metabelian, supersoluble, monomial
Aliases: D36⋊6C6, Dic18⋊6C6, C62.40D6, (C2×C36)⋊1C6, (C4×D9)⋊4C6, C9⋊D4⋊3C6, D36⋊C3⋊5C2, D36⋊5C2⋊C3, (C6×C12).8S3, C36.12(C2×C6), Dic9⋊C6⋊3C2, D18.1(C2×C6), (C3×C12).53D6, C36.C6⋊5C2, C12.100(S3×C6), C9⋊C12.2C22, C18.4(C22×C6), C32.(C4○D12), Dic9.2(C2×C6), 3- 1+2⋊1(C4○D4), (C2×3- 1+2).4C23, (C4×3- 1+2).16C22, (C22×3- 1+2).10C22, (C4×C9⋊C6)⋊4C2, C9⋊1(C3×C4○D4), C6.30(S3×C2×C6), (C2×C4)⋊3(C9⋊C6), C4.16(C2×C9⋊C6), (C2×C6).60(S3×C6), C3.3(C3×C4○D12), C2.5(C22×C9⋊C6), C22.2(C2×C9⋊C6), (C2×C18).10(C2×C6), (C2×C12).27(C3×S3), (C2×C9⋊C6).1C22, (C3×C6).26(C22×S3), (C2×C4×3- 1+2)⋊1C2, SmallGroup(432,355)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D36⋊6C6
G = < a,b,c | a36=b2=c6=1, bab=a-1, cac-1=a25, cbc-1=a6b >
Subgroups: 470 in 128 conjugacy classes, 52 normal (36 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C4○D4, D9, C18, C18, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, 3- 1+2, Dic9, C36, C36, D18, C2×C18, C2×C18, C3×Dic3, C3×C12, S3×C6, C62, C4○D12, C3×C4○D4, C9⋊C6, C2×3- 1+2, C2×3- 1+2, Dic18, C4×D9, D36, C9⋊D4, C2×C36, C2×C36, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, C9⋊C12, C4×3- 1+2, C2×C9⋊C6, C22×3- 1+2, D36⋊5C2, C3×C4○D12, C36.C6, C4×C9⋊C6, D36⋊C3, Dic9⋊C6, C2×C4×3- 1+2, D36⋊6C6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C22×S3, C22×C6, S3×C6, C4○D12, C3×C4○D4, C9⋊C6, S3×C2×C6, C2×C9⋊C6, C3×C4○D12, C22×C9⋊C6, D36⋊6C6
►On 72 points
(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 45)(2 44)(3 43)(4 42)(5 41)(6 40)(7 39)(8 38)(9 37)(10 72)(11 71)(12 70)(13 69)(14 68)(15 67)(16 66)(17 65)(18 64)(19 63)(20 62)(21 61)(22 60)(23 59)(24 58)(25 57)(26 56)(27 55)(28 54)(29 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)
(2 14 26)(3 27 15)(5 17 29)(6 30 18)(8 20 32)(9 33 21)(11 23 35)(12 36 24)(37 55)(38 68 62 56 50 44)(39 45 51 57 63 69)(40 58)(41 71 65 59 53 47)(42 48 54 60 66 72)(43 61)(46 64)(49 67)(52 70)
G:=sub<Sym(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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,72)(11,71)(12,70)(13,69)(14,68)(15,67)(16,66)(17,65)(18,64)(19,63)(20,62)(21,61)(22,60)(23,59)(24,58)(25,57)(26,56)(27,55)(28,54)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46), (2,14,26)(3,27,15)(5,17,29)(6,30,18)(8,20,32)(9,33,21)(11,23,35)(12,36,24)(37,55)(38,68,62,56,50,44)(39,45,51,57,63,69)(40,58)(41,71,65,59,53,47)(42,48,54,60,66,72)(43,61)(46,64)(49,67)(52,70)>;
G:=Group( (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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,72)(11,71)(12,70)(13,69)(14,68)(15,67)(16,66)(17,65)(18,64)(19,63)(20,62)(21,61)(22,60)(23,59)(24,58)(25,57)(26,56)(27,55)(28,54)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46), (2,14,26)(3,27,15)(5,17,29)(6,30,18)(8,20,32)(9,33,21)(11,23,35)(12,36,24)(37,55)(38,68,62,56,50,44)(39,45,51,57,63,69)(40,58)(41,71,65,59,53,47)(42,48,54,60,66,72)(43,61)(46,64)(49,67)(52,70) );
