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## G = C9⋊He3⋊C2order 486 = 2·35

### 3rd semidirect product of C9⋊He3 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C9⋊He3⋊C2
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C9⋊He3 — C9⋊He3⋊C2
 Lower central C32 — C33 — C9⋊He3⋊C2
 Upper central C1 — C3 — C9

Generators and relations for C9⋊He3⋊C2
G = < a,b,c,d,e | a9=b3=c3=d3=e2=1, ab=ba, ac=ca, dad-1=a7, ae=ea, bc=cb, dbd-1=bc-1, ebe=b-1, cd=dc, ece=c-1, de=ed >

Subgroups: 346 in 82 conjugacy classes, 24 normal (18 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, C18, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, C3×C9, He3, 3- 1+2, C33, C33, S3×C9, C32⋊C6, C2×3- 1+2, S3×C32, C3×C3⋊S3, C32⋊C9, C32⋊C9, C32×C9, C3×He3, C3×3- 1+2, C3×3- 1+2, C32⋊C18, C3×C32⋊C6, S3×3- 1+2, C9×C3⋊S3, C9⋊He3, C9⋊He3⋊C2
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, 3- 1+2, C32⋊C6, C2×3- 1+2, S3×C32, C3×C32⋊C6, S3×3- 1+2, C9⋊He3⋊C2

Smallest permutation representation of C9⋊He3⋊C2
On 54 points
Generators in S54
(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)
(1 41 36)(2 42 28)(3 43 29)(4 44 30)(5 45 31)(6 37 32)(7 38 33)(8 39 34)(9 40 35)(10 49 22)(11 50 23)(12 51 24)(13 52 25)(14 53 26)(15 54 27)(16 46 19)(17 47 20)(18 48 21)
(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 49 52)(47 50 53)(48 51 54)
(2 5 8)(3 9 6)(11 14 17)(12 18 15)(19 25 22)(21 24 27)(28 34 31)(30 33 36)(37 40 43)(38 44 41)(46 49 52)(47 53 50)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 10)(8 11)(9 12)(19 44)(20 45)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 53)(29 54)(30 46)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)

G:=sub<Sym(54)| (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), (1,41,36)(2,42,28)(3,43,29)(4,44,30)(5,45,31)(6,37,32)(7,38,33)(8,39,34)(9,40,35)(10,49,22)(11,50,23)(12,51,24)(13,52,25)(14,53,26)(15,54,27)(16,46,19)(17,47,20)(18,48,21), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,49,52)(47,50,53)(48,51,54), (2,5,8)(3,9,6)(11,14,17)(12,18,15)(19,25,22)(21,24,27)(28,34,31)(30,33,36)(37,40,43)(38,44,41)(46,49,52)(47,53,50), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,10)(8,11)(9,12)(19,44)(20,45)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,53)(29,54)(30,46)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)>;

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), (1,41,36)(2,42,28)(3,43,29)(4,44,30)(5,45,31)(6,37,32)(7,38,33)(8,39,34)(9,40,35)(10,49,22)(11,50,23)(12,51,24)(13,52,25)(14,53,26)(15,54,27)(16,46,19)(17,47,20)(18,48,21), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,49,52)(47,50,53)(48,51,54), (2,5,8)(3,9,6)(11,14,17)(12,18,15)(19,25,22)(21,24,27)(28,34,31)(30,33,36)(37,40,43)(38,44,41)(46,49,52)(47,53,50), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,10)(8,11)(9,12)(19,44)(20,45)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,53)(29,54)(30,46)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52) );

