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G = C2×C4×D9order 144 = 24·32

Direct product of C2×C4 and D9

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C2×C4×D9, C363C22, C12.54D6, C18.2C23, C22.9D18, Dic93C22, D18.4C22, C181(C2×C4), (C2×C36)⋊5C2, C91(C22×C4), C6.10(C4×S3), (C2×C6).25D6, (C2×C12).15S3, (C2×Dic9)⋊5C2, C2.1(C22×D9), C6.20(C22×S3), (C2×C18).9C22, (C22×D9).2C2, C3.(S3×C2×C4), SmallGroup(144,38)

Series: Derived Chief Lower central Upper central

C1C9 — C2×C4×D9
C1C3C9C18D18C22×D9 — C2×C4×D9
C9 — C2×C4×D9
C1C2×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 [×2], C2 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×6], S3 [×4], C6, C6 [×2], C2×C4, C2×C4 [×5], C23, C9, Dic3 [×2], C12 [×2], D6 [×6], C2×C6, C22×C4, D9 [×4], C18, C18 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, Dic9 [×2], C36 [×2], D18 [×6], C2×C18, S3×C2×C4, C4×D9 [×4], C2×Dic9, C2×C36, C22×D9, C2×C4×D9
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], C22×C4, D9, C4×S3 [×2], C22×S3, D18 [×3], S3×C2×C4, C4×D9 [×2], 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 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H6A6B6C9A9B9C12A12B12C12D18A···18I36A···36L
order122222223444444446669991212121218···1836···36
size1111999921111999922222222222···22···2

48 irreducible representations

dim11111122222222
type+++++++++++
imageC1C2C2C2C2C4S3D6D6D9C4×S3D18D18C4×D9
kernelC2×C4×D9C4×D9C2×Dic9C2×C36C22×D9D18C2×C12C12C2×C6C2×C4C6C4C22C2
# reps141118121346312

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

3600
010
001
,
100
0310
0031
,
100
0266
03120
,
100
01117
0626
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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