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## G = C2×A4×D9order 432 = 24·33

### Direct product of C2, A4 and D9

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

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