G=PermutationGroup([[(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)], [(1,45),(2,44),(3,43),(4,42),(5,41),(6,40),(7,39),(8,38),(9,37),(10,72),(11,71),(12,70),(13,69),(14,68),(15,67),(16,66),(17,65),(18,64),(19,63),(20,62),(21,61),(22,60),(23,59),(24,58),(25,57),(26,56),(27,55),(28,54),(29,53),(30,52),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46)], [(2,14,26),(3,27,15),(5,17,29),(6,30,18),(8,20,32),(9,33,21),(11,23,35),(12,36,24),(37,55),(38,68,62,56,50,44),(39,45,51,57,63,69),(40,58),(41,71,65,59,53,47),(42,48,54,60,66,72),(43,61),(46,64),(49,67),(52,70)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | 12M | 12N | 18A | ··· | 18I | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 18 | 18 | 2 | 3 | 3 | 1 | 1 | 2 | 18 | 18 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 18 | 18 | 18 | 18 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 18 | 18 | 18 | 18 | 6 | ··· | 6 | 6 | ··· | 6 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D6 | D6 | C4○D4 | C3×S3 | S3×C6 | S3×C6 | C3×C4○D4 | C4○D12 | C3×C4○D12 | C9⋊C6 | C2×C9⋊C6 | C2×C9⋊C6 | D36⋊6C6 |
| kernel | D36⋊6C6 | C36.C6 | C4×C9⋊C6 | D36⋊C3 | Dic9⋊C6 | C2×C4×3- 1+2 | D36⋊5C2 | Dic18 | C4×D9 | D36 | C9⋊D4 | C2×C36 | C6×C12 | C3×C12 | C62 | 3- 1+2 | C2×C12 | C12 | C2×C6 | C9 | C32 | C3 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 1 | 2 | 1 | 4 |
Matrix representation of D36⋊6C6 ►in GL6(𝔽37)
| 6 | 0 | 29 | 7 | 8 | 8 |
| 17 | 31 | 31 | 36 | 36 | 36 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 0 | 0 | 31 |
| 0 | 0 | 0 | 14 | 0 | 0 |
| 1 | 0 | 11 | 31 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 31 |
| 0 | 0 | 0 | 14 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 17 | 31 | 31 | 36 | 36 | 36 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 1 | 11 | 1 | 26 | 18 | 32 |
| 0 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
G:=sub<GL(6,GF(37))| [6,17,0,0,0,0,0,31,6,0,0,0,29,31,0,0,0,0,7,36,0,0,0,14,8,36,0,31,0,0,8,36,0,0,31,0],[1,0,0,0,17,0,0,0,0,0,31,6,11,0,0,8,31,0,31,0,14,0,36,0,0,0,0,0,36,0,0,31,0,0,36,0],[1,0,0,0,0,0,11,26,0,0,0,0,1,0,10,0,0,0,26,0,0,36,0,0,18,0,0,0,11,0,32,0,0,0,0,27] >;
D36⋊6C6 in GAP, Magma, Sage, TeX
D_{36}\rtimes_6C_6 % in TeX
G:=Group("D36:6C6"); // GroupNames label
G:=SmallGroup(432,355);
// by ID
G=gap.SmallGroup(432,355);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,10085,1034,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^36=b^2=c^6=1,b*a*b=a^-1,c*a*c^-1=a^25,c*b*c^-1=a^6*b>;
// generators/relations