metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D18:4D4, C22:3D36, C23.20D18, C9:1C22wrC2, (C2xC4):1D18, (C2xC18):1D4, C2.7(D4xD9), D18:C4:4C2, (C2xD36):2C2, C22:C4:2D9, C18.5(C2xD4), C6.79(S3xD4), (C2xC12).2D6, C2.7(C2xD36), (C2xC6).4D12, (C2xC36):1C22, C3.(D6:D4), C6.34(C2xD12), (C23xD9):1C2, (C22xC6).42D6, (C2xC18).23C23, (C2xDic9):1C22, (C22xD9):1C22, C22.41(C22xD9), (C22xC18).12C22, (C2xC9:D4):1C2, (C9xC22:C4):3C2, (C3xC22:C4).3S3, (C2xC6).180(C22xS3), SmallGroup(288,92)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22:3D36
G = < a,b,c,d | a2=b2=c36=d2=1, cac-1=dad=ab=ba, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1148 in 195 conjugacy classes, 48 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2xC4, C2xC4, D4, C23, C23, C9, Dic3, C12, D6, C2xC6, C2xC6, C2xC6, C22:C4, C22:C4, C2xD4, C24, D9, C18, C18, C18, D12, C2xDic3, C3:D4, C2xC12, C22xS3, C22xC6, C22wrC2, Dic9, C36, D18, D18, C2xC18, C2xC18, C2xC18, D6:C4, C3xC22:C4, C2xD12, C2xC3:D4, S3xC23, D36, C2xDic9, C9:D4, C2xC36, C22xD9, C22xD9, C22xD9, C22xC18, D6:D4, D18:C4, C9xC22:C4, C2xD36, C2xC9:D4, C23xD9, C22:3D36
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, D9, D12, C22xS3, C22wrC2, D18, C2xD12, S3xD4, D36, C22xD9, D6:D4, C2xD36, D4xD9, C22:3D36
►On 72 points
(1 57)(3 59)(5 61)(7 63)(9 65)(11 67)(13 69)(15 71)(17 37)(19 39)(21 41)(23 43)(25 45)(27 47)(29 49)(31 51)(33 53)(35 55)
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(36 56)
(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 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 72)(22 71)(23 70)(24 69)(25 68)(26 67)(27 66)(28 65)(29 64)(30 63)(31 62)(32 61)(33 60)(34 59)(35 58)(36 57)
G:=sub<Sym(72)| (1,57)(3,59)(5,61)(7,63)(9,65)(11,67)(13,69)(15,71)(17,37)(19,39)(21,41)(23,43)(25,45)(27,47)(29,49)(31,51)(33,53)(35,55), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56), (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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)>;
G:=Group( (1,57)(3,59)(5,61)(7,63)(9,65)(11,67)(13,69)(15,71)(17,37)(19,39)(21,41)(23,43)(25,45)(27,47)(29,49)(31,51)(33,53)(35,55), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56), (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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57) );
G=PermutationGroup([[(1,57),(3,59),(5,61),(7,63),(9,65),(11,67),(13,69),(15,71),(17,37),(19,39),(21,41),(23,43),(25,45),(27,47),(29,49),(31,51),(33,53),(35,55)], [(1,57),(2,58),(3,59),(4,60),(5,61),(6,62),(7,63),(8,64),(9,65),(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,55),(36,56)], [(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)], [(1,56),(2,55),(3,54),(4,53),(5,52),(6,51),(7,50),(8,49),(9,48),(10,47),(11,46),(12,45),(13,44),(14,43),(15,42),(16,41),(17,40),(18,39),(19,38),(20,37),(21,72),(22,71),(23,70),(24,69),(25,68),(26,67),(27,66),(28,65),(29,64),(30,63),(31,62),(32,61),(33,60),(34,59),(35,58),(36,57)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 18J | ··· | 18O | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 18 | 18 | 18 | 18 | 36 | 2 | 4 | 4 | 36 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D9 | D12 | D18 | D18 | D36 | S3xD4 | D4xD9 |
| kernel | C22:3D36 | D18:C4 | C9xC22:C4 | C2xD36 | C2xC9:D4 | C23xD9 | C3xC22:C4 | D18 | C2xC18 | C2xC12 | C22xC6 | C22:C4 | C2xC6 | C2xC4 | C23 | C22 | C6 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 3 | 4 | 6 | 3 | 12 | 2 | 6 |
Matrix representation of C22:3D36 ►in GL4(F37) generated by
| 36 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 36 | 0 | 0 | 0 |
| 0 | 0 | 4 | 8 |
| 0 | 0 | 29 | 12 |
| 0 | 36 | 0 | 0 |
| 36 | 0 | 0 | 0 |
| 0 | 0 | 32 | 10 |
| 0 | 0 | 5 | 5 |
G:=sub<GL(4,GF(37))| [36,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[0,36,0,0,1,0,0,0,0,0,4,29,0,0,8,12],[0,36,0,0,36,0,0,0,0,0,32,5,0,0,10,5] >;
C22:3D36 in GAP, Magma, Sage, TeX
C_2^2\rtimes_3D_{36} % in TeX
G:=Group("C2^2:3D36"); // GroupNames label
G:=SmallGroup(288,92);
// by ID
G=gap.SmallGroup(288,92);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,219,58,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^36=d^2=1,c*a*c^-1=d*a*d=a*b=b*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations