C2xHe3.3S3 - GroupNames

G = C₂×He₃.₃S₃ order 324 = 2²·3⁴

Direct product of C₂ and He₃.₃S₃

direct product, non-abelian, supersoluble, monomial

Aliases: C₂×He₃.₃S₃, He₃.₅D₆, 3_-¹⁺²⋊₂D₆, (C₃×C₉)⋊₉D₆, (C₃×C₁₈)⋊₅S₃, (C₂×He₃).₉S₃, He₃.C₃⋊₂C₂², C₆.₈(He₃⋊C₂), (C₂×3_-¹⁺²)⋊₂S₃, (C₃×C₆).₆(C₃⋊S₃), C₃².₂(C₂×C₃⋊S₃), (C₂×He₃.C₃)⋊₁C₂, C₃.₃(C₂×He₃⋊C₂), SmallGroup(324,78)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — He₃.C₃ — C₂×He₃.₃S₃

Chief series

C₁ — C₃ — C₃² — C₃×C₉ — He₃.C₃ — He₃.₃S₃ — C₂×He₃.₃S₃

Lower central

He₃.C₃ — C₂×He₃.₃S₃

Upper central

C₁ — C₂

Generators and relations for C₂×He₃.₃S₃
G = < a,b,c,d,e,f | a²=b³=c³=d³=f²=1, e³=fcf=c^-1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd^-1=fbf=bc^-1, be=eb, cd=dc, ce=ec, ede^-1=b^-1cd, fdf=d^-1, fef=ce² >

Subgroups: 436 in 70 conjugacy classes, 19 normal (13 characteristic)
C₁, C₂, C₂, C₃, C₃, C₂², S₃, C₆, C₆, C₉, C₃², C₃², D₆, C₂×C₆, D₉, C₁₈, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₃×C₉, He₃, 3_-¹⁺², D₁₈, S₃×C₆, C₂×C₃⋊S₃, C₃×D₉, C₃²⋊C₆, C₉⋊C₆, C₃×C₁₈, C₂×He₃, C₂×3_-¹⁺², He₃.C₃, C₆×D₉, C₂×C₃²⋊C₆, C₂×C₉⋊C₆, He₃.₃S₃, C₂×He₃.C₃, C₂×He₃.₃S₃
Quotients: C₁, C₂, C₂², S₃, D₆, C₃⋊S₃, C₂×C₃⋊S₃, He₃⋊C₂, C₂×He₃⋊C₂, He₃.₃S₃, C₂×He₃.₃S₃

