C2xA4xD9 - GroupNames

G = C₂×A₄×D₉ order 432 = 2⁴·3³

Direct product of C₂, A₄ and D₉

direct product, metabelian, soluble, monomial, A-group

Aliases: C₂×A₄×D₉, C₁₈⋊₃(C₂×A₄), (A₄×C₁₈)⋊₃C₂, (C₆×A₄).₈S₃, C₆.₁₁(S₃×A₄), C₉⋊₃(C₂²×A₄), (C₃×A₄).₄D₆, C₂³⋊₂(C₃×D₉), C₂²⋊₂(C₆×D₉), (C₉×A₄)⋊₄C₂², (C₂²×C₁₈)⋊₁C₆, (C₂²×D₉)⋊₅C₆, (C₂³×D₉)⋊₁C₃, C₃.₂(C₂×S₃×A₄), (C₂×C₁₈)⋊₃(C₂×C₆), (C₂×C₆).₆(S₃×C₆), (C₂²×C₆).₁₅(C₃×S₃), SmallGroup(432,540)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₈ — C₂×A₄×D₉

Chief series

C₁ — C₃ — C₉ — C₂×C₁₈ — C₉×A₄ — A₄×D₉ — C₂×A₄×D₉

Lower central

C₂×C₁₈ — C₂×A₄×D₉

Upper central

C₁ — C₂

Generators and relations for C₂×A₄×D₉
G = < a,b,c,d,e,f | a²=b²=c²=d³=e⁹=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, dbd^-1=bc=cb, be=eb, bf=fb, dcd^-1=b, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 976 in 125 conjugacy classes, 27 normal (21 characteristic)
C₁, C₂, C₂, C₃, C₃, C₂², C₂², S₃, C₆, C₆, C₂³, C₂³, C₉, C₉, C₃², A₄, A₄, D₆, C₂×C₆, C₂×C₆, C₂⁴, D₉, D₉, C₁₈, C₁₈, C₃×S₃, C₃×C₆, C₂×A₄, C₂×A₄, C₂²×S₃, C₂²×C₆, C₃×C₉, C₃.A₄, D₁₈, D₁₈, C₂×C₁₈, C₂×C₁₈, C₃×A₄, S₃×C₆, C₂²×A₄, S₃×C₂³, C₃×D₉, C₃×C₁₈, C₂×C₃.A₄, C₂²×D₉, C₂²×D₉, C₂²×C₁₈, S₃×A₄, C₆×A₄, C₉×A₄, C₆×D₉, C₂³×D₉, C₂×S₃×A₄, A₄×D₉, A₄×C₁₈, C₂×A₄×D₉
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, A₄, D₆, C₂×C₆, D₉, C₃×S₃, C₂×A₄, D₁₈, S₃×C₆, C₂²×A₄, C₃×D₉, S₃×A₄, C₆×D₉, C₂×S₃×A₄, A₄×D₉, C₂×A₄×D₉

Smallest permutation representation of C₂×A₄×D₉
►On 54 points

Generators in S₅₄

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

(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)

(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)

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

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

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

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

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

48 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	3D	3E	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	6K	9A	9B	9C	9D	···	9I	18A	18B	18C	18D	···	18I	18J	···	18O
order	1	2	2	2	2	2	2	2	3	3	3	3	3	6	6	6	6	6	6	6	6	6	6	6	9	9	9	9	···	9	18	18	18	18	···	18	18	···	18
size	1	1	3	3	9	9	27	27	2	4	4	8	8	2	4	4	6	6	8	8	36	36	36	36	2	2	2	8	···	8	2	2	2	6	···	6	8	···	8

48 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	3	3	3	6	6	6	6
type	+	+	+				+	+	+		+				+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	D₆	D₉	C₃×S₃	D₁₈	S₃×C₆	C₃×D₉	C₆×D₉	A₄	C₂×A₄	C₂×A₄	S₃×A₄	C₂×S₃×A₄	A₄×D₉	C₂×A₄×D₉
kernel	C₂×A₄×D₉	A₄×D₉	A₄×C₁₈	C₂³×D₉	C₂²×D₉	C₂²×C₁₈	C₆×A₄	C₃×A₄	C₂×A₄	C₂²×C₆	A₄	C₂×C₆	C₂³	C₂²	D₁₈	D₉	C₁₈	C₆	C₃	C₂	C₁
# reps	1	2	1	2	4	2	1	1	3	2	3	2	6	6	1	2	1	1	1	3	3

Matrix representation of C₂×A₄×D₉ ►in GL₅(𝔽₁₉)

1	0	0	0	0
0	1	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	18	0	0
0	0	7	1	0
0	0	8	5	18

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	12	18	0
0	0	12	0	18

1	0	0	0	0
0	1	0	0	0
0	0	1	3	0
0	0	0	7	12
0	0	0	0	11

5	7	0	0	0
12	17	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

5	7	0	0	0
2	14	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

G:=sub<GL(5,GF(19))| [1,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,18,7,8,0,0,0,1,5,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,1,12,12,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,3,7,0,0,0,0,12,11],[5,12,0,0,0,7,17,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[5,2,0,0,0,7,14,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

C₂×A₄×D₉ in GAP, Magma, Sage, TeX

C_2\times A_4\times D_9

% in TeX

G:=Group("C2xA4xD9");

// GroupNames label

G:=SmallGroup(432,540);

// by ID

G=gap.SmallGroup(432,540);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-3,-3,269,123,10085,292,14118]);

// Polycyclic

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

// generators/relations

G = C2×A4×D9 order 432 = 24·33

Direct product of C2, A4 and D9

G = C₂×A₄×D₉ order 432 = 2⁴·3³

Direct product of C₂, A₄ and D₉