S3xC3.He3 - GroupNames

G = S₃×C₃.He₃ order 486 = 2·3⁵

Direct product of S₃ and C₃.He₃

direct product, metabelian, supersoluble, monomial

Aliases: S₃×C₃.He₃, C₃.₈(S₃×He₃), (C₃×S₃).₅He₃, (C₃²×C₉).₂C₆, C₃³.₂₈(C₃×C₆), C₃².₁₇(C₂×He₃), (S₃×3_-¹⁺²).C₃, (S₃×C₃²).₄C₃², C₃².₂₅(S₃×C₃²), (C₃×3_-¹⁺²).₃C₆, 3_-¹⁺².₃(C₃×S₃), (S₃×C₃×C₉).₂C₃, C₃⋊(C₂×C₃.He₃), (C₃×C₉).₄₅(C₃×S₃), (C₃×C₃.He₃)⋊₃C₂, SmallGroup(486,124)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — S₃×C₃.He₃

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃×3_-¹⁺² — C₃×C₃.He₃ — S₃×C₃.He₃

Lower central

C₃ — C₃² — C₃³ — S₃×C₃.He₃

Upper central

C₁ — C₃ — C₃² — C₃.He₃

Generators and relations for S₃×C₃.He₃
G = < a,b,c,d,e,f | a³=b²=c³=e³=1, d³=c^-1, f³=c, 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=cde^-1, fef^-1=c^-1e >

Subgroups: 218 in 72 conjugacy classes, 24 normal (15 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₃², C₃², C₁₈, C₃×S₃, C₃×S₃, C₃×C₆, C₃×C₉, C₃×C₉, 3_-¹⁺², 3_-¹⁺², C₃³, S₃×C₉, C₃×C₁₈, C₂×3_-¹⁺², S₃×C₃², C₃.He₃, C₃.He₃, C₃²×C₉, C₃×3_-¹⁺², C₂×C₃.He₃, S₃×C₃×C₉, S₃×3_-¹⁺², C₃×C₃.He₃, S₃×C₃.He₃
Quotients: C₁, C₂, C₃, S₃, C₆, C₃², C₃×S₃, C₃×C₆, He₃, C₂×He₃, S₃×C₃², C₃.He₃, C₂×C₃.He₃, S₃×He₃, S₃×C₃.He₃

Smallest permutation representation of S₃×C₃.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 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 19)(8 20)(9 21)(10 32)(11 33)(12 34)(13 35)(14 36)(15 28)(16 29)(17 30)(18 31)(37 50)(38 51)(39 52)(40 53)(41 54)(42 46)(43 47)(44 48)(45 49)

(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 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)

(10 13 16)(11 14 17)(12 15 18)(28 31 34)(29 32 35)(30 33 36)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)

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

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

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

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

51 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	3F	3G	3H	3I	6A	6B	6C	6D	9A	···	9F	9G	···	9L	9M	···	9R	9S	···	9X	18A	···	18F	18G	···	18L
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	3	···	3	6	···	6	9	···	9	18	···	18	9	···	9	27	···	27

51 irreducible representations

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

Matrix representation of S₃×C₃.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

1	0	0	0	0
18	18	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	7	0	0
0	0	0	7	0
0	0	0	0	7

1	0	0	0	0
0	1	0	0	0
0	0	16	0	0
0	0	0	16	0
0	0	0	0	17

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

7	0	0	0	0
0	7	0	0	0
0	0	0	16	0
0	0	0	0	17
0	0	17	0	0

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],[1,18,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,17],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,11],[7,0,0,0,0,0,7,0,0,0,0,0,0,0,17,0,0,16,0,0,0,0,0,17,0] >;

S₃×C₃.He₃ in GAP, Magma, Sage, TeX

S_3\times C_3.{\rm He}_3

% in TeX

G:=Group("S3xC3.He3");

// GroupNames label

G:=SmallGroup(486,124);

// by ID

G=gap.SmallGroup(486,124);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,224,176,303,11669]);

// Polycyclic

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

// generators/relations

G = S3×C3.He3 order 486 = 2·35

Direct product of S3 and C3.He3

G = S₃×C₃.He₃ order 486 = 2·3⁵

Direct product of S₃ and C₃.He₃