metabelian, supersoluble, monomial
Aliases: C6.13D36, C36.31D6, Dic18⋊1S3, C12.10D18, C3⋊C8⋊2D9, C12.45S32, C4.9(S3×D9), (C3×C9)⋊2SD16, (C3×C18).2D4, C3⋊2(C72⋊C2), (C3×C12).70D6, (C3×C6).30D12, C36⋊S3.1C2, C9⋊1(Q8⋊2S3), (C3×Dic18)⋊1C2, C18.2(C3⋊D4), (C3×C36).2C22, C6.2(C3⋊D12), C2.5(C3⋊D36), C32.3(C24⋊C2), C3.1(C32⋊5SD16), (C9×C3⋊C8)⋊3C2, (C3×C3⋊C8).5S3, SmallGroup(432,63)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.D36
G = < a,b,c | a36=c2=1, b6=a18, bab-1=cac=a-1, cbc=a27b5 >
Subgroups: 704 in 76 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, SD16, D9, C18, C18, C3⋊S3, C3×C6, C3⋊C8, C24, Dic6, D12, C3×Q8, C3×C9, Dic9, C36, C36, D18, C3×Dic3, C3×C12, C2×C3⋊S3, C24⋊C2, Q8⋊2S3, C9⋊S3, C3×C18, C72, Dic18, D36, C3×C3⋊C8, C3×Dic6, C12⋊S3, C3×Dic9, C3×C36, C2×C9⋊S3, C72⋊C2, C32⋊5SD16, C9×C3⋊C8, C3×Dic18, C36⋊S3, C6.D36
Quotients: C1, C2, C22, S3, D4, D6, SD16, D9, D12, C3⋊D4, D18, S32, C24⋊C2, Q8⋊2S3, D36, C3⋊D12, S3×D9, C72⋊C2, C32⋊5SD16, C3⋊D36, C6.D36
►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 7 39 13 69 19 63 25 57 31 51)(2 44 8 38 14 68 20 62 26 56 32 50)(3 43 9 37 15 67 21 61 27 55 33 49)(4 42 10 72 16 66 22 60 28 54 34 48)(5 41 11 71 17 65 23 59 29 53 35 47)(6 40 12 70 18 64 24 58 30 52 36 46)
(1 34)(2 33)(3 32)(4 31)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)(17 18)(35 36)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(54 72)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)
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), (1,45,7,39,13,69,19,63,25,57,31,51)(2,44,8,38,14,68,20,62,26,56,32,50)(3,43,9,37,15,67,21,61,27,55,33,49)(4,42,10,72,16,66,22,60,28,54,34,48)(5,41,11,71,17,65,23,59,29,53,35,47)(6,40,12,70,18,64,24,58,30,52,36,46), (1,34)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(35,36)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(54,72)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)>;
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), (1,45,7,39,13,69,19,63,25,57,31,51)(2,44,8,38,14,68,20,62,26,56,32,50)(3,43,9,37,15,67,21,61,27,55,33,49)(4,42,10,72,16,66,22,60,28,54,34,48)(5,41,11,71,17,65,23,59,29,53,35,47)(6,40,12,70,18,64,24,58,30,52,36,46), (1,34)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(35,36)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(54,72)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64) );
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)], [(1,45,7,39,13,69,19,63,25,57,31,51),(2,44,8,38,14,68,20,62,26,56,32,50),(3,43,9,37,15,67,21,61,27,55,33,49),(4,42,10,72,16,66,22,60,28,54,34,48),(5,41,11,71,17,65,23,59,29,53,35,47),(6,40,12,70,18,64,24,58,30,52,36,46)], [(1,34),(2,33),(3,32),(4,31),(5,30),(6,29),(7,28),(8,27),(9,26),(10,25),(11,24),(12,23),(13,22),(14,21),(15,20),(16,19),(17,18),(35,36),(37,53),(38,52),(39,51),(40,50),(41,49),(42,48),(43,47),(44,46),(54,72),(55,71),(56,70),(57,69),(58,68),(59,67),(60,66),(61,65),(62,64)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 8A | 8B | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 18A | 18B | 18C | 18D | 18E | 18F | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 36G | ··· | 36L | 72A | ··· | 72L |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 108 | 2 | 2 | 4 | 2 | 36 | 2 | 2 | 4 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 36 | 36 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | SD16 | D9 | C3⋊D4 | D12 | D18 | C24⋊C2 | D36 | C72⋊C2 | S32 | Q8⋊2S3 | C3⋊D12 | S3×D9 | C32⋊5SD16 | C3⋊D36 | C6.D36 |
| kernel | C6.D36 | C9×C3⋊C8 | C3×Dic18 | C36⋊S3 | Dic18 | C3×C3⋊C8 | C3×C18 | C36 | C3×C12 | C3×C9 | C3⋊C8 | C18 | C3×C6 | C12 | C32 | C6 | C3 | C12 | C9 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 2 | 3 | 4 | 6 | 12 | 1 | 1 | 1 | 3 | 2 | 3 | 6 |
Matrix representation of C6.D36 ►in GL6(𝔽73)
| 21 | 3 | 0 | 0 | 0 | 0 |
| 23 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 52 |
| 0 | 0 | 0 | 0 | 11 | 45 |
| 47 | 18 | 0 | 0 | 0 | 0 |
| 7 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 2 | 1 |
| 52 | 70 | 0 | 0 | 0 | 0 |
| 25 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 2 | 1 |
G:=sub<GL(6,GF(73))| [21,23,0,0,0,0,3,52,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,11,0,0,0,0,52,45],[47,7,0,0,0,0,18,26,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,72,2,0,0,0,0,0,1],[52,25,0,0,0,0,70,21,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,72,2,0,0,0,0,0,1] >;
C6.D36 in GAP, Magma, Sage, TeX
C_6.D_{36} % in TeX
G:=Group("C6.D36"); // GroupNames label
G:=SmallGroup(432,63);
// by ID
G=gap.SmallGroup(432,63);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,36,254,58,571,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^36=c^2=1,b^6=a^18,b*a*b^-1=c*a*c=a^-1,c*b*c=a^27*b^5>;
// generators/relations