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## G = C27○He3order 243 = 35

### Central product of C27 and He3

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

Aliases: C27He3, C27.C32, He3.3C9, C9.4C33, C273- 1+2, 3- 1+2.3C9, C27⋊C34C3, (C3×C27)⋊5C3, C9.1(C3×C9), C27(C27⋊C3), C27(C9○He3), C9○He3.3C3, C32.6(C3×C9), C3.7(C32×C9), (C3×C9).28C32, SmallGroup(243,50)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C3 — C27○He3
 Chief series C1 — C3 — C9 — C3×C9 — C9○He3 — C27○He3
 Lower central C1 — C3 — C27○He3
 Upper central C1 — C27 — C27○He3
 Jennings C1 — C3 — C3 — C3 — C3 — C3 — C3 — C9 — C9 — C27○He3

Generators and relations for C27○He3
G = < a,b,c,d | a27=b3=d3=1, c1=a18, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=a9b, cd=dc >

Subgroups: 63 in 55 conjugacy classes, 51 normal (7 characteristic)
C1, C3, C3, C9, C9, C32, C27, C27, C3×C9, He3, 3- 1+2, C3×C27, C27⋊C3, C9○He3, C27○He3
Quotients: C1, C3, C9, C32, C3×C9, C33, C32×C9, C27○He3

Smallest permutation representation of C27○He3
On 81 points
Generators in S81
(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)
(1 57 52)(2 58 53)(3 59 54)(4 60 28)(5 61 29)(6 62 30)(7 63 31)(8 64 32)(9 65 33)(10 66 34)(11 67 35)(12 68 36)(13 69 37)(14 70 38)(15 71 39)(16 72 40)(17 73 41)(18 74 42)(19 75 43)(20 76 44)(21 77 45)(22 78 46)(23 79 47)(24 80 48)(25 81 49)(26 55 50)(27 56 51)
(1 19 10)(2 20 11)(3 21 12)(4 22 13)(5 23 14)(6 24 15)(7 25 16)(8 26 17)(9 27 18)(28 46 37)(29 47 38)(30 48 39)(31 49 40)(32 50 41)(33 51 42)(34 52 43)(35 53 44)(36 54 45)(55 73 64)(56 74 65)(57 75 66)(58 76 67)(59 77 68)(60 78 69)(61 79 70)(62 80 71)(63 81 72)
(1 57 43)(2 58 44)(3 59 45)(4 60 46)(5 61 47)(6 62 48)(7 63 49)(8 64 50)(9 65 51)(10 66 52)(11 67 53)(12 68 54)(13 69 28)(14 70 29)(15 71 30)(16 72 31)(17 73 32)(18 74 33)(19 75 34)(20 76 35)(21 77 36)(22 78 37)(23 79 38)(24 80 39)(25 81 40)(26 55 41)(27 56 42)

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

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

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

C27○He3 is a maximal subgroup of   He3.5D9  He3.5C18

99 conjugacy classes

 class 1 3A 3B 3C ··· 3J 9A ··· 9F 9G ··· 9V 27A ··· 27R 27S ··· 27BN order 1 3 3 3 ··· 3 9 ··· 9 9 ··· 9 27 ··· 27 27 ··· 27 size 1 1 1 3 ··· 3 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3

99 irreducible representations

 dim 1 1 1 1 1 1 3 type + image C1 C3 C3 C3 C9 C9 C27○He3 kernel C27○He3 C3×C27 C27⋊C3 C9○He3 He3 3- 1+2 C1 # reps 1 8 16 2 6 48 18

Matrix representation of C27○He3 in GL3(𝔽109) generated by

 48 0 0 0 48 0 0 0 48
,
 1 44 0 0 108 1 0 108 0
,
 45 0 0 0 45 0 0 0 45
,
 63 47 0 64 46 45 63 46 0
G:=sub<GL(3,GF(109))| [48,0,0,0,48,0,0,0,48],[1,0,0,44,108,108,0,1,0],[45,0,0,0,45,0,0,0,45],[63,64,63,47,46,46,0,45,0] >;

C27○He3 in GAP, Magma, Sage, TeX

C_{27}\circ {\rm He}_3
% in TeX

G:=Group("C27oHe3");
// GroupNames label

G:=SmallGroup(243,50);
// by ID

G=gap.SmallGroup(243,50);
# by ID

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

G:=Group<a,b,c,d|a^27=b^3=d^3=1,c^1=a^18,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^9*b,c*d=d*c>;
// generators/relations

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