metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D36:1C4, C42:4D9, C4.17D36, C36.33D4, Dic18:1C4, C12.52D12, C9:1C4wrC2, (C4xC36):6C2, C4.6(C4xD9), C12.54(C4xS3), C36.16(C2xC4), (C4xC12).11S3, C6.8(D6:C4), (C2xC4).71D18, (C2xC18).27D4, C4.Dic9:1C2, C2.3(D18:C4), C3.(C42:4S3), (C2xC12).391D6, D36:5C2.1C2, C18.1(C22:C4), (C2xC36).94C22, C22.7(C9:D4), (C2xC6).65(C3:D4), SmallGroup(288,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42:4D9
G = < a,b,c,d | a4=b4=c9=d2=1, dad=ab=ba, ac=ca, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 292 in 66 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, Q8, C9, Dic3, C12, C12, D6, C2xC6, C42, M4(2), C4oD4, D9, C18, C18, C3:C8, Dic6, C4xS3, D12, C3:D4, C2xC12, C2xC12, C4wrC2, Dic9, C36, C36, D18, C2xC18, C4.Dic3, C4xC12, C4oD12, C9:C8, Dic18, C4xD9, D36, C9:D4, C2xC36, C2xC36, C42:4S3, C4.Dic9, C4xC36, D36:5C2, C42:4D9
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, D6, C22:C4, D9, C4xS3, D12, C3:D4, C4wrC2, D18, D6:C4, C4xD9, D36, C9:D4, C42:4S3, D18:C4, C42:4D9
►On 72 points
(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 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 55 46 64)(38 56 47 65)(39 57 48 66)(40 58 49 67)(41 59 50 68)(42 60 51 69)(43 61 52 70)(44 62 53 71)(45 63 54 72)
(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 39)(2 38)(3 37)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 52)(11 51)(12 50)(13 49)(14 48)(15 47)(16 46)(17 54)(18 53)(19 61)(20 60)(21 59)(22 58)(23 57)(24 56)(25 55)(26 63)(27 62)(28 70)(29 69)(30 68)(31 67)(32 66)(33 65)(34 64)(35 72)(36 71)
G:=sub<Sym(72)| (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), (1,32,14,23)(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)(37,55,46,64)(38,56,47,65)(39,57,48,66)(40,58,49,67)(41,59,50,68)(42,60,51,69)(43,61,52,70)(44,62,53,71)(45,63,54,72), (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), (1,39)(2,38)(3,37)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,52)(11,51)(12,50)(13,49)(14,48)(15,47)(16,46)(17,54)(18,53)(19,61)(20,60)(21,59)(22,58)(23,57)(24,56)(25,55)(26,63)(27,62)(28,70)(29,69)(30,68)(31,67)(32,66)(33,65)(34,64)(35,72)(36,71)>;
G:=Group( (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), (1,32,14,23)(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)(37,55,46,64)(38,56,47,65)(39,57,48,66)(40,58,49,67)(41,59,50,68)(42,60,51,69)(43,61,52,70)(44,62,53,71)(45,63,54,72), (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), (1,39)(2,38)(3,37)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,52)(11,51)(12,50)(13,49)(14,48)(15,47)(16,46)(17,54)(18,53)(19,61)(20,60)(21,59)(22,58)(23,57)(24,56)(25,55)(26,63)(27,62)(28,70)(29,69)(30,68)(31,67)(32,66)(33,65)(34,64)(35,72)(36,71) );
G=PermutationGroup([[(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)], [(1,32,14,23),(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),(37,55,46,64),(38,56,47,65),(39,57,48,66),(40,58,49,67),(41,59,50,68),(42,60,51,69),(43,61,52,70),(44,62,53,71),(45,63,54,72)], [(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)], [(1,39),(2,38),(3,37),(4,45),(5,44),(6,43),(7,42),(8,41),(9,40),(10,52),(11,51),(12,50),(13,49),(14,48),(15,47),(16,46),(17,54),(18,53),(19,61),(20,60),(21,59),(22,58),(23,57),(24,56),(25,55),(26,63),(27,62),(28,70),(29,69),(30,68),(31,67),(32,66),(33,65),(34,64),(35,72),(36,71)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 6A | 6B | 6C | 8A | 8B | 9A | 9B | 9C | 12A | ··· | 12L | 18A | ··· | 18I | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 36 | 2 | 1 | 1 | 2 | ··· | 2 | 36 | 2 | 2 | 2 | 36 | 36 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D6 | D9 | C4xS3 | D12 | C3:D4 | C4wrC2 | D18 | C4xD9 | D36 | C9:D4 | C42:4S3 | C42:4D9 |
| kernel | C42:4D9 | C4.Dic9 | C4xC36 | D36:5C2 | Dic18 | D36 | C4xC12 | C36 | C2xC18 | C2xC12 | C42 | C12 | C12 | C2xC6 | C9 | C2xC4 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 2 | 4 | 3 | 6 | 6 | 6 | 8 | 24 |
Matrix representation of C42:4D9 ►in GL2(F37) generated by
| 28 | 25 |
| 2 | 2 |
| 9 | 23 |
| 27 | 28 |
| 25 | 1 |
| 6 | 21 |
| 25 | 7 |
| 6 | 12 |
G:=sub<GL(2,GF(37))| [28,2,25,2],[9,27,23,28],[25,6,1,21],[25,6,7,12] >;
C42:4D9 in GAP, Magma, Sage, TeX
C_4^2\rtimes_4D_9
% in TeX
G:=Group("C4^2:4D9"); // GroupNames label
G:=SmallGroup(288,12);
// by ID
G=gap.SmallGroup(288,12);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,675,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^9=d^2=1,d*a*d=a*b=b*a,a*c=c*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations