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## G = C9⋊He3order 243 = 35

### The semidirect product of C9 and He3 acting via He3/C32=C3

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

Aliases: C9⋊He3, C33.28C32, C32.22C33, C3213- 1+2, C32⋊C97C3, (C32×C9)⋊7C3, C3.5(C3×He3), (C3×He3).4C3, (C3×C9).5C32, C3.4(C9○He3), (C3×3- 1+2)⋊3C3, C3.5(C3×3- 1+2), SmallGroup(243,39)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C9⋊He3
G = < a,b,c,d | a9=b3=c3=d3=1, ab=ba, ac=ca, dad-1=a7, bc=cb, dbd-1=bc-1, cd=dc >

Subgroups: 189 in 83 conjugacy classes, 39 normal (9 characteristic)
C1, C3, C3, C3, C9, C9, C32, C32, C32, C3×C9, C3×C9, C3×C9, He3, 3- 1+2, C33, C33, C32⋊C9, C32×C9, C3×He3, C3×3- 1+2, C9⋊He3
Quotients: C1, C3, C32, He3, 3- 1+2, C33, C3×He3, C3×3- 1+2, C9○He3, C9⋊He3

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

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

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

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

C9⋊He3 is a maximal subgroup of   D9⋊He3  C9⋊He3⋊C2  C9⋊He32C2

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 3 type + image C1 C3 C3 C3 C3 He3 3- 1+2 C9○He3 kernel C9⋊He3 C32⋊C9 C32×C9 C3×He3 C3×3- 1+2 C9 C32 C3 # reps 1 16 2 2 6 6 6 12

Matrix representation of C9⋊He3 in GL6(𝔽19)

 0 6 0 0 0 0 0 0 6 0 0 0 6 0 0 0 0 0 0 0 0 0 0 4 0 0 0 4 0 0 0 0 0 0 4 0
,
 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0
,
 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11
,
 0 11 0 0 0 0 0 0 7 0 0 0 1 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 11 0 0 0 1 0 0

G:=sub<GL(6,GF(19))| [0,0,6,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,4,0,0],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[0,0,1,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,7,0,0,0,0,0,0,11,0] >;

C9⋊He3 in GAP, Magma, Sage, TeX

C_9\rtimes {\rm He}_3
% in TeX

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

G:=SmallGroup(243,39);
// by ID

G=gap.SmallGroup(243,39);
# by ID

G:=PCGroup([5,-3,3,3,-3,3,301,96,1352]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^3=c^3=d^3=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
// generators/relations

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