Character table of C₂×He₃.₃S₃

class	1	2A	2B	2C	3A	3B	3C	3D	6A	6B	6C	6D	6E	6F	6G	6H	9A	9B	9C	9D	9E	18A	18B	18C	18D	18E
size	1	1	27	27	2	3	3	18	2	3	3	18	27	27	27	27	6	6	6	18	18	6	6	6	18	18
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	-1	1	1	1	1	1	-1	-1	-1	-1	-1	1	1	-1	1	1	1	1	1	-1	-1	-1	-1	-1	linear of order 2
ρ₃	1	-1	1	-1	1	1	1	1	-1	-1	-1	-1	1	-1	-1	1	1	1	1	1	1	-1	-1	-1	-1	-1	linear of order 2
ρ₄	1	1	-1	-1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	1	1	1	1	1	1	1	linear of order 2
ρ₅	2	2	0	0	2	2	2	-1	2	2	2	-1	0	0	0	0	2	2	2	-1	-1	2	2	2	-1	-1	orthogonal lifted from S₃
ρ₆	2	-2	0	0	2	2	2	2	-2	-2	-2	-2	0	0	0	0	-1	-1	-1	-1	-1	1	1	1	1	1	orthogonal lifted from D₆
ρ₇	2	2	0	0	2	2	2	-1	2	2	2	-1	0	0	0	0	-1	-1	-1	2	-1	-1	-1	-1	-1	2	orthogonal lifted from S₃
ρ₈	2	-2	0	0	2	2	2	-1	-2	-2	-2	1	0	0	0	0	-1	-1	-1	-1	2	1	1	1	-2	1	orthogonal lifted from D₆
ρ₉	2	-2	0	0	2	2	2	-1	-2	-2	-2	1	0	0	0	0	-1	-1	-1	2	-1	1	1	1	1	-2	orthogonal lifted from D₆
ρ₁₀	2	2	0	0	2	2	2	-1	2	2	2	-1	0	0	0	0	-1	-1	-1	-1	2	-1	-1	-1	2	-1	orthogonal lifted from S₃
ρ₁₁	2	-2	0	0	2	2	2	-1	-2	-2	-2	1	0	0	0	0	2	2	2	-1	-1	-2	-2	-2	1	1	orthogonal lifted from D₆
ρ₁₂	2	2	0	0	2	2	2	2	2	2	2	2	0	0	0	0	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₃	3	-3	-1	1	3	^-3+3√-3/₂	^-3-3√-3/₂	0	-3	^3+3√-3/₂	^3-3√-3/₂	0	ζ₆	ζ₃	ζ₃²	ζ₆⁵	0	0	0	0	0	0	0	0	0	0	complex lifted from C₂×He₃⋊C₂
ρ₁₄	3	3	1	1	3	^-3-3√-3/₂	^-3+3√-3/₂	0	3	^-3+3√-3/₂	^-3-3√-3/₂	0	ζ₃	ζ₃²	ζ₃	ζ₃²	0	0	0	0	0	0	0	0	0	0	complex lifted from He₃⋊C₂
ρ₁₅	3	-3	1	-1	3	^-3+3√-3/₂	^-3-3√-3/₂	0	-3	^3+3√-3/₂	^3-3√-3/₂	0	ζ₃²	ζ₆⁵	ζ₆	ζ₃	0	0	0	0	0	0	0	0	0	0	complex lifted from C₂×He₃⋊C₂
ρ₁₆	3	-3	1	-1	3	^-3-3√-3/₂	^-3+3√-3/₂	0	-3	^3-3√-3/₂	^3+3√-3/₂	0	ζ₃	ζ₆	ζ₆⁵	ζ₃²	0	0	0	0	0	0	0	0	0	0	complex lifted from C₂×He₃⋊C₂
ρ₁₇	3	3	-1	-1	3	^-3+3√-3/₂	^-3-3√-3/₂	0	3	^-3-3√-3/₂	^-3+3√-3/₂	0	ζ₆	ζ₆⁵	ζ₆	ζ₆⁵	0	0	0	0	0	0	0	0	0	0	complex lifted from He₃⋊C₂
ρ₁₈	3	3	1	1	3	^-3+3√-3/₂	^-3-3√-3/₂	0	3	^-3-3√-3/₂	^-3+3√-3/₂	0	ζ₃²	ζ₃	ζ₃²	ζ₃	0	0	0	0	0	0	0	0	0	0	complex lifted from He₃⋊C₂
ρ₁₉	3	-3	-1	1	3	^-3-3√-3/₂	^-3+3√-3/₂	0	-3	^3-3√-3/₂	^3+3√-3/₂	0	ζ₆⁵	ζ₃²	ζ₃	ζ₆	0	0	0	0	0	0	0	0	0	0	complex lifted from C₂×He₃⋊C₂
ρ₂₀	3	3	-1	-1	3	^-3-3√-3/₂	^-3+3√-3/₂	0	3	^-3+3√-3/₂	^-3-3√-3/₂	0	ζ₆⁵	ζ₆	ζ₆⁵	ζ₆	0	0	0	0	0	0	0	0	0	0	complex lifted from He₃⋊C₂
ρ₂₁	6	-6	0	0	-3	0	0	0	3	0	0	0	0	0	0	0	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	0	0	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	0	0	orthogonal faithful
ρ₂₂	6	6	0	0	-3	0	0	0	-3	0	0	0	0	0	0	0	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	0	0	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	0	0	orthogonal lifted from He₃.₃S₃
ρ₂₃	6	-6	0	0	-3	0	0	0	3	0	0	0	0	0	0	0	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	0	0	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	0	0	orthogonal faithful
ρ₂₄	6	-6	0	0	-3	0	0	0	3	0	0	0	0	0	0	0	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	0	0	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	0	0	orthogonal faithful
ρ₂₅	6	6	0	0	-3	0	0	0	-3	0	0	0	0	0	0	0	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	0	0	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	0	0	orthogonal lifted from He₃.₃S₃
ρ₂₆	6	6	0	0	-3	0	0	0	-3	0	0	0	0	0	0	0	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	0	0	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	0	0	orthogonal lifted from He₃.₃S₃

Smallest permutation representation of C₂×He₃.₃S₃
►On 54 points

Generators in S₅₄

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

(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 47 26)(11 48 27)(12 49 19)(13 50 20)(14 51 21)(15 52 22)(16 53 23)(17 54 24)(18 46 25)

(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)

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

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

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

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

Matrix representation of C₂×He₃.₃S₃ ►in GL₆(𝔽₁₉)

18	0	0	0	0	0
0	18	0	0	0	0
0	0	18	0	0	0
0	0	0	18	0	0
0	0	0	0	18	0
0	0	0	0	0	18

0	0	1	0	0	0
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

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

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

1	3	1	3	15	1
16	4	16	4	18	16
15	1	1	3	1	3
18	16	16	4	16	4
1	3	15	1	1	3
16	4	18	16	16	4

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

G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[18,18,0,0,0,0,1,0,0,0,0,0,0,0,18,18,0,0,0,0,1,0,0,0,0,0,0,0,18,18,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,18,18,0,0,0,0,0,0,18,18,0,0,0,0,1,0],[1,16,15,18,1,16,3,4,1,16,3,4,1,16,1,16,15,18,3,4,3,4,1,16,15,18,1,16,1,16,1,16,3,4,3,4],[0,18,0,0,0,0,18,0,0,0,0,0,0,0,18,0,0,0,0,0,1,1,0,0,0,0,0,0,1,1,0,0,0,0,0,18] >;

C₂×He₃.₃S₃ in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3._3S_3

% in TeX

G:=Group("C2xHe3.3S3");

// GroupNames label

G:=SmallGroup(324,78);

// by ID

G=gap.SmallGroup(324,78);

# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,146,5763,303,237,7564,1096,7781]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂×He₃.₃S₃ in TeX

G = C2×He3.3S3 order 324 = 22·34

Direct product of C2 and He3.3S3

G = C₂×He₃.₃S₃ order 324 = 2²·3⁴

Direct product of C₂ and He₃.₃S₃