Copied to
clipboard

## G = C9○He3⋊4S3order 486 = 2·35

### 2nd semidirect product of C9○He3 and S3 acting via S3/C3=C2

Aliases: C9○He34S3, (C32×C9)⋊26S3, C3⋊(He3.4C6), He3.14(C3×S3), (C3×He3).26C6, C33.48(C3×S3), He35S3.5C3, C9.2(C33⋊C2), (C3×C9)⋊8(C3⋊S3), (C3×C9○He3)⋊5C2, C32.15(C3×C3⋊S3), C3.7(C3×C33⋊C2), SmallGroup(486,246)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C3×He3 — C9○He3⋊4S3
 Chief series C1 — C3 — C32 — He3 — C3×He3 — C3×C9○He3 — C9○He3⋊4S3
 Lower central C3×He3 — C9○He3⋊4S3
 Upper central C1 — C9

Generators and relations for C9○He34S3
G = < a,b,c,d,e,f | a9=b3=d3=e3=f2=1, c1=a6, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd-1=ebe-1=a3b, fbf=b-1, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 812 in 231 conjugacy classes, 60 normal (10 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, C32, C18, C3×S3, C3⋊S3, C3×C9, C3×C9, C3×C9, He3, 3- 1+2, C33, S3×C9, He3⋊C2, C3×C3⋊S3, C32×C9, C3×He3, C3×3- 1+2, C9○He3, C9○He3, C9×C3⋊S3, He3.4C6, He35S3, C3×C9○He3, C9○He34S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C3×C3⋊S3, C33⋊C2, He3.4C6, C3×C33⋊C2, C9○He34S3

Smallest permutation representation of C9○He34S3
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 44 36)(2 45 28)(3 37 29)(4 38 30)(5 39 31)(6 40 32)(7 41 33)(8 42 34)(9 43 35)(10 54 22)(11 46 23)(12 47 24)(13 48 25)(14 49 26)(15 50 27)(16 51 19)(17 52 20)(18 53 21)
(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)
(1 36 38)(2 28 39)(3 29 40)(4 30 41)(5 31 42)(6 32 43)(7 33 44)(8 34 45)(9 35 37)(10 25 54)(11 26 46)(12 27 47)(13 19 48)(14 20 49)(15 21 50)(16 22 51)(17 23 52)(18 24 53)
(1 30 44)(2 31 45)(3 32 37)(4 33 38)(5 34 39)(6 35 40)(7 36 41)(8 28 42)(9 29 43)(10 22 48)(11 23 49)(12 24 50)(13 25 51)(14 26 52)(15 27 53)(16 19 54)(17 20 46)(18 21 47)
(1 17)(2 18)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(9 16)(19 43)(20 44)(21 45)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(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), (1,44,36)(2,45,28)(3,37,29)(4,38,30)(5,39,31)(6,40,32)(7,41,33)(8,42,34)(9,43,35)(10,54,22)(11,46,23)(12,47,24)(13,48,25)(14,49,26)(15,50,27)(16,51,19)(17,52,20)(18,53,21), (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), (1,36,38)(2,28,39)(3,29,40)(4,30,41)(5,31,42)(6,32,43)(7,33,44)(8,34,45)(9,35,37)(10,25,54)(11,26,46)(12,27,47)(13,19,48)(14,20,49)(15,21,50)(16,22,51)(17,23,52)(18,24,53), (1,30,44)(2,31,45)(3,32,37)(4,33,38)(5,34,39)(6,35,40)(7,36,41)(8,28,42)(9,29,43)(10,22,48)(11,23,49)(12,24,50)(13,25,51)(14,26,52)(15,27,53)(16,19,54)(17,20,46)(18,21,47), (1,17)(2,18)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(9,16)(19,43)(20,44)(21,45)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(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), (1,44,36)(2,45,28)(3,37,29)(4,38,30)(5,39,31)(6,40,32)(7,41,33)(8,42,34)(9,43,35)(10,54,22)(11,46,23)(12,47,24)(13,48,25)(14,49,26)(15,50,27)(16,51,19)(17,52,20)(18,53,21), (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), (1,36,38)(2,28,39)(3,29,40)(4,30,41)(5,31,42)(6,32,43)(7,33,44)(8,34,45)(9,35,37)(10,25,54)(11,26,46)(12,27,47)(13,19,48)(14,20,49)(15,21,50)(16,22,51)(17,23,52)(18,24,53), (1,30,44)(2,31,45)(3,32,37)(4,33,38)(5,34,39)(6,35,40)(7,36,41)(8,28,42)(9,29,43)(10,22,48)(11,23,49)(12,24,50)(13,25,51)(14,26,52)(15,27,53)(16,19,54)(17,20,46)(18,21,47), (1,17)(2,18)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(9,16)(19,43)(20,44)(21,45)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(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)], [(1,44,36),(2,45,28),(3,37,29),(4,38,30),(5,39,31),(6,40,32),(7,41,33),(8,42,34),(9,43,35),(10,54,22),(11,46,23),(12,47,24),(13,48,25),(14,49,26),(15,50,27),(16,51,19),(17,52,20),(18,53,21)], [(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)], [(1,36,38),(2,28,39),(3,29,40),(4,30,41),(5,31,42),(6,32,43),(7,33,44),(8,34,45),(9,35,37),(10,25,54),(11,26,46),(12,27,47),(13,19,48),(14,20,49),(15,21,50),(16,22,51),(17,23,52),(18,24,53)], [(1,30,44),(2,31,45),(3,32,37),(4,33,38),(5,34,39),(6,35,40),(7,36,41),(8,28,42),(9,29,43),(10,22,48),(11,23,49),(12,24,50),(13,25,51),(14,26,52),(15,27,53),(16,19,54),(17,20,46),(18,21,47)], [(1,17),(2,18),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15),(9,16),(19,43),(20,44),(21,45),(22,37),(23,38),(24,39),(25,40),(26,41),(27,42),(28,53),(29,54),(30,46),(31,47),(32,48),(33,49),(34,50),(35,51),(36,52)]])

