 53)(10 72 29)(11 64 30)(12 65 31)(13 66 32)(14 67 33)(15 68 34)(16 69 35)(17 70 36)(18 71 28)(37 77 59)(38 78 60)(39 79 61)(40 80 62)(41 81 63)(42 73 55)(43 74 56)(44 75 57)(45 76 58)

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

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

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

C3×C9○He3 is a maximal subgroup of   C9○He33S3  C9○He34S3

99 conjugacy classes

 class 1 3A ··· 3H 3I ··· 3AF 9A ··· 9R 9S ··· 9BN order 1 3 ··· 3 3 ··· 3 9 ··· 9 9 ··· 9 size 1 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3

99 irreducible representations

 dim 1 1 1 1 1 3 type + image C1 C3 C3 C3 C3 C9○He3 kernel C3×C9○He3 C32×C9 C3×He3 C3×3- 1+2 C9○He3 C3 # reps 1 8 2 16 54 18

Matrix representation of C3×C9○He3 in GL4(𝔽19) generated by

 11 0 0 0 0 7 0 0 0 0 7 0 0 0 0 7
,
 1 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 11 0 0 0 0 1 0 0 0 18 7 0 0 7 0 11
,
 1 0 0 0 0 7 0 0 0 0 7 0 0 0 0 7
,
 1 0 0 0 0 1 13 0 0 18 18 11 0 7 8 0
G:=sub<GL(4,GF(19))| [11,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[11,0,0,0,0,1,18,7,0,0,7,0,0,0,0,11],[1,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[1,0,0,0,0,1,18,7,0,13,18,8,0,0,11,0] >;

C3×C9○He3 in GAP, Magma, Sage, TeX

C_3\times C_9\circ {\rm He}_3
% in TeX

G:=Group("C3xC9oHe3");
// GroupNames label

G:=SmallGroup(243,64);
// by ID

G=gap.SmallGroup(243,64);
# by ID

G:=PCGroup([5,-3,3,3,3,-3,841,147]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^9=c^3=e^3=1,d^1=b^6,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,d*e=e*d>;
// generators/relations

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