C9oHe3:4S3 - GroupNames

G = C₉○He₃⋊₄S₃ order 486 = 2·3⁵

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

Aliases: C₉○He₃⋊₄S₃, (C₃²×C₉)⋊₂₆S₃, C₃⋊(He₃.₄C₆), He₃.₁₄(C₃×S₃), (C₃×He₃).₂₆C₆, C₃³.₄₈(C₃×S₃), He₃⋊₅S₃.₅C₃, 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○He₃⋊₄S₃
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, He₃⋊₅S₃, C₃×C₉○He₃, C₉○He₃⋊₄S₃
Quotients: C₁, C₂, C₃, S₃, C₆, C₃×S₃, C₃⋊S₃, C₃×C₃⋊S₃, C₃³⋊C₂, He₃.₄C₆, C₃×C₃³⋊C₂, C₉○He₃⋊₄S₃

Smallest permutation representation of C₉○He₃⋊₄S₃
►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○He₃⋊₄S₃ ►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○He₃⋊₄S₃ 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

G = C9○He3⋊4S3 order 486 = 2·35

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

G = C₉○He₃⋊₄S₃ order 486 = 2·3⁵

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