Copied to
clipboard

G = C2xA4xD9order 432 = 24·33

Direct product of C2, A4 and D9

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

Aliases: C2xA4xD9, C18:3(C2xA4), (A4xC18):3C2, (C6xA4).8S3, C6.11(S3xA4), C9:3(C22xA4), (C3xA4).4D6, C23:2(C3xD9), C22:2(C6xD9), (C9xA4):4C22, (C22xC18):1C6, (C22xD9):5C6, (C23xD9):1C3, C3.2(C2xS3xA4), (C2xC18):3(C2xC6), (C2xC6).6(S3xC6), (C22xC6).15(C3xS3), SmallGroup(432,540)

Series: Derived Chief Lower central Upper central

C1C2xC18 — C2xA4xD9
C1C3C9C2xC18C9xA4A4xD9 — C2xA4xD9
C2xC18 — C2xA4xD9
C1C2

Generators and relations for C2xA4xD9
 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, C2xC6, C2xC6, C24, D9, D9, C18, C18, C3xS3, C3xC6, C2xA4, C2xA4, C22xS3, C22xC6, C3xC9, C3.A4, D18, D18, C2xC18, C2xC18, C3xA4, S3xC6, C22xA4, S3xC23, C3xD9, C3xC18, C2xC3.A4, C22xD9, C22xD9, C22xC18, S3xA4, C6xA4, C9xA4, C6xD9, C23xD9, C2xS3xA4, A4xD9, A4xC18, C2xA4xD9
Quotients: C1, C2, C3, C22, S3, C6, A4, D6, C2xC6, D9, C3xS3, C2xA4, D18, S3xC6, C22xA4, C3xD9, S3xA4, C6xD9, C2xS3xA4, A4xD9, C2xA4xD9

Smallest permutation representation of C2xA4xD9
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 2A2B2C2D2E2F2G3A3B3C3D3E6A6B6C6D6E6F6G6H6I6J6K9A9B9C9D···9I18A18B18C18D···18I18J···18O
order1222222233333666666666669999···918181818···1818···18
size1133992727244882446688363636362228···82226···68···8

48 irreducible representations

dim111111222222223336666
type++++++++++++++
imageC1C2C2C3C6C6S3D6D9C3xS3D18S3xC6C3xD9C6xD9A4C2xA4C2xA4S3xA4C2xS3xA4A4xD9C2xA4xD9
kernelC2xA4xD9A4xD9A4xC18C23xD9C22xD9C22xC18C6xA4C3xA4C2xA4C22xC6A4C2xC6C23C22D18D9C18C6C3C2C1
# reps121242113232661211133

Matrix representation of C2xA4xD9 in GL5(F19)

10000
01000
001800
000180
000018
,
10000
01000
001800
00710
008518
,
10000
01000
00100
0012180
0012018
,
10000
01000
00130
000712
000011
,
57000
1217000
00100
00010
00001
,
57000
214000
00100
00010
00001

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] >;

C2xA4xD9 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

׿
x
:
Z
F
o
wr
Q
<