non-abelian, supersoluble, monomial
Aliases: C62.9D6, He3⋊6(C4○D4), He3⋊2D4⋊3C2, He3⋊6D4⋊2C2, He3⋊2Q8⋊5C2, C32⋊7D4⋊2S3, C3⋊Dic3.4D6, C32⋊4(D4⋊2S3), C3.2(D6.4D6), C32⋊C12.4C22, (C2×He3).13C23, C22.2(C32⋊D6), He3⋊3C4.15C22, (C22×He3).9C22, C6.87(C2×S32), (C2×C6).54S32, (C2×C3⋊S3).4D6, C6.S32⋊4C2, (C2×He3⋊3C4)⋊5C2, C2.14(C2×C32⋊D6), (C3×C6).13(C22×S3), (C2×C32⋊C6).3C22, SmallGroup(432,319)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.9D6
G = < a,b,c,d | a6=b6=1, c6=d2=b3, ab=ba, cac-1=a-1b, dad-1=a-1b4, cbc-1=b-1, bd=db, dcd-1=c5 >
Subgroups: 879 in 156 conjugacy classes, 35 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, Q8, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, He3, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C62, C62, C4○D12, D4⋊2S3, C32⋊C6, C2×He3, C2×He3, S3×Dic3, C6.D6, C3⋊D12, C32⋊2Q8, C6×Dic3, C3×C3⋊D4, C32⋊7D4, C32⋊C12, He3⋊3C4, C2×C32⋊C6, C22×He3, D6.3D6, He3⋊2Q8, C6.S32, He3⋊2D4, He3⋊6D4, C2×He3⋊3C4, C62.9D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, D4⋊2S3, C2×S32, C32⋊D6, D6.4D6, C2×C32⋊D6, C62.9D6
►On 72 points
(2 55 30 8 49 36)(3 50 31)(4 10)(5 33 52)(6 28 53 12 34 59)(9 56 25)(11 27 58)(13 47 66)(14 42 67 20 48 61)(16 63 38 22 69 44)(17 70 39)(18 24)(19 41 72)(23 64 45)(26 32)(40 46)(51 57)(65 71)
(1 54 29 7 60 35)(2 36 49 8 30 55)(3 56 31 9 50 25)(4 26 51 10 32 57)(5 58 33 11 52 27)(6 28 53 12 34 59)(13 72 47 19 66 41)(14 42 67 20 48 61)(15 62 37 21 68 43)(16 44 69 22 38 63)(17 64 39 23 70 45)(18 46 71 24 40 65)
(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 23 7 17)(2 16 8 22)(3 21 9 15)(4 14 10 20)(5 19 11 13)(6 24 12 18)(25 37 31 43)(26 42 32 48)(27 47 33 41)(28 40 34 46)(29 45 35 39)(30 38 36 44)(49 69 55 63)(50 62 56 68)(51 67 57 61)(52 72 58 66)(53 65 59 71)(54 70 60 64)
G:=sub<Sym(72)| (2,55,30,8,49,36)(3,50,31)(4,10)(5,33,52)(6,28,53,12,34,59)(9,56,25)(11,27,58)(13,47,66)(14,42,67,20,48,61)(16,63,38,22,69,44)(17,70,39)(18,24)(19,41,72)(23,64,45)(26,32)(40,46)(51,57)(65,71), (1,54,29,7,60,35)(2,36,49,8,30,55)(3,56,31,9,50,25)(4,26,51,10,32,57)(5,58,33,11,52,27)(6,28,53,12,34,59)(13,72,47,19,66,41)(14,42,67,20,48,61)(15,62,37,21,68,43)(16,44,69,22,38,63)(17,64,39,23,70,45)(18,46,71,24,40,65), (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), (1,23,7,17)(2,16,8,22)(3,21,9,15)(4,14,10,20)(5,19,11,13)(6,24,12,18)(25,37,31,43)(26,42,32,48)(27,47,33,41)(28,40,34,46)(29,45,35,39)(30,38,36,44)(49,69,55,63)(50,62,56,68)(51,67,57,61)(52,72,58,66)(53,65,59,71)(54,70,60,64)>;
G:=Group( (2,55,30,8,49,36)(3,50,31)(4,10)(5,33,52)(6,28,53,12,34,59)(9,56,25)(11,27,58)(13,47,66)(14,42,67,20,48,61)(16,63,38,22,69,44)(17,70,39)(18,24)(19,41,72)(23,64,45)(26,32)(40,46)(51,57)(65,71), (1,54,29,7,60,35)(2,36,49,8,30,55)(3,56,31,9,50,25)(4,26,51,10,32,57)(5,58,33,11,52,27)(6,28,53,12,34,59)(13,72,47,19,66,41)(14,42,67,20,48,61)(15,62,37,21,68,43)(16,44,69,22,38,63)(17,64,39,23,70,45)(18,46,71,24,40,65), (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), (1,23,7,17)(2,16,8,22)(3,21,9,15)(4,14,10,20)(5,19,11,13)(6,24,12,18)(25,37,31,43)(26,42,32,48)(27,47,33,41)(28,40,34,46)(29,45,35,39)(30,38,36,44)(49,69,55,63)(50,62,56,68)(51,67,57,61)(52,72,58,66)(53,65,59,71)(54,70,60,64) );
