non-abelian, supersoluble, monomial
Aliases: C62.47D6, (C6×C12)⋊4S3, He3⋊8(C4○D4), He3⋊7D4⋊5C2, He3⋊5D4⋊8C2, He3⋊4Q8⋊8C2, (C3×C12).55D6, C32⋊6(C4○D12), (C4×He3).40C22, (C2×He3).32C23, C3.2(C12.59D6), He3⋊3C4.17C22, (C22×He3).34C22, (C2×C4×He3)⋊5C2, C12.93(C2×C3⋊S3), (C4×He3⋊C2)⋊8C2, C6.64(C22×C3⋊S3), (C2×C12).24(C3⋊S3), (C2×C4)⋊3(He3⋊C2), (C3×C6).42(C22×S3), C4.16(C2×He3⋊C2), C22.2(C2×He3⋊C2), C2.5(C22×He3⋊C2), (C2×He3⋊C2).17C22, (C2×C6).27(C2×C3⋊S3), SmallGroup(432,387)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.47D6
G = < a,b,c,d | a6=b6=1, c6=d2=b3, ab=ba, cac-1=ab2, dad-1=a-1b, bc=cb, bd=db, dcd-1=c5 >
Subgroups: 797 in 220 conjugacy classes, 53 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C4○D4, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, He3, C3×Dic3, C3×C12, S3×C6, C62, C4○D12, C3×C4○D4, He3⋊C2, C2×He3, C2×He3, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, He3⋊3C4, C4×He3, C2×He3⋊C2, C22×He3, C3×C4○D12, He3⋊4Q8, C4×He3⋊C2, He3⋊5D4, He3⋊7D4, C2×C4×He3, C62.47D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C3⋊S3, C22×S3, C2×C3⋊S3, C4○D12, He3⋊C2, C22×C3⋊S3, C2×He3⋊C2, C12.59D6, C22×He3⋊C2, C62.47D6
►On 72 points
(1 26 61 20 50 46)(2 35 70 17 59 43)(3 32 67 18 56 40)(4 29 64 19 53 37)(5 60 48 21 36 63)(6 57 45 22 33 72)(7 54 42 23 30 69)(8 51 39 24 27 66)(9 49 41 13 25 68)(10 58 38 14 34 65)(11 55 47 15 31 62)(12 52 44 16 28 71)
(1 21 14 3 23 16)(2 22 15 4 24 13)(5 10 18 7 12 20)(6 11 19 8 9 17)(25 35 33 31 29 27)(26 36 34 32 30 28)(37 39 41 43 45 47)(38 40 42 44 46 48)(49 59 57 55 53 51)(50 60 58 56 54 52)(61 63 65 67 69 71)(62 64 66 68 70 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 18 3 20)(2 19 4 17)(5 21 7 23)(6 22 8 24)(9 13 11 15)(10 14 12 16)(25 64 31 70)(26 69 32 63)(27 62 33 68)(28 67 34 61)(29 72 35 66)(30 65 36 71)(37 49 43 55)(38 54 44 60)(39 59 45 53)(40 52 46 58)(41 57 47 51)(42 50 48 56)
G:=sub<Sym(72)| (1,26,61,20,50,46)(2,35,70,17,59,43)(3,32,67,18,56,40)(4,29,64,19,53,37)(5,60,48,21,36,63)(6,57,45,22,33,72)(7,54,42,23,30,69)(8,51,39,24,27,66)(9,49,41,13,25,68)(10,58,38,14,34,65)(11,55,47,15,31,62)(12,52,44,16,28,71), (1,21,14,3,23,16)(2,22,15,4,24,13)(5,10,18,7,12,20)(6,11,19,8,9,17)(25,35,33,31,29,27)(26,36,34,32,30,28)(37,39,41,43,45,47)(38,40,42,44,46,48)(49,59,57,55,53,51)(50,60,58,56,54,52)(61,63,65,67,69,71)(62,64,66,68,70,72), (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), (1,18,3,20)(2,19,4,17)(5,21,7,23)(6,22,8,24)(9,13,11,15)(10,14,12,16)(25,64,31,70)(26,69,32,63)(27,62,33,68)(28,67,34,61)(29,72,35,66)(30,65,36,71)(37,49,43,55)(38,54,44,60)(39,59,45,53)(40,52,46,58)(41,57,47,51)(42,50,48,56)>;
G:=Group( (1,26,61,20,50,46)(2,35,70,17,59,43)(3,32,67,18,56,40)(4,29,64,19,53,37)(5,60,48,21,36,63)(6,57,45,22,33,72)(7,54,42,23,30,69)(8,51,39,24,27,66)(9,49,41,13,25,68)(10,58,38,14,34,65)(11,55,47,15,31,62)(12,52,44,16,28,71), (1,21,14,3,23,16)(2,22,15,4,24,13)(5,10,18,7,12,20)(6,11,19,8,9,17)(25,35,33,31,29,27)(26,36,34,32,30,28)(37,39,41,43,45,47)(38,40,42,44,46,48)(49,59,57,55,53,51)(50,60,58,56,54,52)(61,63,65,67,69,71)(62,64,66,68,70,72), (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), (1,18,3,20)(2,19,4,17)(5,21,7,23)(6,22,8,24)(9,13,11,15)(10,14,12,16)(25,64,31,70)(26,69,32,63)(27,62,33,68)(28,67,34,61)(29,72,35,66)(30,65,36,71)(37,49,43,55)(38,54,44,60)(39,59,45,53)(40,52,46,58)(41,57,47,51)(42,50,48,56) );
G=PermutationGroup([[(1,26,61,20,50,46),(2,35,70,17,59,43),(3,32,67,18,56,40),(4,29,64,19,53,37),(5,60,48,21,36,63),(6,57,45,22,33,72),(7,54,42,23,30,69),(8,51,39,24,27,66),(9,49,41,13,25,68),(10,58,38,14,34,65),(11,55,47,15,31,62),(12,52,44,16,28,71)], [(1,21,14,3,23,16),(2,22,15,4,24,13),(5,10,18,7,12,20),(6,11,19,8,9,17),(25,35,33,31,29,27),(26,36,34,32,30,28),(37,39,41,43,45,47),(38,40,42,44,46,48),(49,59,57,55,53,51),(50,60,58,56,54,52),(61,63,65,67,69,71),(62,64,66,68,70,72)], [(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)], [(1,18,3,20),(2,19,4,17),(5,21,7,23),(6,22,8,24),(9,13,11,15),(10,14,12,16),(25,64,31,70),(26,69,32,63),(27,62,33,68),(28,67,34,61),(29,72,35,66),(30,65,36,71),(37,49,43,55),(38,54,44,60),(39,59,45,53),(40,52,46,58),(41,57,47,51),(42,50,48,56)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | ··· | 6P | 6Q | 6R | 6S | 6T | 12A | 12B | 12C | 12D | 12E | 12F | 12G | ··· | 12V | 12W | 12X | 12Y | 12Z |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 18 | 18 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 1 | 2 | 18 | 18 | 1 | 1 | 2 | 2 | 6 | ··· | 6 | 18 | 18 | 18 | 18 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | ··· | 6 | 18 | 18 | 18 | 18 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 6 |
| type | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | C4○D12 | He3⋊C2 | C2×He3⋊C2 | C2×He3⋊C2 | C62.47D6 |
| kernel | C62.47D6 | He3⋊4Q8 | C4×He3⋊C2 | He3⋊5D4 | He3⋊7D4 | C2×C4×He3 | C6×C12 | C3×C12 | C62 | He3 | C32 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 8 | 4 | 2 | 16 | 4 | 8 | 4 | 4 |
Matrix representation of C62.47D6 ►in GL5(𝔽13)
| 9 | 2 | 0 | 0 | 0 |
| 11 | 11 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 0 | 5 | 0 | 0 | 0 |
| 8 | 5 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 12 |
| 0 | 0 | 0 | 10 | 10 |
| 0 | 0 | 0 | 0 | 12 |
| 11 | 11 | 0 | 0 | 0 |
| 9 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 9 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(5,GF(13))| [9,11,0,0,0,2,11,0,0,0,0,0,12,1,8,0,0,12,0,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[0,8,0,0,0,5,5,0,0,0,0,0,4,0,0,0,0,0,10,0,0,0,12,10,12],[11,9,0,0,0,11,2,0,0,0,0,0,0,9,0,0,0,3,0,0,0,0,3,1,12] >;
C62.47D6 in GAP, Magma, Sage, TeX
C_6^2._{47}D_6 % in TeX
G:=Group("C6^2.47D6"); // GroupNames label
G:=SmallGroup(432,387);
// by ID
G=gap.SmallGroup(432,387);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,1124,4037,537]);
// 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*b^2,d*a*d^-1=a^-1*b,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations