C3xC3^2.A4 - GroupNames

G = C₃×C₃².A₄ order 324 = 2²·3⁴

Direct product of C₃ and C₃².A₄

direct product, metabelian, soluble, monomial

Aliases: C₃×C₃².A₄, C₃³.₂A₄, C₆².₂₇C₃², C₃.A₄⋊₃C₃², (C₃×C₆²).₄C₃, C₃.₉(C₃²×A₄), (C₂×C₆).₉C₃³, C₃².₂₃(C₃×A₄), (C₂×C₆)⋊₂3_-¹⁺², C₂²⋊₃(C₃×3_-¹⁺²), (C₃×C₃.A₄)⋊₉C₃, SmallGroup(324,134)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₃×C₃².A₄

Chief series

C₁ — C₂² — C₂×C₆ — C₃.A₄ — C₃×C₃.A₄ — C₃×C₃².A₄

Lower central

C₂² — C₂×C₆ — C₃×C₃².A₄

Upper central

C₁ — C₃² — C₃³

Generators and relations for C₃×C₃².A₄
G = < a,b,c,d,e,f | a³=b³=c³=d²=e²=1, f³=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf^-1=bc^-1, cd=dc, ce=ec, cf=fc, fdf^-1=de=ed, fef^-1=d >

Subgroups: 250 in 104 conjugacy classes, 42 normal (9 characteristic)
C₁, C₂, C₃, C₃, C₃, C₂², C₆, C₉, C₃², C₃², C₃², C₂×C₆, C₂×C₆, C₂×C₆, C₃×C₆, C₃×C₉, 3_-¹⁺², C₃³, C₃.A₄, C₆², C₆², C₆², C₃²×C₆, C₃×3_-¹⁺², C₃×C₃.A₄, C₃².A₄, C₃×C₆², C₃×C₃².A₄
Quotients: C₁, C₃, C₃², A₄, 3_-¹⁺², C₃³, C₃×A₄, C₃×3_-¹⁺², C₃².A₄, C₃²×A₄, C₃×C₃².A₄

Smallest permutation representation of C₃×C₃².A₄
►On 54 points

Generators in S₅₄

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

(2 8 5)(3 6 9)(10 16 13)(11 14 17)(19 25 22)(20 23 26)(28 34 31)(29 32 35)(37 40 43)(39 45 42)(47 53 50)(48 51 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)

(1 52)(3 54)(4 46)(6 48)(7 49)(9 51)(11 29)(12 30)(14 32)(15 33)(17 35)(18 36)(20 37)(21 38)(23 40)(24 41)(26 43)(27 44)

(1 52)(2 53)(4 46)(5 47)(7 49)(8 50)(10 28)(12 30)(13 31)(15 33)(16 34)(18 36)(19 45)(21 38)(22 39)(24 41)(25 42)(27 44)

(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)

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

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

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

60 conjugacy classes

class	1	2	3A	···	3H	3I	···	3N	6A	···	6Z	9A	···	9R
order	1	2	3	···	3	3	···	3	6	···	6	9	···	9
size	1	3	1	···	1	3	···	3	3	···	3	12	···	12

60 irreducible representations

dim	1	1	1	1	3	3	3	3
type	+				+
image	C₁	C₃	C₃	C₃	A₄	3_-¹⁺²	C₃×A₄	C₃².A₄
kernel	C₃×C₃².A₄	C₃×C₃.A₄	C₃².A₄	C₃×C₆²	C₃³	C₂×C₆	C₃²	C₃
# reps	1	6	18	2	1	6	8	18

Matrix representation of C₃×C₃².A₄ ►in GL₆(𝔽₁₉)

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

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

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

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

G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,18],[0,0,11,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,7,0,0,0,0,0,0,7,0] >;

C₃×C₃².A₄ in GAP, Magma, Sage, TeX

C_3\times C_3^2.A_4

% in TeX

G:=Group("C3xC3^2.A4");

// GroupNames label

G:=SmallGroup(324,134);

// by ID

G=gap.SmallGroup(324,134);

# by ID

G:=PCGroup([6,-3,-3,-3,-3,-2,2,162,650,4864,8753]);

// Polycyclic

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

// generators/relations

G = C3×C32.A4 order 324 = 22·34

Direct product of C3 and C32.A4

G = C₃×C₃².A₄ order 324 = 2²·3⁴

Direct product of C₃ and C₃².A₄