S3xC9oHe3 - GroupNames

G = S₃×C₉○He₃ order 486 = 2·3⁵

Direct product of S₃ and C₉○He₃

direct product, metabelian, supersoluble, monomial

Aliases: S₃×C₉○He₃, (S₃×C₉).C₃², (C₃²×C₉)⋊₂₅C₆, (S₃×He₃).₂C₃, C₃.₇(S₃×C₃³), C₉.₂(S₃×C₃²), He₃.₁₅(C₃×S₃), (C₃×S₃).₄C₃³, (C₃×He₃).₂₄C₆, C₃³.₃₀(C₃×C₆), (S₃×C₃²).₆C₃², C₃².₂₆(S₃×C₃²), C₃².₁₆(C₃²×C₆), (S₃×3_-¹⁺²)⋊₃C₃, 3_-¹⁺²⋊₁₀(C₃×S₃), (C₃×3_-¹⁺²)⋊₂₂C₆, (S₃×C₃×C₉)⋊₅C₃, C₃⋊(C₂×C₉○He₃), (C₃×C₉)⋊₂₈(C₃×S₃), (C₃×C₉○He₃)⋊₃C₂, (C₃×C₉).₁₆(C₃×C₆), SmallGroup(486,226)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — S₃×C₉○He₃

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²×C₉ — C₃×C₉○He₃ — S₃×C₉○He₃

Lower central

C₃ — C₃² — S₃×C₉○He₃

Upper central

C₁ — C₉ — C₉○He₃

Generators and relations for S₃×C₉○He₃
G = < a,b,c,d,e,f | a³=b²=c⁹=d³=f³=1, e¹=c⁶, bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf^-1=c³d, ef=fe >

Subgroups: 452 in 219 conjugacy classes, 87 normal (15 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₉, C₉, C₃², C₃², C₃², C₁₈, C₃×S₃, C₃×S₃, C₃×C₆, C₃×C₉, C₃×C₉, C₃×C₉, He₃, He₃, 3_-¹⁺², 3_-¹⁺², C₃³, S₃×C₉, S₃×C₉, C₃×C₁₈, C₂×He₃, C₂×3_-¹⁺², S₃×C₃², C₃²×C₉, C₃×He₃, C₃×3_-¹⁺², C₉○He₃, C₉○He₃, S₃×C₃×C₉, S₃×He₃, S₃×3_-¹⁺², C₂×C₉○He₃, C₃×C₉○He₃, S₃×C₉○He₃
Quotients: C₁, C₂, C₃, S₃, C₆, C₃², C₃×S₃, C₃×C₆, C₃³, S₃×C₃², C₃²×C₆, C₉○He₃, C₂×C₉○He₃, S₃×C₃³, S₃×C₉○He₃

Smallest permutation representation of S₃×C₉○He₃
►On 54 points

Generators in S₅₄

(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)

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

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

(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 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)(28 31 34)(29 32 35)(30 33 36)(46 52 49)(47 53 50)(48 54 51)

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

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

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

99 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	3F	···	3M	3N	···	3U	6A	6B	6C	···	6J	9A	···	9F	9G	···	9L	9M	···	9AB	9AC	···	9AR	18A	···	18F	18G	···	18V
order	1	2	3	3	3	3	3	3	···	3	3	···	3	6	6	6	···	6	9	···	9	9	···	9	9	···	9	9	···	9	18	···	18	18	···	18
size	1	3	1	1	2	2	2	3	···	3	6	···	6	3	3	9	···	9	1	···	1	2	···	2	3	···	3	6	···	6	3	···	3	9	···	9

99 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	3	3	6
type	+	+							+
image	C₁	C₂	C₃	C₃	C₃	C₆	C₆	C₆	S₃	C₃×S₃	C₃×S₃	C₃×S₃	C₉○He₃	C₂×C₉○He₃	S₃×C₉○He₃
kernel	S₃×C₉○He₃	C₃×C₉○He₃	S₃×C₃×C₉	S₃×He₃	S₃×3_-¹⁺²	C₃²×C₉	C₃×He₃	C₃×3_-¹⁺²	C₉○He₃	C₃×C₉	He₃	3_-¹⁺²	S₃	C₃	C₁
# reps	1	1	8	2	16	8	2	16	1	8	2	16	6	6	6

Matrix representation of S₃×C₉○He₃ ►in GL₅(𝔽₁₉)

0	1	0	0	0
18	18	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

18	0	0	0	0
1	1	0	0	0
0	0	18	0	0
0	0	0	18	0
0	0	0	0	18

1	0	0	0	0
0	1	0	0	0
0	0	9	0	0
0	0	0	9	0
0	0	0	0	9

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

7	0	0	0	0
0	7	0	0	0
0	0	11	0	0
0	0	0	1	0
0	0	0	0	7

G:=sub<GL(5,GF(19))| [0,18,0,0,0,1,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[18,1,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[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],[7,0,0,0,0,0,7,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,7] >;

S₃×C₉○He₃ in GAP, Magma, Sage, TeX

S_3\times C_9\circ {\rm He}_3

% in TeX

G:=Group("S3xC9oHe3");

// GroupNames label

G:=SmallGroup(486,226);

// by ID

G=gap.SmallGroup(486,226);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,176,873,11669]);

// Polycyclic

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

// generators/relations

G = S3×C9○He3 order 486 = 2·35

Direct product of S3 and C9○He3

G = S₃×C₉○He₃ order 486 = 2·3⁵

Direct product of S₃ and C₉○He₃