63 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F ··· 3Q 6A 6B 9A ··· 9F 9G ··· 9L 9M ··· 9AJ 18A ··· 18F order 1 2 3 3 3 3 3 3 ··· 3 6 6 9 ··· 9 9 ··· 9 9 ··· 9 18 ··· 18 size 1 27 1 1 2 2 2 6 ··· 6 27 27 1 ··· 1 2 ··· 2 6 ··· 6 27 ··· 27

63 irreducible representations

 dim 1 1 1 1 2 2 2 2 3 6 type + + + + image C1 C2 C3 C6 S3 S3 C3×S3 C3×S3 He3.4C6 C9○He3⋊4S3 kernel C9○He3⋊4S3 C3×C9○He3 He3⋊5S3 C3×He3 C32×C9 C9○He3 He3 C33 C3 C1 # reps 1 1 2 2 4 9 18 8 12 6

Matrix representation of C9○He34S3 in GL5(𝔽19)

 1 0 0 0 0 0 1 0 0 0 0 0 6 0 0 0 0 0 6 0 0 0 0 0 6
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 11 0 0 0 0 0 11
,
 18 1 0 0 0 18 0 0 0 0 0 0 0 11 0 0 0 0 0 7 0 0 1 0 0
,
 0 18 0 0 0 1 18 0 0 0 0 0 0 11 0 0 0 0 0 7 0 0 1 0 0
,
 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0

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

C9○He34S3 in GAP, Magma, Sage, TeX

C_9\circ {\rm He}_3\rtimes_4S_3
% in TeX

G:=Group("C9oHe3:4S3");
// GroupNames label

G:=SmallGroup(486,246);
// by ID

G=gap.SmallGroup(486,246);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,218,867,3244,382]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^9=b^3=d^3=e^3=f^2=1,c^1=a^6,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,d*b*d^-1=e*b*e^-1=a^3*b,f*b*f=b^-1,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=d^-1,f*e*f=e^-1>;
// generators/relations

׿
×
𝔽