G=PermutationGroup([[(2,55,30,8,49,36),(3,50,31),(4,10),(5,33,52),(6,28,53,12,34,59),(9,56,25),(11,27,58),(13,47,66),(14,42,67,20,48,61),(16,63,38,22,69,44),(17,70,39),(18,24),(19,41,72),(23,64,45),(26,32),(40,46),(51,57),(65,71)], [(1,54,29,7,60,35),(2,36,49,8,30,55),(3,56,31,9,50,25),(4,26,51,10,32,57),(5,58,33,11,52,27),(6,28,53,12,34,59),(13,72,47,19,66,41),(14,42,67,20,48,61),(15,62,37,21,68,43),(16,44,69,22,38,63),(17,64,39,23,70,45),(18,46,71,24,40,65)], [(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)], [(1,23,7,17),(2,16,8,22),(3,21,9,15),(4,14,10,20),(5,19,11,13),(6,24,12,18),(25,37,31,43),(26,42,32,48),(27,47,33,41),(28,40,34,46),(29,45,35,39),(30,38,36,44),(49,69,55,63),(50,62,56,68),(51,67,57,61),(52,72,58,66),(53,65,59,71),(54,70,60,64)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | ··· | 6J | 6K | 6L | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 18 | 18 | 2 | 6 | 6 | 12 | 9 | 9 | 18 | 18 | 18 | 2 | 2 | 2 | 6 | 6 | 12 | ··· | 12 | 36 | 36 | 18 | 18 | 18 | 18 | 36 | 36 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | S32 | D4⋊2S3 | C2×S32 | D6.4D6 | C32⋊D6 | C2×C32⋊D6 | C62.9D6 |
| kernel | C62.9D6 | He3⋊2Q8 | C6.S32 | He3⋊2D4 | He3⋊6D4 | C2×He3⋊3C4 | C32⋊7D4 | C3⋊Dic3 | C2×C3⋊S3 | C62 | He3 | C2×C6 | C32 | C6 | C3 | C22 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C62.9D6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 4 | 0 | 9 | 0 | 0 | 0 |
| 1 | 0 | 0 | 12 | 0 | 0 |
| 5 | 0 | 0 | 0 | 4 | 0 |
| 2 | 0 | 0 | 0 | 0 | 10 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 10 | 0 | 0 | 10 | 0 | 0 |
| 10 | 0 | 0 | 0 | 10 | 0 |
| 10 | 0 | 0 | 0 | 0 | 10 |
| 12 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 1 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 12 | 0 | 12 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 1 |
| 12 | 12 | 0 | 0 | 0 | 1 |
| 8 | 0 | 3 | 0 | 0 | 0 |
| 0 | 8 | 5 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 8 |
| 0 | 0 | 8 | 0 | 8 | 0 |
| 0 | 0 | 8 | 8 | 0 | 0 |
G:=sub<GL(6,GF(13))| [1,1,4,1,5,2,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,12,0,0,0,0,0,0,4,0,0,0,0,0,0,10],[4,0,0,10,10,10,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[12,0,0,12,12,12,0,0,0,0,0,12,0,0,0,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,2,12,12,1,1,1],[8,0,0,0,0,0,0,8,0,0,0,0,3,5,5,8,8,8,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8,0,0] >;
C62.9D6 in GAP, Magma, Sage, TeX
C_6^2._9D_6
% in TeX
G:=Group("C6^2.9D6"); // GroupNames label
G:=SmallGroup(432,319);
// by ID
G=gap.SmallGroup(432,319);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,571,4037,537,14118,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^6=d^2=b^3,a*b=b*a,c*a*c^-1=a^-1*b,d*a*d^-1=a^-1*b^4,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations