C3xHe3.4S3 - GroupNames

G = C₃×He₃.₄S₃ order 486 = 2·3⁵

Direct product of C₃ and He₃.₄S₃

direct product, metabelian, supersoluble, monomial

Aliases: C₃×He₃.₄S₃, C₉⋊S₃⋊₉C₃², C₉○He₃⋊₈C₆, (C₃²×C₉)⋊₂₀C₆, (C₃²×C₉)⋊₂₂S₃, C₉.₅(S₃×C₃²), He₃.₁₂(C₃×S₃), (C₃×He₃).₂₄S₃, C₃³.₆₉(C₃×S₃), C₃³.₄₃(C₃⋊S₃), C₃².₁₉(S₃×C₃²), 3_-¹⁺²⋊₈(C₃×S₃), (C₃×3_-¹⁺²)⋊₂₀S₃, (C₃×C₉⋊S₃)⋊₅C₃, (C₃×C₉)⋊₁₅(C₃×S₃), (C₃×C₉)⋊₁₄(C₃×C₆), (C₃×C₉○He₃)⋊₁C₂, C₃.₄(C₃²×C₃⋊S₃), C₃².₁₂(C₃×C₃⋊S₃), SmallGroup(486,234)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₉ — C₃×He₃.₄S₃

Chief series

C₁ — C₃ — C₃² — C₃×C₉ — C₃²×C₉ — C₃×C₉○He₃ — C₃×He₃.₄S₃

Lower central

C₃×C₉ — C₃×He₃.₄S₃

Upper central

C₁ — C₃

Generators and relations for C₃×He₃.₄S₃
G = < a,b,c,d,e,f | a³=b³=c³=d³=f²=1, e³=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd^-1=bc^-1, be=eb, fbf=b^-1, cd=dc, ce=ec, fcf=c^-1, de=ed, df=fd, fef=c^-1e² >

Subgroups: 636 in 186 conjugacy classes, 44 normal (17 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₉, C₉, C₃², C₃², C₃², D₉, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₉, C₃×C₉, C₃×C₉, He₃, He₃, 3_-¹⁺², 3_-¹⁺², C₃³, C₃³, C₃×D₉, C₃²⋊C₆, C₉⋊C₆, C₉⋊S₃, S₃×C₃², C₃×C₃⋊S₃, C₃²×C₉, C₃²×C₉, C₃×He₃, C₃×3_-¹⁺², C₃×3_-¹⁺², C₉○He₃, C₉○He₃, C₃²×D₉, C₃×C₃²⋊C₆, C₃×C₉⋊C₆, C₃×C₉⋊S₃, He₃.₄S₃, C₃×C₉○He₃, C₃×He₃.₄S₃
Quotients: C₁, C₂, C₃, S₃, C₆, C₃², C₃×S₃, C₃⋊S₃, C₃×C₆, S₃×C₃², C₃×C₃⋊S₃, He₃.₄S₃, C₃²×C₃⋊S₃, C₃×He₃.₄S₃

Smallest permutation representation of C₃×He₃.₄S₃
►On 54 points

Generators in S₅₄

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

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

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

(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(28 31 34)(29 32 35)(30 33 36)(37 43 40)(38 44 41)(39 45 42)

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

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

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

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

63 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	3F	···	3K	3L	···	3T	6A	···	6H	9A	···	9I	9J	···	9AG
order	1	2	3	3	3	3	3	3	···	3	3	···	3	6	···	6	9	···	9	9	···	9
size	1	27	1	1	2	2	2	3	···	3	6	···	6	27	···	27	2	···	2	6	···	6

63 irreducible representations

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

Matrix representation of C₃×He₃.₄S₃ ►in GL₆(𝔽₁₉)

11	0	0	0	0	0
0	11	0	0	0	0
0	0	11	0	0	0
0	0	0	11	0	0
0	0	0	0	11	0
0	0	0	0	0	11

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

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

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

16	0	0	0	0	0
0	16	0	0	0	0
0	0	16	0	0	0
0	0	0	6	0	0
0	0	0	0	6	0
0	0	0	0	0	6

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

G:=sub<GL(6,GF(19))| [11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[1,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,11],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,6],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;

C₃×He₃.₄S₃ in GAP, Magma, Sage, TeX

C_3\times {\rm He}_3._4S_3

% in TeX

G:=Group("C3xHe3.4S3");

// GroupNames label

G:=SmallGroup(486,234);

// by ID

G=gap.SmallGroup(486,234);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,4755,453,3244,3250,11669]);

// Polycyclic

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

// generators/relations

G = C3×He3.4S3 order 486 = 2·35

Direct product of C3 and He3.4S3

G = C₃×He₃.₄S₃ order 486 = 2·3⁵

Direct product of C₃ and He₃.₄S₃