C2xC9.S4 - GroupNames

G = C₂×C₉.S₄ order 432 = 2⁴·3³

Direct product of C₂ and C₉.S₄

direct product, non-abelian, soluble, monomial

Aliases: C₂×C₉.S₄, C₂³⋊D₂₇, C₂²⋊D₅₄, C₁₈.₄S₄, C₉.(C₂×S₄), (C₂×C₁₈).D₆, (C₂×C₆).D₁₈, C₉.A₄⋊C₂², C₆.₄(C₃.S₄), (C₂²×C₆).₃D₉, (C₂²×C₁₈).₃S₃, (C₂×C₉.A₄)⋊C₂, C₃.(C₂×C₃.S₄), SmallGroup(432,224)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₉.A₄ — C₂×C₉.S₄

Chief series

C₁ — C₂² — C₂×C₆ — C₂×C₁₈ — C₉.A₄ — C₉.S₄ — C₂×C₉.S₄

Lower central

C₉.A₄ — C₂×C₉.S₄

Upper central

C₁ — C₂

Generators and relations for C₂×C₉.S₄
G = < a,b,c,d,e,f | a²=b⁹=c²=d²=f²=1, e³=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b^-1, ece^-1=fcf=cd=dc, ede^-1=c, df=fd, fef=b^-1e² >

Subgroups: 814 in 79 conjugacy classes, 17 normal (15 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂×C₄, D₄, C₂³, C₂³, C₉, Dic₃, D₆, C₂×C₆, C₂×C₆, C₂×D₄, D₉, C₁₈, C₁₈, C₂×Dic₃, C₃⋊D₄, C₂²×S₃, C₂²×C₆, C₂₇, Dic₉, D₁₈, C₂×C₁₈, C₂×C₁₈, C₂×C₃⋊D₄, D₂₇, C₅₄, C₂×Dic₉, C₉⋊D₄, C₂²×D₉, C₂²×C₁₈, C₉.A₄, D₅₄, C₂×C₉⋊D₄, C₉.S₄, C₂×C₉.A₄, C₂×C₉.S₄
Quotients: C₁, C₂, C₂², S₃, D₆, D₉, S₄, D₁₈, C₂×S₄, D₂₇, C₃.S₄, D₅₄, C₂×C₃.S₄, C₉.S₄, C₂×C₉.S₄

Smallest permutation representation of C₂×C₉.S₄
►On 54 points

Generators in S₅₄

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

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

(1 45)(2 46)(4 48)(5 49)(7 51)(8 52)(10 54)(11 28)(13 30)(14 31)(16 33)(17 34)(19 36)(20 37)(22 39)(23 40)(25 42)(26 43)

(2 46)(3 47)(5 49)(6 50)(8 52)(9 53)(11 28)(12 29)(14 31)(15 32)(17 34)(18 35)(20 37)(21 38)(23 40)(24 41)(26 43)(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)

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

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

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

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

42 conjugacy classes

class	1	2A	2B	2C	2D	2E	3	4A	4B	6A	6B	6C	9A	9B	9C	18A	18B	18C	18D	···	18I	27A	···	27I	54A	···	54I
order	1	2	2	2	2	2	3	4	4	6	6	6	9	9	9	18	18	18	18	···	18	27	···	27	54	···	54
size	1	1	3	3	54	54	2	54	54	2	6	6	2	2	2	2	2	2	6	···	6	8	···	8	8	···	8

42 irreducible representations

dim

type

image

C₁

C₂

S₃

D₆

D₉

D₁₈

D₂₇

D₅₄

S₄

C₂×S₄

C₃.S₄

C₂×C₃.S₄

C₉.S₄

C₂×C₉.S₄

kernel

C₂×C₉.S₄

C₉.S₄

C₂×C₉.A₄

C₂²×C₁₈

C₂×C₁₈

C₂²×C₆

C₂×C₆

C₂³

C₂²

C₁₈

C₉

C₆

C₃

C₂

C₁

# reps

Matrix representation of C₂×C₉.S₄ ►in GL₅(𝔽₁₀₉)

108	0	0	0	0
0	108	0	0	0
0	0	108	0	0
0	0	0	108	0
0	0	0	0	108

5	54	0	0	0
14	86	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

1	0	0	0	0
0	1	0	0	0
0	0	108	0	0
0	0	0	1	0
0	0	0	108	108

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	108	0
0	0	108	0	108

82	34	0	0	0
29	24	0	0	0
0	0	0	1	0
0	0	108	108	107
0	0	0	0	1

106	20	0	0	0
65	3	0	0	0
0	0	1	0	0
0	0	108	108	107
0	0	0	0	1

G:=sub<GL(5,GF(109))| [108,0,0,0,0,0,108,0,0,0,0,0,108,0,0,0,0,0,108,0,0,0,0,0,108],[5,14,0,0,0,54,86,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,108,0,0,0,0,0,1,108,0,0,0,0,108],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,108,0,0,0,108,0,0,0,0,0,108],[82,29,0,0,0,34,24,0,0,0,0,0,0,108,0,0,0,1,108,0,0,0,0,107,1],[106,65,0,0,0,20,3,0,0,0,0,0,1,108,0,0,0,0,108,0,0,0,0,107,1] >;

C₂×C₉.S₄ in GAP, Magma, Sage, TeX

C_2\times C_9.S_4

% in TeX

G:=Group("C2xC9.S4");

// GroupNames label

G:=SmallGroup(432,224);

// by ID

G=gap.SmallGroup(432,224);

# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,422,331,1683,192,2524,9077,2287,5298,3989]);

// Polycyclic

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

// generators/relations

G = C2×C9.S4 order 432 = 24·33

Direct product of C2 and C9.S4

G = C₂×C₉.S₄ order 432 = 2⁴·3³

Direct product of C₂ and C₉.S₄