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)], [(1,41,36),(2,42,28),(3,43,29),(4,44,30),(5,45,31),(6,37,32),(7,38,33),(8,39,34),(9,40,35),(10,49,22),(11,50,23),(12,51,24),(13,52,25),(14,53,26),(15,54,27),(16,46,19),(17,47,20),(18,48,21)], [(1,7,4),(2,8,5),(3,9,6),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,34,31),(29,35,32),(30,36,33),(37,43,40),(38,44,41),(39,45,42),(46,49,52),(47,50,53),(48,51,54)], [(2,5,8),(3,9,6),(11,14,17),(12,18,15),(19,25,22),(21,24,27),(28,34,31),(30,33,36),(37,40,43),(38,44,41),(46,49,52),(47,53,50)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,10),(8,11),(9,12),(19,44),(20,45),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(27,43),(28,53),(29,54),(30,46),(31,47),(32,48),(33,49),(34,50),(35,51),(36,52)]])

42 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 6A 6B 6C 6D 9A 9B 9C ··· 9J 9K 9L 9M 9N 9O 9P 9Q 9R 18A ··· 18F order 1 2 3 3 3 3 3 3 3 3 3 3 3 3 6 6 6 6 9 9 9 ··· 9 9 9 9 9 9 9 9 9 18 ··· 18 size 1 9 1 1 2 2 2 6 6 6 9 9 18 18 9 9 27 27 3 3 6 ··· 6 9 9 9 9 18 18 18 18 27 ··· 27

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 3 3 6 6 6 6 type + + + + image C1 C2 C3 C3 C3 C6 C6 C6 S3 C3×S3 C3×S3 3- 1+2 C2×3- 1+2 C32⋊C6 C3×C32⋊C6 S3×3- 1+2 C9⋊He3⋊C2 kernel C9⋊He3⋊C2 C9⋊He3 C32⋊C18 C3×C32⋊C6 C9×C3⋊S3 C32⋊C9 C32×C9 C3×He3 C3×3- 1+2 C3×C9 C33 C3⋊S3 C32 C9 C3 C3 C1 # reps 1 1 4 2 2 4 2 2 1 6 2 2 2 1 2 2 6

Matrix representation of C9⋊He3⋊C2 in GL6(𝔽19)

 5 0 0 0 0 0 12 14 14 16 16 16 0 5 0 0 0 0 0 0 0 0 0 5 0 0 0 5 0 0 0 0 0 0 5 0
,
 1 0 0 0 0 0 0 0 1 0 0 0 10 18 18 7 7 7 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0
,
 7 0 0 1 1 1 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
,
 11 11 18 0 18 7 0 7 0 0 0 0 0 0 1 0 0 0 0 0 0 11 0 0 0 0 0 0 7 0 0 0 0 0 0 1
,
 12 0 0 18 18 18 0 0 0 0 1 0 0 0 0 0 0 1 10 18 18 7 7 7 0 1 0 0 0 0 0 0 1 0 0 0

G:=sub<GL(6,GF(19))| [5,12,0,0,0,0,0,14,5,0,0,0,0,14,0,0,0,0,0,16,0,0,5,0,0,16,0,0,0,5,0,16,0,5,0,0],[1,0,10,0,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,7,0,1,0,0,0,7,0,0,1,0,0,7,1,0,0],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,1,0,0,11,0,0,1,0,0,0,11,0,1,0,0,0,0,11],[11,0,0,0,0,0,11,7,0,0,0,0,18,0,1,0,0,0,0,0,0,11,0,0,18,0,0,0,7,0,7,0,0,0,0,1],[12,0,0,10,0,0,0,0,0,18,1,0,0,0,0,18,0,1,18,0,0,7,0,0,18,1,0,7,0,0,18,0,1,7,0,0] >;

C9⋊He3⋊C2 in GAP, Magma, Sage, TeX

C_9\rtimes {\rm He}_3\rtimes C_2
% in TeX

G:=Group("C9:He3:C2");
// GroupNames label

G:=SmallGroup(486,107);
// by ID

G=gap.SmallGroup(486,107);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,68,3244,3250,11669]);
// Polycyclic

G:=Group<a,b,c,d,e|a^9=b^3=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^7,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,d*e=e*d>;
// generators/relations

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