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## G = C2×C4×D9order 144 = 24·32

### Direct product of C2×C4 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C2×C4×D9
 Chief series C1 — C3 — C9 — C18 — D18 — C22×D9 — C2×C4×D9
 Lower central C9 — C2×C4×D9
 Upper central C1 — C2×C4

Generators and relations for C2×C4×D9
G = < a,b,c,d | a2=b4=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 271 in 81 conjugacy classes, 43 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C9, Dic3, C12, D6, C2×C6, C22×C4, D9, C18, C18, C4×S3, C2×Dic3, C2×C12, C22×S3, Dic9, C36, D18, C2×C18, S3×C2×C4, C4×D9, C2×Dic9, C2×C36, C22×D9, C2×C4×D9
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, D9, C4×S3, C22×S3, D18, S3×C2×C4, C4×D9, C22×D9, C2×C4×D9

Smallest permutation representation of C2×C4×D9
On 72 points
Generators in S72
(1 41)(2 42)(3 43)(4 44)(5 45)(6 37)(7 38)(8 39)(9 40)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)
(1 32 14 23)(2 33 15 24)(3 34 16 25)(4 35 17 26)(5 36 18 27)(6 28 10 19)(7 29 11 20)(8 30 12 21)(9 31 13 22)(37 64 46 55)(38 65 47 56)(39 66 48 57)(40 67 49 58)(41 68 50 59)(42 69 51 60)(43 70 52 61)(44 71 53 62)(45 72 54 63)
(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)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)
(1 49)(2 48)(3 47)(4 46)(5 54)(6 53)(7 52)(8 51)(9 50)(10 44)(11 43)(12 42)(13 41)(14 40)(15 39)(16 38)(17 37)(18 45)(19 71)(20 70)(21 69)(22 68)(23 67)(24 66)(25 65)(26 64)(27 72)(28 62)(29 61)(30 60)(31 59)(32 58)(33 57)(34 56)(35 55)(36 63)

G:=sub<Sym(72)| (1,41)(2,42)(3,43)(4,44)(5,45)(6,37)(7,38)(8,39)(9,40)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72), (1,32,14,23)(2,33,15,24)(3,34,16,25)(4,35,17,26)(5,36,18,27)(6,28,10,19)(7,29,11,20)(8,30,12,21)(9,31,13,22)(37,64,46,55)(38,65,47,56)(39,66,48,57)(40,67,49,58)(41,68,50,59)(42,69,51,60)(43,70,52,61)(44,71,53,62)(45,72,54,63), (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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,49)(2,48)(3,47)(4,46)(5,54)(6,53)(7,52)(8,51)(9,50)(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,45)(19,71)(20,70)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,72)(28,62)(29,61)(30,60)(31,59)(32,58)(33,57)(34,56)(35,55)(36,63)>;

G:=Group( (1,41)(2,42)(3,43)(4,44)(5,45)(6,37)(7,38)(8,39)(9,40)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72), (1,32,14,23)(2,33,15,24)(3,34,16,25)(4,35,17,26)(5,36,18,27)(6,28,10,19)(7,29,11,20)(8,30,12,21)(9,31,13,22)(37,64,46,55)(38,65,47,56)(39,66,48,57)(40,67,49,58)(41,68,50,59)(42,69,51,60)(43,70,52,61)(44,71,53,62)(45,72,54,63), (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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,49)(2,48)(3,47)(4,46)(5,54)(6,53)(7,52)(8,51)(9,50)(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,45)(19,71)(20,70)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,72)(28,62)(29,61)(30,60)(31,59)(32,58)(33,57)(34,56)(35,55)(36,63) );

G=PermutationGroup([[(1,41),(2,42),(3,43),(4,44),(5,45),(6,37),(7,38),(8,39),(9,40),(10,46),(11,47),(12,48),(13,49),(14,50),(15,51),(16,52),(17,53),(18,54),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(25,61),(26,62),(27,63),(28,64),(29,65),(30,66),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72)], [(1,32,14,23),(2,33,15,24),(3,34,16,25),(4,35,17,26),(5,36,18,27),(6,28,10,19),(7,29,11,20),(8,30,12,21),(9,31,13,22),(37,64,46,55),(38,65,47,56),(39,66,48,57),(40,67,49,58),(41,68,50,59),(42,69,51,60),(43,70,52,61),(44,71,53,62),(45,72,54,63)], [(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),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72)], [(1,49),(2,48),(3,47),(4,46),(5,54),(6,53),(7,52),(8,51),(9,50),(10,44),(11,43),(12,42),(13,41),(14,40),(15,39),(16,38),(17,37),(18,45),(19,71),(20,70),(21,69),(22,68),(23,67),(24,66),(25,65),(26,64),(27,72),(28,62),(29,61),(30,60),(31,59),(32,58),(33,57),(34,56),(35,55),(36,63)]])

C2×C4×D9 is a maximal subgroup of
D18⋊C8  C422D9  Dic94D4  C23.9D18  D18⋊D4  C4⋊C47D9  D36⋊C4  D18.D4  C4⋊D36  D18⋊Q8  D182Q8  C362D4  D183Q8
C2×C4×D9 is a maximal quotient of
C422D9  C23.16D18  Dic94D4  Dic93Q8  C4⋊C47D9  D36⋊C4  D36.2C4  D36.C4

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 9A 9B 9C 12A 12B 12C 12D 18A ··· 18I 36A ··· 36L order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 9 9 9 12 12 12 12 18 ··· 18 36 ··· 36 size 1 1 1 1 9 9 9 9 2 1 1 1 1 9 9 9 9 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D6 D6 D9 C4×S3 D18 D18 C4×D9 kernel C2×C4×D9 C4×D9 C2×Dic9 C2×C36 C22×D9 D18 C2×C12 C12 C2×C6 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 8 1 2 1 3 4 6 3 12

Matrix representation of C2×C4×D9 in GL3(𝔽37) generated by

 36 0 0 0 1 0 0 0 1
,
 1 0 0 0 31 0 0 0 31
,
 1 0 0 0 26 6 0 31 20
,
 1 0 0 0 11 17 0 6 26
G:=sub<GL(3,GF(37))| [36,0,0,0,1,0,0,0,1],[1,0,0,0,31,0,0,0,31],[1,0,0,0,26,31,0,6,20],[1,0,0,0,11,6,0,17,26] >;

C2×C4×D9 in GAP, Magma, Sage, TeX

C_2\times C_4\times D_9
% in TeX

G:=Group("C2xC4xD9");
// GroupNames label

G:=SmallGroup(144,38);
// by ID

G=gap.SmallGroup(144,38);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,50,2404,208,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^9=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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