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## G = C9.5He3order 243 = 35

### 3rd central extension by C9 of He3

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

Aliases: C9.5He3, He3.1C9, C9.13- 1+2, 3- 1+2.1C9, C27⋊C32C3, (C3×C27)⋊2C3, C9○He3.1C3, C32.3(C3×C9), C3.9(C32⋊C9), (C3×C9).25C32, SmallGroup(243,19)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C32 — C9.5He3
 Chief series C1 — C3 — C9 — C3×C9 — C9○He3 — C9.5He3
 Lower central C1 — C3 — C32 — C9.5He3
 Upper central C1 — C9 — C3×C9 — C9.5He3
 Jennings C1 — C3 — C3 — C3 — C3 — C3 — C3 — C9 — C3×C9 — C9.5He3

Generators and relations for C9.5He3
G = < a,b,c,d | a9=b3=c3=1, d3=a, ab=ba, ac=ca, ad=da, cbc-1=a3b, dbd-1=a3bc-1, dcd-1=a6c >

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

C9.5He3 is a maximal subgroup of   He3.C18  He3.D9  He3.3D9

51 conjugacy classes

 class 1 3A 3B 3C 3D 3E 3F 9A ··· 9F 9G 9H 9I 9J 9K 9L 9M 9N 27A ··· 27R 27S ··· 27AD order 1 3 3 3 3 3 3 9 ··· 9 9 9 9 9 9 9 9 9 27 ··· 27 27 ··· 27 size 1 1 1 3 3 9 9 1 ··· 1 3 3 3 3 9 9 9 9 3 ··· 3 9 ··· 9

51 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 type + image C1 C3 C3 C3 C9 C9 He3 3- 1+2 C9.5He3 kernel C9.5He3 C3×C27 C27⋊C3 C9○He3 He3 3- 1+2 C9 C9 C1 # reps 1 2 4 2 6 12 2 4 18

Matrix representation of C9.5He3 in GL3(𝔽109) generated by

 38 0 0 0 38 0 0 0 38
,
 1 0 0 0 63 0 0 0 45
,
 0 1 0 0 0 1 1 0 0
,
 86 77 77 55 77 55 86 86 55
G:=sub<GL(3,GF(109))| [38,0,0,0,38,0,0,0,38],[1,0,0,0,63,0,0,0,45],[0,0,1,1,0,0,0,1,0],[86,55,86,77,77,86,77,55,55] >;

C9.5He3 in GAP, Magma, Sage, TeX

C_9._5{\rm He}_3
% in TeX

G:=Group("C9.5He3");
// GroupNames label

G:=SmallGroup(243,19);
// by ID

G=gap.SmallGroup(243,19);
# by ID

G:=PCGroup([5,-3,3,-3,3,-3,135,121,1352,457,78]);
// Polycyclic

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

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