direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C4×D9, C36⋊3C22, C12.54D6, C18.2C23, C22.9D18, Dic9⋊3C22, D18.4C22, C18⋊1(C2×C4), (C2×C36)⋊5C2, C9⋊1(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
| C9 — C2×C4×D9 |
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
►On 72 points
(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 C42⋊2D9 Dic9⋊4D4 C23.9D18 D18⋊D4 C4⋊C4⋊7D9 D36⋊C4 D18.D4 C4⋊D36 D18⋊Q8 D18⋊2Q8 C36⋊2D4 D18⋊3Q8
C2×C4×D9 is a maximal quotient of
C42⋊2D9 C23.16D18 Dic9⋊4D4 Dic9⋊3Q8 C4⋊C4⋊7D9 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