direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D36, C4⋊2D18, C18⋊1D4, C6.6D12, C36⋊2C22, C12.44D6, D18⋊1C22, C18.3C23, C22.10D18, C9⋊1(C2×D4), (C2×C4)⋊2D9, C3.(C2×D12), (C2×C36)⋊3C2, (C2×C12).7S3, (C2×C6).26D6, (C22×D9)⋊1C2, C2.4(C22×D9), C6.21(C22×S3), (C2×C18).10C22, SmallGroup(144,39)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D36
G = < a,b,c | a2=b36=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 375 in 81 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C9, C12, D6, C2×C6, C2×D4, D9, C18, C18, D12, C2×C12, C22×S3, C36, D18, D18, C2×C18, C2×D12, D36, C2×C36, C22×D9, C2×D36
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, D12, C22×S3, D18, C2×D12, D36, C22×D9, C2×D36
►On 72 points
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 37)(34 38)(35 39)(36 40)
(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 40)(2 39)(3 38)(4 37)(5 72)(6 71)(7 70)(8 69)(9 68)(10 67)(11 66)(12 65)(13 64)(14 63)(15 62)(16 61)(17 60)(18 59)(19 58)(20 57)(21 56)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)
G:=sub<Sym(72)| (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,37)(34,38)(35,39)(36,40), (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), (1,40)(2,39)(3,38)(4,37)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)>;
G:=Group( (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,37)(34,38)(35,39)(36,40), (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), (1,40)(2,39)(3,38)(4,37)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41) );
G=PermutationGroup([[(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,37),(34,38),(35,39),(36,40)], [(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)], [(1,40),(2,39),(3,38),(4,37),(5,72),(6,71),(7,70),(8,69),(9,68),(10,67),(11,66),(12,65),(13,64),(14,63),(15,62),(16,61),(17,60),(18,59),(19,58),(20,57),(21,56),(22,55),(23,54),(24,53),(25,52),(26,51),(27,50),(28,49),(29,48),(30,47),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41)]])
C2×D36 is a maximal subgroup of
C18.D8 C2.D72 C36.48D4 C42⋊6D9 C42⋊7D9 C22⋊3D36 D18⋊D4 D36⋊C4 D18.D4 C4⋊D36 C8⋊D18 C36⋊7D4 C36⋊D4 C36.23D4 D4⋊D18 C2×D4×D9 D4⋊8D18
C2×D36 is a maximal quotient of
C36⋊2Q8 C42⋊6D9 C42⋊7D9 C22⋊3D36 C22.4D36 C4⋊D36 D18⋊2Q8 D72⋊7C2 C8⋊D18 C8.D18 C36⋊7D4
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 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 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D9 | D12 | D18 | D18 | D36 |
| kernel | C2×D36 | D36 | C2×C36 | C22×D9 | C2×C12 | C18 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 1 | 2 | 2 | 1 | 3 | 4 | 6 | 3 | 12 |
Matrix representation of C2×D36 ►in GL3(𝔽37) generated by
| 36 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 33 | 29 |
| 0 | 8 | 25 |
| 1 | 0 | 0 |
| 0 | 8 | 25 |
| 0 | 33 | 29 |
G:=sub<GL(3,GF(37))| [36,0,0,0,1,0,0,0,1],[1,0,0,0,33,8,0,29,25],[1,0,0,0,8,33,0,25,29] >;
C2×D36 in GAP, Magma, Sage, TeX
C_2\times D_{36} % in TeX
G:=Group("C2xD36"); // GroupNames label
G:=SmallGroup(144,39);
// by ID
G=gap.SmallGroup(144,39);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,218,50,2404,208,3461]);
// Polycyclic
G:=Group<a,b,c|a^2=b^36=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations