C3:S3xC3xC9 - GroupNames

G = C₃:S₃xC₃xC₉ order 486 = 2·3⁵

Direct product of C₃xC₉ and C₃:S₃

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C₃:S₃xC₃xC₉, C₃³:₈C₁₈, C₃⁴.₁₃C₆, (C₃³xC₉):₂C₂, C₃²:₅(S₃xC₉), (C₃²xC₉):₂₇S₃, C₃²:₆(C₃xC₁₈), (C₃²xC₉):₃₈C₆, C₃³.₅₃(C₃xC₆), C₃³.₈₃(C₃xS₃), C₃².₄₆(S₃xC₃²), C₃:(S₃xC₃xC₉), (C₃xC₉):₂₉(C₃xS₃), C₃.₇(C₃²xC₃:S₃), (C₃²xC₃:S₃).₄C₃, (C₃xC₃:S₃).₆C₃², C₃².₅₆(C₃xC₃:S₃), SmallGroup(486,228)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃:S₃xC₃xC₉

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²xC₉ — C₃³xC₉ — C₃:S₃xC₃xC₉

Lower central

C₃² — C₃:S₃xC₃xC₉

Upper central

C₁ — C₃xC₉

Generators and relations for C₃:S₃xC₃xC₉
G = < a,b,c,d,e | a³=b⁹=c³=d³=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c^-1, ede=d^-1 >

Subgroups: 616 in 288 conjugacy classes, 70 normal (12 characteristic)
C₁, C₂, C₃, C₃, C₃, S₃, C₆, C₉, C₉, C₃², C₃², C₃², C₁₈, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₉, C₃xC₉, C₃xC₉, C₃³, C₃³, C₃³, S₃xC₉, C₃xC₁₈, S₃xC₃², C₃xC₃:S₃, C₃xC₃:S₃, C₃²xC₉, C₃²xC₉, C₃⁴, S₃xC₃xC₉, C₉xC₃:S₃, C₃²xC₃:S₃, C₃³xC₉, C₃:S₃xC₃xC₉
Quotients: C₁, C₂, C₃, S₃, C₆, C₉, C₃², C₁₈, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₉, S₃xC₉, C₃xC₁₈, S₃xC₃², C₃xC₃:S₃, S₃xC₃xC₉, C₉xC₃:S₃, C₃²xC₃:S₃, C₃:S₃xC₃xC₉

Smallest permutation representation of C₃:S₃xC₃xC₉
►On 54 points

Generators in S₅₄

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

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

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

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

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

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

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

162 conjugacy classes

class	1	2	3A	···	3H	3I	···	3AR	6A	···	6H	9A	···	9R	9S	···	9CL	18A	···	18R
order	1	2	3	···	3	3	···	3	6	···	6	9	···	9	9	···	9	18	···	18
size	1	9	1	···	1	2	···	2	9	···	9	1	···	1	2	···	2	9	···	9

162 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2
type	+	+							+
image	C₁	C₂	C₃	C₃	C₆	C₆	C₉	C₁₈	S₃	C₃xS₃	C₃xS₃	S₃xC₉
kernel	C₃:S₃xC₃xC₉	C₃³xC₉	C₉xC₃:S₃	C₃²xC₃:S₃	C₃²xC₉	C₃⁴	C₃xC₃:S₃	C₃³	C₃²xC₉	C₃xC₉	C₃³	C₃²
# reps	1	1	6	2	6	2	18	18	4	24	8	72

Matrix representation of C₃:S₃xC₃xC₉ ►in GL₄(F₁₉) generated by

11	0	0	0
0	11	0	0
0	0	7	0
0	0	0	7

6	0	0	0
0	6	0	0
0	0	4	0
0	0	0	4

11	0	0	0
0	7	0	0
0	0	7	0
0	0	0	11

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

0	1	0	0
1	0	0	0
0	0	0	1
0	0	1	0

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

C₃:S₃xC₃xC₉ in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times C_3\times C_9

% in TeX

G:=Group("C3:S3xC3xC9");

// GroupNames label

G:=SmallGroup(486,228);

// by ID

G=gap.SmallGroup(486,228);

# by ID

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

// Polycyclic

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

// generators/relations

G = C3:S3xC3xC9 order 486 = 2·35

Direct product of C3xC9 and C3:S3

G = C₃:S₃xC₃xC₉ order 486 = 2·3⁵

Direct product of C₃xC₉ and C₃:S₃